research article the combined poisson inma( )...

Research ArticleThe Combined Poisson INMA(119902) Models forTime Series of Counts

Kaizhi Yu1 and Hong Zou2

1Statistics School Southwestern University of Finance and Economics Chengdu 611130 China2School of Economics Southwestern University of Finance and Economics Chengdu 611130 China

Correspondence should be addressed to Kaizhi Yu kaizhiyuswufegmailcom

Received 11 October 2014 Revised 14 January 2015 Accepted 14 January 2015

Academic Editor Angelo Ciaramella

Copyright copy 2015 K Yu and H Zou This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Anew stationary 119902th-order integer-valuedmoving average processwith Poisson innovation is introduced based ondecision randomvector Some statistical properties of the process are established Estimators of the parameters of the process are obtained using themethod of moments Some numerical results of the estimators are presented to assess the performance of moment estimators

1 Introduction

In natural and social sciences time series of correlatedcounting are met very often In particular in economicsand medicine many interesting variables are nonnegativecount data for example the number of shareholders in largeFinnish and Swedish stocks big numbers even for frequentlytraded stocks the number of arrivals per week to the emer-gency service of the hospital and monthly polio incidencecounts in Germany Most of the research on count processesassumes that the count data are independent and identicallydistributed However in practice observations may be auto-correlated and this may adversely affect the performanceof traditional model developed under the assumption ofindependence In recent years count data time series modelshave been devised to avoid making restrictive assumptionson the distribution of the error term Regression models fortime series count data have been proposed [1 2] On the otherhand there have been attempts to develop suitable classes ofmodels that resemble the structure and properties of the usuallinear ARMA models For instances Al-Osh and Alzaid [3]proposed an integer-valued moving average model (INMA)for discrete data An integer-valuedGARCHmodel was givento study overdispersed counts [4] Silva et al [5] consideredthe problem of forecasting in INAR(1) model Randomcoefficient INAR models were introduced by Zheng et al [67] The signed thinning operator was developed by Kachour

and Truquet [8] A new stationary first-order integer-valuedautoregressive process with geometric marginal distributionbased on the generalized binomial thinning was introducedby Ristic et al [9] In this analysis of counts the class ofinteger-valued moving average models plays an importantrole

The nonnegative integer-valued moving average processof the order 119902 (INMAR(119902)) was introduced by Al-Osh andAlzaid [3] The INMA(119902) process is defined by the recursion

119883119905= 1205791∘119905120576119905minus1

+ sdot sdot sdot + 120579119902∘119905120576119905minus119902

+ 120576119905 (1)

where 120579119896∘119905120576119905minus119896

= sum120576119905minus119896

119894=1V119894and the V

119894 designated by counting

series is a sequence of iid Bernoulli random variableswith 119864(V

119894) = 120579 independent of 120576

119905minus119896 The parameters

1205791 120579

119902isin (0 1) The thinning operation ldquo∘

119905rdquo indicates the

corresponding thinning is associated with time The terms120579119894∘119905120576119905and 120579

119895∘119905120576119905are independent The choice of appropri-

ate marginal distributions is still problematic for getting aparticular distribution of 119883

119905 To overcome these difficulties

Weiszlig [10] introduced combined INAR(119901) models by usingldquodecisionrdquo random variables Ristic et al [11] consideredpiecewise functions for count data model Therefore weadopt the similar approach to deal with the problem inINMA(119902) models In this paper we propose a combinedINMA(119902) model by allowing the parameters value to varywith ldquodecisionrdquo random vector

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2015 Article ID 457842 7 pageshttpdxdoiorg1011552015457842

2 Journal of Applied Mathematics

The paper is organized as follows In Section 2 wespecify the model and derive some statistical propertiesSection 3 concerns unknown parameter estimation by Yule-Walker method In Section 4 we conduct some Monte Carlosimulations Finally Section 5 concludes

2 Definition and Basic Properties ofthe PCINMA(119902) Process

Definition 1 (PCINMA(119902) model) A stochastic process 119883119905

is said to be the Poisson combined INMA(119902) process if itsatisfies the following recursive equations

119883119905= 1198631199051

(120579∘119905120576119905minus1

) + sdot sdot sdot + 119863119905119902

(120579∘119905120576119905minus119902

) + 120576119905 (2)

where 120576119905 119905 isin Z is a sequence of independent and identically

distributed Poisson random variables with parameter 120582 and120579 isin (0 1) D

119905 119905 isin Z is an iid process of ldquodecisionrdquo

random vector D119905

= (1198631199051

119863119905119902

) sim Mult(1 1206011 120601

119902)

independent of 120576119905 119905 isin Z Moreover the counting series

120579∘119905120576119905minus119896

119896 = 1 119902 are independent of D119905 and 120576

119905at time 119905

To our knowledge few efforts have been devoted to studyingcombined INMA models In this paper we aim to fill thisgap Definition 1 shows that 119883

119905is equal to 120579∘

119905120576119905minus1

+ 120576119905with

probability 1206011or equal to 120579∘

119905120576119905minus2

+ 120576119905with probability 120601

2

or else equal to 120579∘119905120576119905minus119902

+120576119905with probability 120601

119902(= 1minus120601

1minus1206012minus

sdot sdot sdot minus 120601119902minus1

)

The moments will be useful in obtaining the appropriateestimating equations for parameter estimation

Theorem 2 The numerical characteristics of 119883119905 are as

follows(i)

120583119883

= 119864 (119883119905) = 120582 (120579 + 1) (3)

(ii)

1205902

119883= Var (119883

119905) = 120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2 (4)

(iii)

120574119883

(119896) = cov (119883119905 119883119905minus119896

)

=

1205791205822120601119896+ 12057921205822

119902

sum

119895=119896+1

120601119895120601119895minus119896

1 le 119896 le 119902 minus 1

1205791205822120601119896 119896 = 119902

0 119896 ge 119902 + 1

(5)

Proof (i) Consider

119864 (119883119905) = 119864 [119863

1199051(120579∘119905120576119905minus1

+ 120576119905)] + sdot sdot sdot + 119864 [119863

119905119902(120579∘119905120576119905minus119902

+ 120576119905)]

= 119864 (1198631199051

) 119864 (120579∘119905120576119905minus1

+ 120576119905) + sdot sdot sdot + 119864 (119863

119905119902)

sdot 119864 (120579∘119905120576119905minus119902

+ 120576119905)

= 1206011(120582120579 + 120582) + sdot sdot sdot + 120601

119902(120582120579 + 120582) = 120582 (120579 + 1)

(6)

(ii) Moreover

Var (119883119905) = Var(

119902

sum

119895=1

119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

=

119902

sum

119895=1

Var (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

+ 2sum

119894lt119895

cov (119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905) 119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

(7)

Note that Var(119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905)) in first summation on the

right-hand side is

= Var (119863119905119895

) 119864 (120579∘119905120576119905minus119895

+ 120576119905)2

+ 1198642(119863119905119895

)Var (120579∘119905120576119905minus119895

+ 120576119905)

= 120601119895(1 minus 120601

119895) 119864 ((120579∘

119905120576119905minus119895

)2

+ 1205762

119905+ 2 (120579∘

119905120576119905minus119895

) 120576119905)

+ 1206012

119895(Var (120579∘

119905120576119905minus119895

) + Var (120576119905) + 2 cov (120579∘

119905120576119905minus119895

120576119905))

= 120601119895(1 minus 120601

119895) [1205792119864 (1205762

119905minus119895) + 120579 (1 minus 120579) 119864 (120576

119905minus119895)] + 119864 (120576

2

119905)

+ 2119864 (120579∘119905120576119905minus119895

) 119864 (120576119905)

+ 1206012

119895[1205792 Var (120576

119905minus119895) + 120579 (1 minus 120579) 119864 (120576

119905minus119895)] + 120590

2

120576

= 120601119895(1 minus 120601

119895) [(120579 + 1)

21205822+ 120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2]

+ 1206012

119895[(1205792+ 1) 120582

2+ 120579 (1 minus 120579) 120582]

= 120601119895(1 minus 120601

119895) (120579 + 1)

21205822+ 120601119895[120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2]

(8)

By the same arguments as above it follows that

cov (119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905) 119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

