Chapter 1

INTRODUCTION

1.1 Order Statistics and its Importance

Order statistics as a branch of statistics deals with the properties and applications

of ordered random variables and of functions involving them. The sample maximum,

sample minimum, sample median, sample range etc., are simple examples of order d

random variables which are of much use in day-to-day life.

Let Xl, X 2 , ... , X n be a random sample of size n drawn from a population. If the

observations are arranged in non-decreasing order as X I :n ::; X 2:n ::; ... ::; X n :nl then

X I:n, X 2:n, ... ,Xn :n are known as the order statistics of the sample and in particular

for any integer r such that 1 ::; r ::; n, X r :n is called the r th order statistic. L t

Xl, X 2 , ... , X n be a random sample of size n drawn from a population with abso­

lutely continuous distribution function F(x) and probability density function (pdf)

f(x). Then the pdf fxr:n(x) of the r th order statistic X r :n for 1 ~ r ::; n is giv n by

(see David and Nagaraja, 2003, p.lO)n'

fX r :n(x) = (r _ l)!(n _ r)! [F(x)r- l [l - F(x)t-r f(x), -00 < x < 00.

The joint pdf fXr,.o:n (Xl, X2) of the order statistics X r:n and X s:n for 1 ::; r < 8 ::; n is

given byn'

fXr,s:n (Xl, X2) = (r _ 1)'(8 _ r'- l)'(n _ 8)! [F(XIW-1[F(X2) - F(X1W-

r-

1

X [1 - F(X2)]n-s f(Xl)f(X2)' -CX) < Xl < X2 < 00.

1

Order statistics and functions of order statistics play a very important role in

statistical theory and methodology. In many cases, methods based on order statistics

become more efficient than others. In certain other situations, they are us d because

of their simplicity or their robustness, even at the cost of some minor loss of effi i n y.

Developments in order statistics until 1962 were summarized in the edited volume

of Sarhan and Greenberg (1962). The two volumes published by Harter (1970a,1970b)

presented numerous tables, which facilitate the use of order statistics in tests of hy­

pothesis and in the construction of estimators for the unknown parameters involved

in many parent distributions. These two volumes have been further revised and ex­

panded by Harter and Balakrishnan (1996, 1997). Outliers are naturally observed as

extreme observations in the data. Barnett and Lewis (1994) have given an excellent

survey of all developments concerning outliers. In Arnold and Balakrishnan (1989),

an extensive collection of results relating to recurrence relations, bounds and approx­

imations for the moments of order statistics are enlisted. Balakrishnan and Cohen

(1991) elaborate various methods of estimation based on complete and censored sam­

ples. Arnold et al. (1992) provide a course of fundamental study in order statistics.

Balakrishnan and Rao (1998 a, 1998 b) have edited two handbooks, the first of whi h

focuses on theory and methods of order statistics and the second dealing primarily

with applications of order statistics. David and Nagaraja (2003) have elaborately

elucidated the latest developments in both theory and applications of order statistics.

Let Xl, X 2 , ... , X n be independent random variables with distribution functions

F1 (x), F2 (x), ... , Fn(x) respectively and corresponding pdf's II (x), f2(x), ... ,fn(x)

respectively. Then the pdf of the r th order statistic Xr :n of these indep ndent

2

non-identically distributed (inid) random variables is given by

[

~l(X) 1 - ~l(X) fl(X)]1 ...

f (x) = per:::X

r:n (r - l)!(n - r)! ~n(x) 1 - ~n(x) fn(x) ,

r-1 n-r 1

(1.1.1)

where per A of a square matrix A is the permanent of A and is defined as ju t like th '

determinant of the matrix A except that in the permanent all terms in its expansion

are taken with positive sign and in a permanent if a symbol a is marked below a

column vector I x' then it means that the column vector I x' has a copies in it.

For r < s, the joint pdf of X r :n and X s:n is given by

1fXr,s:n (x, y) = C( . )perr,s. n [

~lix) ~l(Y) ~ ~l(X) 1- ~l(Y) h~X) h~Y)]

~n(x) ~n(Y) - ~n(x) 1 - ~n(Y) fn(x) fn(Y)r-1 s-r-1 n-s 1 1

(1.1.2)

where C(r, s : n) = (r - l)!(s - r - l)!(n - s)! and -00 < x < Y < 00.

The permanent expressions for the order statistics densities were introduced by Vaughan

and Venables (1972). For further works on the theory of order statistics of inid ran­

dom variables, see Beg (1991), Balakrishnan (1988