= cov (119863119905119894

(120579∘119905120576119905minus119894

) 119863119905119895

(120579∘119905120576119905minus119895

))

+ cov (119863119905119894

(120579∘119905120576119905minus119894

) 119863119905119895

120576119905)

+ cov (119863119905119894120576119905 119863119905119895

(120572∘119905120576119905minus119895

)) + cov (119863119905119894120576119905 119863119905119895

120576119905)

= cov (119863119905119894 119863119905119895

) 119864 (120579∘119905120576119905minus119894

) 119864 (120579∘119905120576119905minus119895

)

+ cov (119863119905119894 119863119905119895

) 119864 (120579∘119905120576119905minus119894

) 119864 (120576119905)

+ cov (119863119905119894 119863119905119895

) 119864 (120576119905) 119864 (120579∘

119905120576119905minus119895

)

+ cov (119863119905119894 119863119905119895

) 1198642(120576119905)

= minus120601119894120601119895(12057921205822+ 1205791205822+ 1205791205822+ 1205822) = minus120601

119894120601119895(120579 + 1)

21205822

(9)

Journal of Applied Mathematics 3

Therefore

Var (119883119905) = Var(

119902

sum

119895=1

119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

=

119902

sum

119895=1

(120601119895(1 minus 120601

119895) (120579 + 1)

21205822

+120601119895[120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2])

minus 2sum

119894lt119895

120601119894120601119895(120579 + 1)

21205822

= (120579 + 1)21205822(1 minus

119902

sum

119895=1

1206012

119895)

+ [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2]

minus 2 (120579 + 1)21205822sum

119894lt119895

120601119894120601119895

= 120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2

(10)

(iii) For 1 le 119896 le 119902 minus 1 the autocovariance function of 119883119905

is

cov (119883119905 119883119905minus119896

)

= cov(

119902

sum

119894=1

119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

= cov(119863119905119896

(120579∘119905120576119905minus119896

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

+ 120576119905)

119863119905minus119896119895minus119896

(120579∘119905minus119896

120576119905minus119896minus(119895minus119896)

+ 120576119905minus119896

))

=

119902

sum

119895=1

cov (119863119905119896

(120579∘119905120576119905minus119896

) 119863119905minus119896119895

120576119905minus119896

)

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

) 119863119905minus119896119895minus119896

(120579∘119905minus119896

120576119905minus119895

))

=

119902

sum

119895=1

cov (120579∘119905120576119905minus119896

120576119905minus119896

) 119864 (119863119905119896

) 119864 (119863119905minus119896119895

)

+

119902

sum

119895=119896+1

cov (120579∘119905minus119895

120576119905minus119895

120579∘119905minus119896

120576119905minus119895

) 119864 (119863119905119895

) 119864 (119863119905minus119896119895minus119896

)

=

119902

sum

119895=1

120579Var (120576119905minus119896

) 120601119896120601119895+

119902

sum

119895=119896+1

1205792 Var (120576

119905minus119895) 120601119895120601119895minus119896

= 1205791205822120601119896+ 12057921205822

119902

sum

119895=119896+1

120601119895120601119895minus119896

(11)

If 119896 = 119902 by using a similar approach we get cov(119883119905

119883119905minus119902

) = 1205791205822120601119902 For 119896 ge 119902 + 1 all the terms 120579∘

119905120576119905minus1

120579∘119905120576119905minus119902

120576119905 120579∘119905minus119896

120576119905minus119896minus1

120579∘119905minus119896

120576119905minus119896minus119902

and 120576119905minus119896

involved in 119883119905

and 119883119905minus119896

are mutually independent Therefore the autoco-variance function of 119883

119905is equal to zero for 119896 ge 119902 + 1

Theorem 3 Let 119883119905be the process defined by the equation in

(2) Then

(i) 119883119905 is a covariance stationary process

(ii) 119864(119883119896

119905) le 119862 lt infin 119896 = 1 2 3 4 for some constant 119862 gt

0

Proof (i) The first conclusion is immediate from the defini-tion of covariance stationary process

(ii) For 119896 = 1 it is straightforwardFor 119896 = 2 it follows that

119864 (1198832

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))2

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))2

(12)

Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))2

= 119864 (119863119905119895

)2

119864 (120579∘119905120576119905minus119895

+ 120576119905)2

= 119864 (119863119905119895

)2

(119864 (120579∘119905120576119905minus119895

)2

+ 119864 (120576119905)2

+ 2119864 [(120579∘119905120576119905minus119895

) 120576119905])

= 120601119895(1 minus 120601

119895)

sdot [1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582 + (120582 + 120582

2) + 2120579120582

2]

= 120601119895(1 minus 120601

119895) (120579 + 1) 120582 [(120579 + 1) 120582 + 1]

(13)

Then we have

119864 (1198832

119905) le

1

4(120579 + 1) 120582 [(120579 + 1) 120582 + 1] lt infin (14)

Similarly for 119896 = 3

119864 (1198833

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))3

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))3

(15)


Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))3

= 119864 (119863119905119895

)3

119864 (120579∘119905120576119905minus119895

+ 120576119905)3

= 119864 (119863119905119895

)3

(119864 (120579∘119905120576119905minus119895

)3

+ 119864 (120576119905)3

+ 3119864 [(120579∘119905120576119905minus119895

)2

120576119905] + 119864 [(120579∘

119905120576119905minus119895

) 1205762

119905])

= 1206013

119895[1205793120591 + 3120579

2(1 minus 120579) (120582 + 120582

2)

+ (120579 minus 31205792(1 minus 120579) minus 120579

3) 120582]

+ 1206013

119895120591 + 3120601

3

119895[1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582] 120582

+ 31206013

119895120579120582 (120582 + 120582

2)

= 1206013

119895120579120582 [120579

2(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2+ 6120582 + 1]

+ 120591

(16)

where 120591 = 1205823+ 31205822+ 120582 Let 1206013max = max(1206013

1 120601

3

119902) Then

119864 (1198833

119905)

le 1206013

max 120579120582 [1205792(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2

+ 6120582 + 1] + 120591 lt infin

(17)

After some tedious calculations we also can show the resultholds for 119896 = 4 We skip the details Next we will presentergodic theorem for stationary process 119883

119905 There are a

variety of ergodic theorems differing in their assumptionsand in the modes of convergence Here the convergence is inmean square The next two lemmas will be useful in provingthe ergodicity of the samplemean and sample autocovariancefunction of 119883

119905

Lemma 4 If 119885119905 is stationary with mean 120583

119885and autocovari-

ance function 120574119885(sdot) then as 119879 rarr infin Var(119885

119879) = 119864(119885

119879minus

120583119885)2

rarr 0 if 120574119885(119879) rarr 0 where 119885

119879= (1119879)sum

119879

119905=1119883119905

Proof See Theorem 711 in Brockwell and Davis [13]

Lemma 5 Suppose 119885119905 is a covariance stationary process

having covariance function 120574119885(V) = 119864(119885

119905+V119885119905) and a mean ofzero If lim

119879rarrinfin(1119879)sum

119879

119897=1[119864(119885119905119885119905+V119885119905+119897119885119905+119897+V) minus 120574

2

119885(V)] =

0 Then for any fixed V = 0 1 2

lim119879rarrinfin

1

119879

119879

sum

119897=1

119864 [120574119885(V) minus 120574

2

119885(V)]2

= 0 (18)

where 120574119885(V) is the sample covariance function 120574

119885(V) =

(1119879)sum119879minusV119905=1

119885119905+V119885119905

Proof See Theorem 52 in Karlin and Taylor [12]

Lemma 6 Let process 119884119905= 119883119905minus 120583119883be a transformation of

119883119905 then the following results hold

(i) 119884119905 is a covariance stationary process with zero mean

(ii) 120574119884(119896) = cov(119884

119905 119884119905minus119896

) = 120574119883(119896) 119896 = 1 2 3

(iii) 119864(119884119896

119905) le 119862

lowastlt infin 119896 = 1 2 3 4 for some constant

119862lowast

gt 0(iv) 119884

119905 is ergodic in autocovariance function

Proof The only part of this lemma that is not obvious is part(iv) The proof of properties (iv) is as follows

To prove that 119884119905 is ergodic in autocovariance function

it suffices to show that

lim119879rarrinfin

1

119879

119879

sum

119897=1

[119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] = 0 (19)

according to Lemma 5 Thus we will discuss two cases Forsimplicity in notation we define 119888

2= 119864(119884

2

119905) 1198883

= 119864(1198843

119905)

1198884= 119864(119884

4

119905) and 119877

119884(V) = 119864(119884

119905119884119905+V)

Case 1 For V = 0 119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864(119884

2

1199051198842

119905+119897) 1205742119884(V) =

1205742

119883(V) = 120590

2

119883

If 1 le 119897 le 119902 using the Schwarz inequality we get

119864 (1198842

1199051198842

119905+119897) le radic119864 (

10038161003816100381610038161198842

119905

10038161003816100381610038162

) 119864 (100381610038161003816100381610038161198842119905+119897

10038161003816100381610038161003816

2

) = 1198884le 119862lowast (20)

If 119897 ge 119902 + 1 note that 119884119905+119897

and 119884119905are irrelevant and then

119864 (1198842

1199051198842

119905+119897) = 119864 (119884

2

119905) 119864 (119884

2

119905+119897) = 1198882

2= 1205742

119883(V) (21)

Therefore (1119879)sum119879

119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le

(1119879)[2(1198884minus 1205902

119883)] rarr 0 for 119879 rarr infin

Case 2 For V ge 1 1198772119884(V) = [119864(119884

119905119884119905+V)]2= 1205742

119883(V)

If 1 le 119897 le 119902 + V using Schwarz inequality twice

119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V)

le radic119864 (1003816100381610038161003816119884119905119884119905+V

10038161003816100381610038162

) 119864 (1003816100381610038161003816119884119905+119897119884119905+119897+V

10038161003816100381610038162

)

le14radic119864 (1198844

119905) 119864 (1198844

119905+V) 119864 (1198844119905+119897

) 119864 (1198844119905+119897+V) = 119888

4

(22)

If 119897 ge 119902 + V + 1 note that 119883119905119883119905+V与119883

119905+119897119883119905+119897+V uncorrelation

thus we have

119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864 (119883

119905119883119905+V) 119864 (119884

119905+119897119884119905+119897+V) = 119877

2

119884(V) (23)

Then (1119879)sum119879

119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le (1119879)(2 +

V)[1198884minus 1198772

119884(V)] rarr 0 for 119879 rarr infin

This proves Lemma 6

Theorem 7 Let 119883119905 be a PCINMA(119902) process according to

Definition 1 Then the stochastic process 119883119905 is ergodic in the

mean and autocovariance function


Proof For notational simplicity we define 119883119879(119896) =

(1119879)sum119879minus119896

119905=1119883119905+119896

120574119884(119896) = (1119879)sum

119879minus119896

119905=1119884119905+119896

119884119905 and 120574

119883(119896) =

(1119879)sum119879minus119896

119905=1(119883119905+119896

minus 119883119879)(119883119905minus 119883119879) And we assume here that

the sample consists of 119879 + 119896 observations on 119883(i) Note that 120574

119883(119896) rarr 0 for 119896 rarr infin From the result of

Lemma 4 we obtain Var(119883119879) = 119864(119883

119879minus 120583119883)2

rarr 0Then 119883

119879converges in probability to 120583

119883 Therefore the

process 119883119905 is ergodic in the mean

Next we prove that the 119883119905 is ergodic for secondmoment

by induction(ii) First we prove 119883

119879(119896)119875

997888rarr 120583119883 Suppose 120576

1gt 0 is given

119875 (10038161003816100381610038161003816119883119879(119896) minus 120583

119883

10038161003816100381610038161003816ge 1205761)

le 119875(10038161003816100381610038161003816119883119879(119896) minus 119883

10038161003816100381610038161003816ge

1205761

2)

+ 119875(10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

= 119875(

1003816100381610038161003816100381610038161003816100381610038161003816

1

119879

119896

sum

119905=1

119883119905

1003816100381610038161003816100381610038161003816100381610038161003816

ge1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

le

119896

sum

119905=1

119875(1

119879

10038161003816100381610038161198831199051003816100381610038161003816 ge

1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

(24)

Using the Markov inequality sum119896

119905=1119875((1119879)|119883

119905| ge 120576

12) le

sum119896

119905=1(119864(119883119905)(12)119879120576

1) rarr 0 for 119879 rarr infin

Since 119883119905 is ergodic in the mean thus 119875(|119883

119879minus 120583119883| ge

12057612) rarr 0 for 119879 rarr infinTherefore 119883

119879(119896)119875

997888rarr 120583119883 for 119879 rarr infin

Now we prove the second result 120574119883(119896) minus 120574

119883(119896)119875

997888rarr 0Consider any 120576 gt 0

119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

= 119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896) + 120574

119884(119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

le 119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2) + 119875(

1003816100381610038161003816120574119884 (119896) minus 120574119883

(119896)1003816100381610038161003816 ge

120576

2)

(25)

Note that

119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2)

= 119875(100381610038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883) + (119883

2

119879minus 1205832

119883)100381610038161003816100381610038161003816ge

120576

2)

le 119875(10038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883)10038161003816100381610038161003816ge

120576

4)

+ 119875(1003816100381610038161003816100381610038161198832

119879minus 1205832

119883

100381610038161003816100381610038161003816ge

120576

4)

(26)

Since the sample mean 119883119879converges in probability to 120583

119883

according to Slutskyrsquos theorem we get (119883119879(119896) + 119883

119879)(119883119879

minus

120583119883)119875

997888rarr 0 1198832119879minus 1205832

119883

119875

997888rarr 0

Then we have 119875(|120574119883(119896)minus120574

119884(119896)| ge 1205762)

119875

997888rarr 0 for 119879 rarr infinFrom the (iv) of Lemma 6 we obtain

120574119884(119896) minus 120574

119883(119896) = 120574

119884(119896) minus 120574

119884(119896)119901


(27)

And consequently 119875(|120574119883(119896) minus 120574

119883(119896)| ge 120576)

119875

997888rarr 0 for 119879 rarr infinThis leads to the desired conclusion

3 Estimation of the Unknown Parameters

In this section we discuss approaches to the estimation ofthe unknown parameters And we assume we have 119879 obser-vations 119883

1 119883

119879 from a Poisson combined INMA(119902)

process in which the order parameter 119902 is known One ofthe main interests in the literature of INMA process is toestimate the unknown parameters Using the sample covari-ance function we get the estimators of unknown parameters(1206011 120601

119902 120579 120582) through solving the following equations

120574 (0) minus [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2] = 0

120574 (1) minus (12057912058221206011+ 12057921205822

119902

sum

119895=2

120601119895120601119895minus1

) = 0

120574 (119902 minus 1) minus (1205791205822120601119902minus1

+ 120579212058221206011199021206011) = 0

120574 (119902) minus 1205791205822120601119902= 0

(28)

The idea behind these estimators is that of equating popu-lation moments to sample moments and then solving for theparameters in terms of the samplemomentsThese estimatorsare typically called the Yule-Walker estimators As the 120574(119896)

consistently estimates the true autocovariance function 120574(119896)

[13] the Yule-Walker estimators are consistent FollowingBrockwell and Davis (2009) and Billingsley [14] it is easy toshow that under somemild moment conditions the marginalmean estimator 119883

119879is asymptotically normally distributed

4 Monte Carlo Simulation Study

We provide some simulations results to show the empiricalperformance of these estimators Owing to the nonlinearitythe estimator expressions of unknown parameters are quitecomplicatedThe aim of simulation study is to assess the finitesample performances of the moments estimators Considerthe following model

119883119905= 1198631199051

(120579∘119905120576119905minus1

) + 1198631199052

(120579∘119905120576119905minus2

) + 1198631199053

(120579∘119905120576119905minus3

) + 120576119905

(29)

where 120576119905 is a sequence of iid Poisson random variables

with parameter 120582 and 120579 isin (0 1) The random vectors(1198631199051

1198631199052

1198631199053

) are multinomial distribution with parame-ters (1 120601

1 1206012 1206013) independent of 120576

119905 The parameters values

considered are


Table 1 Sample mean and mean square error (in brackets) formodels A B and C

Model Sample sizeParameter 50 100 400 700

A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10

The length of this discrete-valued time series 119879 is 50 100400 and 700 For each realization of these estimators 500independent replicates were simulatedThe numerical resultsof the estimators for different true values of the parameters(1206011 1206012 1206013) 120579 and 120582 are presented in Table 1

All the biases of 1206011 1206012 and 120579 are negative whereas

the biases of 120582 are positive It can be seen that as thesample size increases the estimates seem to converge tothe true parameter values For example when increasingsample size 119879 the bias and MSE both converge to zero Thereason might be that the Yule-Walker method is based onsufficient statistics On the other hand the performances ofthe estimators of 120582 are weaker than for the ones of 120601

1and 120579

5 Conclusion

In this paper we introduce a class of self-exciting thresholdinteger-valued moving average models driven by decisionrandom vector Basic probabilistic and statistical proper-ties of this class of models are discussed Specifically themethod of estimation under analysis is the Yule-Walker

Their performance is compared through a simulation studyPotential issues of future research include extending theresults for general INARMA(119901 119902) models including an arbi-trary distribution of binomial thinning parameter as well asautoregressive and moving average parameters This remainsa topic of future research

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Professor Peter X-K Songfrom University of Michigan for his help and suggestionsThe authors also thank an anonymous referee for helpfulsuggestions This work is supported by the National NaturalScience Foundation of China (no 71171166 and no 71201126)the Science Foundation of Ministry of Education of China(no 12XJC910001) and the Fundamental Research Funds forthe Central Universities (nos JBK150949 JBK150501 andJBK120405)

References

[1] S L Zeger ldquoA regression model for time series of countsrdquoBiometrika vol 75 no 4 pp 621ndash629 1988

[2] K Brannas and P Johansson ldquoTime series count data regres-sionrdquoCommunications in StatisticsTheory andMethods vol 23no 10 pp 2907ndash2925 1994

[3] M Al-Osh and A A Alzaid ldquoInteger-valued moving average(INMA) processrdquo Statistische Papers vol 29 no 4 pp 281ndash3001988

[4] R Ferland A Latour and D Oraichi ldquoInteger-valued GARCHprocessrdquo Journal of Time Series Analysis vol 27 no 6 pp 923ndash942 2006

[5] N Silva I Pereira and M E Silva ldquoForecasting in INAR(1)modelrdquo REVSTAT Statistical Journal vol 7 no 1 pp 119ndash1342009

[6] H Zheng I V Basawa and S Datta ldquoInference for pth-orderrandom coefficient integer-valued autoregressive processesrdquoJournal of Time Series Analysis vol 27 no 3 pp 411ndash440 2006

[7] H Zheng I V Basawa and S Datta ldquoFirst-order randomcoefficient integer-valued autoregressive processesrdquo Journal ofStatistical Planning and Inference vol 137 no 1 pp 212ndash2292007

[8] M Kachour and L Truquet ldquoA p-Order signed integer-valuedautoregressive (SINAR(p)) modelrdquo Journal of Time Series Anal-ysis vol 32 no 3 pp 223ndash236 2011

[9] M M Ristic A S Nastic and A V Miletic Ilic ldquoA geometrictime series model with dependent Bernoulli counting seriesrdquoJournal of Time Series Analysis vol 34 no 4 pp 466ndash476 2013

[10] C H Weiszlig ldquoThe combined INAR (p) models for time series ofcountsrdquo Statistics amp Probability Letters vol 78 no 13 pp 1817ndash1822 2008

[11] M M Ristic A S Nastic and H S Bakouch ldquoEstimation inan integer-valued autoregressive process with negative binomialmarginals (NBINAR(1))rdquo Communications in Statistics Theoryand Methods vol 41 no 4 pp 606ndash618 2012


[12] S Karlin and H E Taylor A First Course in Stochastic ProcessesAcademic Press New York NY USA 1975

[13] P J Brockwell andR ADavisTime SeriesTheory andMethodsSpringer New York NY USA 1991

[14] P Billingsley ldquoStatistical methods in Markov chainsrdquo Annals ofMathematical Statistics vol 32 pp 12ndash40 1961

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The paper is organized as follows In Section 2 wespecify the model and derive some statistical propertiesSection 3 concerns unknown parameter estimation by Yule-Walker method In Section 4 we conduct some Monte Carlosimulations Finally Section 5 concludes

2 Definition and Basic Properties ofthe PCINMA(119902) Process

Definition 1 (PCINMA(119902) model) A stochastic process 119883119905

is said to be the Poisson combined INMA(119902) process if itsatisfies the following recursive equations

119883119905= 1198631199051

(120579∘119905120576119905minus1

) + sdot sdot sdot + 119863119905119902

(120579∘119905120576119905minus119902

) + 120576119905 (2)

where 120576119905 119905 isin Z is a sequence of independent and identically

distributed Poisson random variables with parameter 120582 and120579 isin (0 1) D

119905 119905 isin Z is an iid process of ldquodecisionrdquo

random vector D119905

= (1198631199051

119863119905119902

) sim Mult(1 1206011 120601

119902)

independent of 120576119905 119905 isin Z Moreover the counting series

120579∘119905120576119905minus119896

119896 = 1 119902 are independent of D119905 and 120576

119905at time 119905

To our knowledge few efforts have been devoted to studyingcombined INMA models In this paper we aim to fill thisgap Definition 1 shows that 119883

119905is equal to 120579∘

119905120576119905minus1

+ 120576119905with

probability 1206011or equal to 120579∘

119905120576119905minus2

+ 120576119905with probability 120601

2

or else equal to 120579∘119905120576119905minus119902

+120576119905with probability 120601

119902(= 1minus120601

1minus1206012minus

sdot sdot sdot minus 120601119902minus1

)

The moments will be useful in obtaining the appropriateestimating equations for parameter estimation

Theorem 2 The numerical characteristics of 119883119905 are as

follows(i)

120583119883

= 119864 (119883119905) = 120582 (120579 + 1) (3)

(ii)

1205902

119883= Var (119883

119905) = 120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2 (4)

(iii)

120574119883

(119896) = cov (119883119905 119883119905minus119896

)

=

1205791205822120601119896+ 12057921205822

119902

sum

119895=119896+1

120601119895120601119895minus119896

1 le 119896 le 119902 minus 1

1205791205822120601119896 119896 = 119902

0 119896 ge 119902 + 1

(5)

Proof (i) Consider

119864 (119883119905) = 119864 [119863

1199051(120579∘119905120576119905minus1

+ 120576119905)] + sdot sdot sdot + 119864 [119863

119905119902(120579∘119905120576119905minus119902

+ 120576119905)]

= 119864 (1198631199051

) 119864 (120579∘119905120576119905minus1

+ 120576119905) + sdot sdot sdot + 119864 (119863

119905119902)

sdot 119864 (120579∘119905120576119905minus119902

+ 120576119905)

= 1206011(120582120579 + 120582) + sdot sdot sdot + 120601

119902(120582120579 + 120582) = 120582 (120579 + 1)

(6)

(ii) Moreover

Var (119883119905) = Var(

119902

sum

119895=1

119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

=

119902

sum

119895=1

Var (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

+ 2sum

119894lt119895

cov (119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905) 119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

(7)

Note that Var(119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905)) in first summation on the

right-hand side is

= Var (119863119905119895

) 119864 (120579∘119905120576119905minus119895

+ 120576119905)2

+ 1198642(119863119905119895

)Var (120579∘119905120576119905minus119895

+ 120576119905)

= 120601119895(1 minus 120601

119895) 119864 ((120579∘

119905120576119905minus119895

)2

+ 1205762

119905+ 2 (120579∘

119905120576119905minus119895

) 120576119905)

+ 1206012

119895(Var (120579∘

119905120576119905minus119895

) + Var (120576119905) + 2 cov (120579∘

119905120576119905minus119895

120576119905))

= 120601119895(1 minus 120601

119895) [1205792119864 (1205762

119905minus119895) + 120579 (1 minus 120579) 119864 (120576

119905minus119895)] + 119864 (120576

2

119905)

+ 2119864 (120579∘119905120576119905minus119895

) 119864 (120576119905)

+ 1206012

119895[1205792 Var (120576

119905minus119895) + 120579 (1 minus 120579) 119864 (120576

119905minus119895)] + 120590

2

120576

= 120601119895(1 minus 120601

119895) [(120579 + 1)

21205822+ 120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2]

+ 1206012

119895[(1205792+ 1) 120582

2+ 120579 (1 minus 120579) 120582]

= 120601119895(1 minus 120601

119895) (120579 + 1)

21205822+ 120601119895[120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2]

(8)

By the same arguments as above it follows that

cov (119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905) 119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

= cov (119863119905119894

(120579∘119905120576119905minus119894

) 119863119905119895

(120579∘119905120576119905minus119895

))

+ cov (119863119905119894

(120579∘119905120576119905minus119894

) 119863119905119895

120576119905)

+ cov (119863119905119894120576119905 119863119905119895

(120572∘119905120576119905minus119895

)) + cov (119863119905119894120576119905 119863119905119895

120576119905)

= cov (119863119905119894 119863119905119895

) 119864 (120579∘119905120576119905minus119894

) 119864 (120579∘119905120576119905minus119895

)

+ cov (119863119905119894 119863119905119895

) 119864 (120579∘119905120576119905minus119894

) 119864 (120576119905)

+ cov (119863119905119894 119863119905119895

) 119864 (120576119905) 119864 (120579∘

119905120576119905minus119895

)

+ cov (119863119905119894 119863119905119895

) 1198642(120576119905)

= minus120601119894120601119895(12057921205822+ 1205791205822+ 1205791205822+ 1205822) = minus120601

119894120601119895(120579 + 1)

21205822

(9)


Therefore

Var (119883119905) = Var(

119902

sum

119895=1

119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

=

119902

sum

119895=1

(120601119895(1 minus 120601

119895) (120579 + 1)

21205822

+120601119895[120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2])

minus 2sum

119894lt119895

120601119894120601119895(120579 + 1)

21205822

= (120579 + 1)21205822(1 minus

119902

sum

119895=1

1206012

119895)

+ [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2]

minus 2 (120579 + 1)21205822sum

119894lt119895

120601119894120601119895

= 120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2

(10)


is

cov (119883119905 119883119905minus119896

)

= cov(

119902

sum

119894=1

119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

= cov(119863119905119896

(120579∘119905120576119905minus119896

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

+ 120576119905)

119863119905minus119896119895minus119896

(120579∘119905minus119896


+ 120576119905minus119896

))

=

119902

sum

119895=1

cov (119863119905119896

(120579∘119905120576119905minus119896

) 119863119905minus119896119895

120576119905minus119896

)

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

) 119863119905minus119896119895minus119896

(120579∘119905minus119896

120576119905minus119895

))

=

119902

sum

119895=1

cov (120579∘119905120576119905minus119896

120576119905minus119896

) 119864 (119863119905119896

) 119864 (119863119905minus119896119895

)

+

119902

sum

119895=119896+1

cov (120579∘119905minus119895

120576119905minus119895

120579∘119905minus119896

120576119905minus119895

) 119864 (119863119905119895

) 119864 (119863119905minus119896119895minus119896

)

=

119902

sum

119895=1

120579Var (120576119905minus119896

) 120601119896120601119895+

119902

sum

119895=119896+1

1205792 Var (120576

119905minus119895) 120601119895120601119895minus119896

= 1205791205822120601119896+ 12057921205822

119902

sum

119895=119896+1

120601119895120601119895minus119896

(11)


119883119905minus119902


119905120576119905minus1

120579∘119905120576119905minus119902

120576119905 120579∘119905minus119896

120576119905minus119896minus1

120579∘119905minus119896

120576119905minus119896minus119902

and 120576119905minus119896


and 119883119905minus119896




(2) Then


(ii) 119864(119883119896


0



119864 (1198832

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))2

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))2

(12)

Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))2

= 119864 (119863119905119895

)2

119864 (120579∘119905120576119905minus119895

+ 120576119905)2

= 119864 (119863119905119895

)2

(119864 (120579∘119905120576119905minus119895

)2

+ 119864 (120576119905)2

+ 2119864 [(120579∘119905120576119905minus119895

) 120576119905])

= 120601119895(1 minus 120601

119895)

sdot [1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582 + (120582 + 120582

2) + 2120579120582

2]

= 120601119895(1 minus 120601

119895) (120579 + 1) 120582 [(120579 + 1) 120582 + 1]

(13)

Then we have

119864 (1198832

119905) le

1

4(120579 + 1) 120582 [(120579 + 1) 120582 + 1] lt infin (14)


119864 (1198833

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))3

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))3

(15)


Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))3

= 119864 (119863119905119895

)3

119864 (120579∘119905120576119905minus119895

+ 120576119905)3

= 119864 (119863119905119895

)3

(119864 (120579∘119905120576119905minus119895

)3

+ 119864 (120576119905)3

+ 3119864 [(120579∘119905120576119905minus119895

)2

120576119905] + 119864 [(120579∘

119905120576119905minus119895

) 1205762

119905])

= 1206013

119895[1205793120591 + 3120579

2(1 minus 120579) (120582 + 120582

2)


3) 120582]

+ 1206013

119895120591 + 3120601

3

119895[1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582] 120582

+ 31206013

119895120579120582 (120582 + 120582

2)

= 1206013

119895120579120582 [120579

2(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2+ 6120582 + 1]

+ 120591

(16)

where 120591 = 1205823+ 31205822+ 120582 Let 1206013max = max(1206013

1 120601

3

119902) Then

119864 (1198833

119905)

le 1206013

max 120579120582 [1205792(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2

+ 6120582 + 1] + 120591 lt infin

(17)


119905 There are a


119905




119879) = 119864(119885

119879minus

120583119885)2


119879= (1119879)sum

119879

119905=1119883119905






119879

119897=1[119864(119885119905119885119905+V119885119905+119897119885119905+119897+V) minus 120574

2

119885(V)] =


lim119879rarrinfin

1

119879

119879

sum

119897=1

119864 [120574119885(V) minus 120574

2

119885(V)]2

= 0 (18)


119885(V) =

(1119879)sum119879minusV119905=1

119885119905+V119885119905





(ii) 120574119884(119896) = cov(119884

119905 119884119905minus119896

) = 120574119883(119896) 119896 = 1 2 3

(iii) 119864(119884119896

119905) le 119862


119862lowast

gt 0(iv) 119884





lim119879rarrinfin

1

119879

119879

sum

119897=1

[119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] = 0 (19)


2= 119864(119884

2

119905) 1198883

= 119864(1198843

119905)

1198884= 119864(119884

4

119905) and 119877

119884(V) = 119864(119884

119905119884119905+V)

Case 1 For V = 0 119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864(119884

2

1199051198842

119905+119897) 1205742119884(V) =

1205742

119883(V) = 120590

2

119883


119864 (1198842

1199051198842

119905+119897) le radic119864 (

10038161003816100381610038161198842

119905

10038161003816100381610038162

) 119864 (100381610038161003816100381610038161198842119905+119897

10038161003816100381610038161003816

2

) = 1198884le 119862lowast (20)

If 119897 ge 119902 + 1 note that 119884119905+119897


119864 (1198842

1199051198842

119905+119897) = 119864 (119884

2

119905) 119864 (119884

2

119905+119897) = 1198882

2= 1205742

119883(V) (21)


119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le

(1119879)[2(1198884minus 1205902


Case 2 For V ge 1 1198772119884(V) = [119864(119884

119905119884119905+V)]2= 1205742

119883(V)


119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V)

le radic119864 (1003816100381610038161003816119884119905119884119905+V

10038161003816100381610038162

) 119864 (1003816100381610038161003816119884119905+119897119884119905+119897+V

10038161003816100381610038162

)

le14radic119864 (1198844

119905) 119864 (1198844

119905+V) 119864 (1198844119905+119897

) 119864 (1198844119905+119897+V) = 119888

4

(22)


119905+119897119883119905+119897+V uncorrelation

thus we have

119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864 (119883

119905119883119905+V) 119864 (119884

119905+119897119884119905+119897+V) = 119877

2

119884(V) (23)

Then (1119879)sum119879

119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le (1119879)(2 +

V)[1198884minus 1198772


This proves Lemma 6






(1119879)sum119879minus119896

119905=1119883119905+119896

120574119884(119896) = (1119879)sum

119879minus119896

119905=1119884119905+119896

119884119905 and 120574

119883(119896) =

(1119879)sum119879minus119896

119905=1(119883119905+119896





119879minus 120583119883)2

rarr 0Then 119883






119879(119896)119875

997888rarr 120583119883 Suppose 120576

1gt 0 is given

119875 (10038161003816100381610038161003816119883119879(119896) minus 120583

119883

10038161003816100381610038161003816ge 1205761)

le 119875(10038161003816100381610038161003816119883119879(119896) minus 119883

10038161003816100381610038161003816ge

1205761

2)

+ 119875(10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

= 119875(

1003816100381610038161003816100381610038161003816100381610038161003816

1

119879

119896

sum

119905=1

119883119905

1003816100381610038161003816100381610038161003816100381610038161003816

ge1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

le

119896

sum

119905=1

119875(1

119879

10038161003816100381610038161198831199051003816100381610038161003816 ge

1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

(24)


119905=1119875((1119879)|119883

119905| ge 120576

12) le

sum119896

119905=1(119864(119883119905)(12)119879120576



119879minus 120583119883| ge


119879(119896)119875



119883(119896)119875


119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

= 119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896) + 120574

119884(119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

le 119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2) + 119875(

1003816100381610038161003816120574119884 (119896) minus 120574119883

(119896)1003816100381610038161003816 ge

120576

2)

(25)

Note that

119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2)

= 119875(100381610038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883) + (119883

2

119879minus 1205832

119883)100381610038161003816100381610038161003816ge

120576

2)

le 119875(10038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883)10038161003816100381610038161003816ge

120576

4)

+ 119875(1003816100381610038161003816100381610038161198832

119879minus 1205832

119883

100381610038161003816100381610038161003816ge

120576

4)

(26)


119883


119879)(119883119879

minus

120583119883)119875

997888rarr 0 1198832119879minus 1205832

119883

119875

997888rarr 0


119884(119896)| ge 1205762)

119875


120574119884(119896) minus 120574

119883(119896) = 120574

119884(119896) minus 120574

119884(119896)119901


(27)


119883(119896)| ge 120576)

119875




1 119883




120574 (0) minus [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2] = 0

120574 (1) minus (12057912058221206011+ 12057921205822

119902

sum

119895=2

120601119895120601119895minus1

) = 0


+ 120579212058221206011199021206011) = 0

120574 (119902) minus 1205791205822120601119902= 0

(28)







119883119905= 1198631199051

(120579∘119905120576119905minus1

) + 1198631199052

(120579∘119905120576119905minus2

) + 1198631199053

(120579∘119905120576119905minus3

) + 120576119905

(29)



1198631199052

1198631199053


1 1206012 1206013) independent of 120576


considered are




A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10




1and 120579

5 Conclusion





Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Therefore

Var (119883119905) = Var(

119902

sum

119895=1

119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))

=

119902

sum

119895=1

(120601119895(1 minus 120601

119895) (120579 + 1)

21205822

+120601119895[120579 (1 minus 120579) 120582 + (120579

2+ 1) 120582

2])

minus 2sum

119894lt119895

120601119894120601119895(120579 + 1)

21205822

= (120579 + 1)21205822(1 minus

119902

sum

119895=1

1206012

119895)

+ [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2]

minus 2 (120579 + 1)21205822sum

119894lt119895

120601119894120601119895

= 120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2

(10)


is

cov (119883119905 119883119905minus119896

)

= cov(

119902

sum

119894=1

119863119905119894

(120579∘119905120576119905minus119894

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

= cov(119863119905119896

(120579∘119905120576119905minus119896

+ 120576119905)

119902

sum

119895=1

119863119905minus119896119895

(120579∘119905minus119896

120576119905minus119896minus119895

+ 120576119905minus119896

))

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

+ 120576119905)

119863119905minus119896119895minus119896

(120579∘119905minus119896


+ 120576119905minus119896

))

=

119902

sum

119895=1

cov (119863119905119896

(120579∘119905120576119905minus119896

) 119863119905minus119896119895

120576119905minus119896

)

+

119902

sum

119895=119896+1

cov (119863119905119895

(120579∘119905minus119895

120576119905minus119895

) 119863119905minus119896119895minus119896

(120579∘119905minus119896

120576119905minus119895

))

=

119902

sum

119895=1

cov (120579∘119905120576119905minus119896

120576119905minus119896

) 119864 (119863119905119896

) 119864 (119863119905minus119896119895

)

+

119902

sum

119895=119896+1

cov (120579∘119905minus119895

120576119905minus119895

120579∘119905minus119896

120576119905minus119895

) 119864 (119863119905119895

) 119864 (119863119905minus119896119895minus119896

)

=

119902

sum

119895=1

120579Var (120576119905minus119896

) 120601119896120601119895+

119902

sum

119895=119896+1

1205792 Var (120576

119905minus119895) 120601119895120601119895minus119896

= 1205791205822120601119896+ 12057921205822

119902

sum

119895=119896+1

120601119895120601119895minus119896

(11)


119883119905minus119902


119905120576119905minus1

120579∘119905120576119905minus119902

120576119905 120579∘119905minus119896

120576119905minus119896minus1

120579∘119905minus119896

120576119905minus119896minus119902

and 120576119905minus119896


and 119883119905minus119896




(2) Then


(ii) 119864(119883119896


0



119864 (1198832

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))2

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))2

(12)

Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))2

= 119864 (119863119905119895

)2

119864 (120579∘119905120576119905minus119895

+ 120576119905)2

= 119864 (119863119905119895

)2

(119864 (120579∘119905120576119905minus119895

)2

+ 119864 (120576119905)2

+ 2119864 [(120579∘119905120576119905minus119895

) 120576119905])

= 120601119895(1 minus 120601

119895)

sdot [1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582 + (120582 + 120582

2) + 2120579120582

2]

= 120601119895(1 minus 120601

119895) (120579 + 1) 120582 [(120579 + 1) 120582 + 1]

(13)

Then we have

119864 (1198832

119905) le

1

4(120579 + 1) 120582 [(120579 + 1) 120582 + 1] lt infin (14)


119864 (1198833

119905) le max 119864 (119863

1199051(120579∘119905120576119905minus1

+ 120576119905))3

119864 (119863119905119902

(120579∘119905120576119905minus119902

+ 120576119905))3

(15)


Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))3

= 119864 (119863119905119895

)3

119864 (120579∘119905120576119905minus119895

+ 120576119905)3

= 119864 (119863119905119895

)3

(119864 (120579∘119905120576119905minus119895

)3

+ 119864 (120576119905)3

+ 3119864 [(120579∘119905120576119905minus119895

)2

120576119905] + 119864 [(120579∘

119905120576119905minus119895

) 1205762

119905])

= 1206013

119895[1205793120591 + 3120579

2(1 minus 120579) (120582 + 120582

2)


3) 120582]

+ 1206013

119895120591 + 3120601

3

119895[1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582] 120582

+ 31206013

119895120579120582 (120582 + 120582

2)

= 1206013

119895120579120582 [120579

2(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2+ 6120582 + 1]

+ 120591

(16)

where 120591 = 1205823+ 31205822+ 120582 Let 1206013max = max(1206013

1 120601

3

119902) Then

119864 (1198833

119905)

le 1206013

max 120579120582 [1205792(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2

+ 6120582 + 1] + 120591 lt infin

(17)


119905 There are a


119905




119879) = 119864(119885

119879minus

120583119885)2


119879= (1119879)sum

119879

119905=1119883119905






119879

119897=1[119864(119885119905119885119905+V119885119905+119897119885119905+119897+V) minus 120574

2

119885(V)] =


lim119879rarrinfin

1

119879

119879

sum

119897=1

119864 [120574119885(V) minus 120574

2

119885(V)]2

= 0 (18)


119885(V) =

(1119879)sum119879minusV119905=1

119885119905+V119885119905





(ii) 120574119884(119896) = cov(119884

119905 119884119905minus119896

) = 120574119883(119896) 119896 = 1 2 3

(iii) 119864(119884119896

119905) le 119862


119862lowast

gt 0(iv) 119884





lim119879rarrinfin

1

119879

119879

sum

119897=1

[119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] = 0 (19)


2= 119864(119884

2

119905) 1198883

= 119864(1198843

119905)

1198884= 119864(119884

4

119905) and 119877

119884(V) = 119864(119884

119905119884119905+V)

Case 1 For V = 0 119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864(119884

2

1199051198842

119905+119897) 1205742119884(V) =

1205742

119883(V) = 120590

2

119883


119864 (1198842

1199051198842

119905+119897) le radic119864 (

10038161003816100381610038161198842

119905

10038161003816100381610038162

) 119864 (100381610038161003816100381610038161198842119905+119897

10038161003816100381610038161003816

2

) = 1198884le 119862lowast (20)

If 119897 ge 119902 + 1 note that 119884119905+119897


119864 (1198842

1199051198842

119905+119897) = 119864 (119884

2

119905) 119864 (119884

2

119905+119897) = 1198882

2= 1205742

119883(V) (21)


119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le

(1119879)[2(1198884minus 1205902


Case 2 For V ge 1 1198772119884(V) = [119864(119884

119905119884119905+V)]2= 1205742

119883(V)


119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V)

le radic119864 (1003816100381610038161003816119884119905119884119905+V

10038161003816100381610038162

) 119864 (1003816100381610038161003816119884119905+119897119884119905+119897+V

10038161003816100381610038162

)

le14radic119864 (1198844

119905) 119864 (1198844

119905+V) 119864 (1198844119905+119897

) 119864 (1198844119905+119897+V) = 119888

4

(22)


119905+119897119883119905+119897+V uncorrelation

thus we have

119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864 (119883

119905119883119905+V) 119864 (119884

119905+119897119884119905+119897+V) = 119877

2

119884(V) (23)

Then (1119879)sum119879

119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le (1119879)(2 +

V)[1198884minus 1198772


This proves Lemma 6






(1119879)sum119879minus119896

119905=1119883119905+119896

120574119884(119896) = (1119879)sum

119879minus119896

119905=1119884119905+119896

119884119905 and 120574

119883(119896) =

(1119879)sum119879minus119896

119905=1(119883119905+119896





119879minus 120583119883)2

rarr 0Then 119883






119879(119896)119875

997888rarr 120583119883 Suppose 120576

1gt 0 is given

119875 (10038161003816100381610038161003816119883119879(119896) minus 120583

119883

10038161003816100381610038161003816ge 1205761)

le 119875(10038161003816100381610038161003816119883119879(119896) minus 119883

10038161003816100381610038161003816ge

1205761

2)

+ 119875(10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

= 119875(

1003816100381610038161003816100381610038161003816100381610038161003816

1

119879

119896

sum

119905=1

119883119905

1003816100381610038161003816100381610038161003816100381610038161003816

ge1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

le

119896

sum

119905=1

119875(1

119879

10038161003816100381610038161198831199051003816100381610038161003816 ge

1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

(24)


119905=1119875((1119879)|119883

119905| ge 120576

12) le

sum119896

119905=1(119864(119883119905)(12)119879120576



119879minus 120583119883| ge


119879(119896)119875



119883(119896)119875


119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

= 119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896) + 120574

119884(119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

le 119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2) + 119875(

1003816100381610038161003816120574119884 (119896) minus 120574119883

(119896)1003816100381610038161003816 ge

120576

2)

(25)

Note that

119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2)

= 119875(100381610038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883) + (119883

2

119879minus 1205832

119883)100381610038161003816100381610038161003816ge

120576

2)

le 119875(10038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883)10038161003816100381610038161003816ge

120576

4)

+ 119875(1003816100381610038161003816100381610038161198832

119879minus 1205832

119883

100381610038161003816100381610038161003816ge

120576

4)

(26)


119883


119879)(119883119879

minus

120583119883)119875

997888rarr 0 1198832119879minus 1205832

119883

119875

997888rarr 0


119884(119896)| ge 1205762)

119875


120574119884(119896) minus 120574

119883(119896) = 120574

119884(119896) minus 120574

119884(119896)119901


(27)


119883(119896)| ge 120576)

119875




1 119883




120574 (0) minus [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2] = 0

120574 (1) minus (12057912058221206011+ 12057921205822

119902

sum

119895=2

120601119895120601119895minus1

) = 0


+ 120579212058221206011199021206011) = 0

120574 (119902) minus 1205791205822120601119902= 0

(28)







119883119905= 1198631199051

(120579∘119905120576119905minus1

) + 1198631199052

(120579∘119905120576119905minus2

) + 1198631199053

(120579∘119905120576119905minus3

) + 120576119905

(29)



1198631199052

1198631199053


1 1206012 1206013) independent of 120576


considered are




A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10




1and 120579

5 Conclusion





Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Note that for 119895 = 1 2 119902

119864 (119863119905119895

(120579∘119905120576119905minus119895

+ 120576119905))3

= 119864 (119863119905119895

)3

119864 (120579∘119905120576119905minus119895

+ 120576119905)3

= 119864 (119863119905119895

)3

(119864 (120579∘119905120576119905minus119895

)3

+ 119864 (120576119905)3

+ 3119864 [(120579∘119905120576119905minus119895

)2

120576119905] + 119864 [(120579∘

119905120576119905minus119895

) 1205762

119905])

= 1206013

119895[1205793120591 + 3120579

2(1 minus 120579) (120582 + 120582

2)


3) 120582]

+ 1206013

119895120591 + 3120601

3

119895[1205792(120582 + 120582

2) + 120579 (1 minus 120579) 120582] 120582

+ 31206013

119895120579120582 (120582 + 120582

2)

= 1206013

119895120579120582 [120579

2(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2+ 6120582 + 1]

+ 120591

(16)

where 120591 = 1205823+ 31205822+ 120582 Let 1206013max = max(1206013

1 120601

3

119902) Then

119864 (1198833

119905)

le 1206013

max 120579120582 [1205792(120591 minus 1 minus 3120582) + 3120582 (1 + 120582) 120579 + 3120582

2

+ 6120582 + 1] + 120591 lt infin

(17)


119905 There are a


119905




119879) = 119864(119885

119879minus

120583119885)2


119879= (1119879)sum

119879

119905=1119883119905






119879

119897=1[119864(119885119905119885119905+V119885119905+119897119885119905+119897+V) minus 120574

2

119885(V)] =


lim119879rarrinfin

1

119879

119879

sum

119897=1

119864 [120574119885(V) minus 120574

2

119885(V)]2

= 0 (18)


119885(V) =

(1119879)sum119879minusV119905=1

119885119905+V119885119905





(ii) 120574119884(119896) = cov(119884

119905 119884119905minus119896

) = 120574119883(119896) 119896 = 1 2 3

(iii) 119864(119884119896

119905) le 119862


119862lowast

gt 0(iv) 119884





lim119879rarrinfin

1

119879

119879

sum

119897=1

[119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] = 0 (19)


2= 119864(119884

2

119905) 1198883

= 119864(1198843

119905)

1198884= 119864(119884

4

119905) and 119877

119884(V) = 119864(119884

119905119884119905+V)

Case 1 For V = 0 119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864(119884

2

1199051198842

119905+119897) 1205742119884(V) =

1205742

119883(V) = 120590

2

119883


119864 (1198842

1199051198842

119905+119897) le radic119864 (

10038161003816100381610038161198842

119905

10038161003816100381610038162

) 119864 (100381610038161003816100381610038161198842119905+119897

10038161003816100381610038161003816

2

) = 1198884le 119862lowast (20)

If 119897 ge 119902 + 1 note that 119884119905+119897


119864 (1198842

1199051198842

119905+119897) = 119864 (119884

2

119905) 119864 (119884

2

119905+119897) = 1198882

2= 1205742

119883(V) (21)


119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le

(1119879)[2(1198884minus 1205902


Case 2 For V ge 1 1198772119884(V) = [119864(119884

119905119884119905+V)]2= 1205742

119883(V)


119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V)

le radic119864 (1003816100381610038161003816119884119905119884119905+V

10038161003816100381610038162

) 119864 (1003816100381610038161003816119884119905+119897119884119905+119897+V

10038161003816100381610038162

)

le14radic119864 (1198844

119905) 119864 (1198844

119905+V) 119864 (1198844119905+119897

) 119864 (1198844119905+119897+V) = 119888

4

(22)


119905+119897119883119905+119897+V uncorrelation

thus we have

119864 (119884119905119884119905+V119884119905+119897119884119905+119897+V) = 119864 (119883

119905119883119905+V) 119864 (119884

119905+119897119884119905+119897+V) = 119877

2

119884(V) (23)

Then (1119879)sum119879

119897=1[119864(119884119905119884119905+V119884119905+119897119884119905+119897+V) minus 120574

2

119884(V)] le (1119879)(2 +

V)[1198884minus 1198772


This proves Lemma 6






(1119879)sum119879minus119896

119905=1119883119905+119896

120574119884(119896) = (1119879)sum

119879minus119896

119905=1119884119905+119896

119884119905 and 120574

119883(119896) =

(1119879)sum119879minus119896

119905=1(119883119905+119896





119879minus 120583119883)2

rarr 0Then 119883






119879(119896)119875

997888rarr 120583119883 Suppose 120576

1gt 0 is given

119875 (10038161003816100381610038161003816119883119879(119896) minus 120583

119883

10038161003816100381610038161003816ge 1205761)

le 119875(10038161003816100381610038161003816119883119879(119896) minus 119883

10038161003816100381610038161003816ge

1205761

2)

+ 119875(10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

= 119875(

1003816100381610038161003816100381610038161003816100381610038161003816

1

119879

119896

sum

119905=1

119883119905

1003816100381610038161003816100381610038161003816100381610038161003816

ge1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

le

119896

sum

119905=1

119875(1

119879

10038161003816100381610038161198831199051003816100381610038161003816 ge

1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

(24)


119905=1119875((1119879)|119883

119905| ge 120576

12) le

sum119896

119905=1(119864(119883119905)(12)119879120576



119879minus 120583119883| ge


119879(119896)119875



119883(119896)119875


119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

= 119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896) + 120574

119884(119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

le 119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2) + 119875(

1003816100381610038161003816120574119884 (119896) minus 120574119883

(119896)1003816100381610038161003816 ge

120576

2)

(25)

Note that

119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2)

= 119875(100381610038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883) + (119883

2

119879minus 1205832

119883)100381610038161003816100381610038161003816ge

120576

2)

le 119875(10038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883)10038161003816100381610038161003816ge

120576

4)

+ 119875(1003816100381610038161003816100381610038161198832

119879minus 1205832

119883

100381610038161003816100381610038161003816ge

120576

4)

(26)


119883


119879)(119883119879

minus

120583119883)119875

997888rarr 0 1198832119879minus 1205832

119883

119875

997888rarr 0


119884(119896)| ge 1205762)

119875


120574119884(119896) minus 120574

119883(119896) = 120574

119884(119896) minus 120574

119884(119896)119901


(27)


119883(119896)| ge 120576)

119875




1 119883




120574 (0) minus [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2] = 0

120574 (1) minus (12057912058221206011+ 12057921205822

119902

sum

119895=2

120601119895120601119895minus1

) = 0


+ 120579212058221206011199021206011) = 0

120574 (119902) minus 1205791205822120601119902= 0

(28)







119883119905= 1198631199051

(120579∘119905120576119905minus1

) + 1198631199052

(120579∘119905120576119905minus2

) + 1198631199053

(120579∘119905120576119905minus3

) + 120576119905

(29)



1198631199052

1198631199053


1 1206012 1206013) independent of 120576


considered are




A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10




1and 120579

5 Conclusion





Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1119879)sum119879minus119896

119905=1119883119905+119896

120574119884(119896) = (1119879)sum

119879minus119896

119905=1119884119905+119896

119884119905 and 120574

119883(119896) =

(1119879)sum119879minus119896

119905=1(119883119905+119896





119879minus 120583119883)2

rarr 0Then 119883






119879(119896)119875

997888rarr 120583119883 Suppose 120576

1gt 0 is given

119875 (10038161003816100381610038161003816119883119879(119896) minus 120583

119883

10038161003816100381610038161003816ge 1205761)

le 119875(10038161003816100381610038161003816119883119879(119896) minus 119883

10038161003816100381610038161003816ge

1205761

2)

+ 119875(10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

= 119875(

1003816100381610038161003816100381610038161003816100381610038161003816

1

119879

119896

sum

119905=1

119883119905

1003816100381610038161003816100381610038161003816100381610038161003816

ge1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

le

119896

sum

119905=1

119875(1

119879

10038161003816100381610038161198831199051003816100381610038161003816 ge

1205761

2) + 119875(

10038161003816100381610038161003816119883119879minus 120583119883

10038161003816100381610038161003816ge

1205761

2)

(24)


119905=1119875((1119879)|119883

119905| ge 120576

12) le

sum119896

119905=1(119864(119883119905)(12)119879120576



119879minus 120583119883| ge


119879(119896)119875



119883(119896)119875


119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

= 119875 (1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896) + 120574

119884(119896) minus 120574

119883(119896)

1003816100381610038161003816 ge 120576)

le 119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2) + 119875(

1003816100381610038161003816120574119884 (119896) minus 120574119883

(119896)1003816100381610038161003816 ge

120576

2)

(25)

Note that

119875(1003816100381610038161003816120574119883 (119896) minus 120574

119884(119896)

1003816100381610038161003816 ge120576

2)

= 119875(100381610038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883) + (119883

2

119879minus 1205832

119883)100381610038161003816100381610038161003816ge

120576

2)

le 119875(10038161003816100381610038161003816(119883119879(119896) + 119883

119879) (119883119879minus 120583119883)10038161003816100381610038161003816ge

120576

4)

+ 119875(1003816100381610038161003816100381610038161198832

119879minus 1205832

119883

100381610038161003816100381610038161003816ge

120576

4)

(26)


119883


119879)(119883119879

minus

120583119883)119875

997888rarr 0 1198832119879minus 1205832

119883

119875

997888rarr 0


119884(119896)| ge 1205762)

119875


120574119884(119896) minus 120574

119883(119896) = 120574

119884(119896) minus 120574

119884(119896)119901


(27)


119883(119896)| ge 120576)

119875




1 119883




120574 (0) minus [120579 (1 minus 120579) 120582 + (1205792+ 1) 120582

2] = 0

120574 (1) minus (12057912058221206011+ 12057921205822

119902

sum

119895=2

120601119895120601119895minus1

) = 0


+ 120579212058221206011199021206011) = 0

120574 (119902) minus 1205791205822120601119902= 0

(28)







119883119905= 1198631199051

(120579∘119905120576119905minus1

) + 1198631199052

(120579∘119905120576119905minus2

) + 1198631199053

(120579∘119905120576119905minus3

) + 120576119905

(29)



1198631199052

1198631199053


1 1206012 1206013) independent of 120576


considered are




A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10




1and 120579

5 Conclusion





Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












A

120601100781 00873 00925 00953(00316) (00084) (00047) (00017)

120601201658 01795 01867 01957(00251) (00193) (00052) (00035)

12057902433 02731 02875 02924(00236) (00074) (00032) (00021)

12058211372 11007 10285 10046(01256) (00632) (00358) (00067)

B

120601102547 02871 02935 02977(00412) (00136) (00087) (00032)

120601203468 03726 03891 03964(00308) (00158) (00072) (00023)

12057904625 04789 04836 04976(00217) (00104) (00044) (00013)

12058235672 31741 30522 30075(03265) (02381) (00943) (00045)

C

120601105648 05783 05867 05973(00213) (00151) (00078) (00024)

120601201732 01847 01907 01976(00342) (00153) (00024) (00005)

12057906347 06824 06947 06953(00145) (00083) (00026) (00011)

120582123627 106538 101486 100024(10764) (03105) (00852) (00053)

(model A) (1206011 1206012 1206013) = (01 02 07) 120579 = 03 120582 = 1

(model B) (1206011 1206012 1206013) = (03 04 03) 120579 = 05 120582 = 3

(model C) (1206011 1206012 1206013) = (06 02 02) 120579 = 07 120582 =

10




1and 120579

5 Conclusion





Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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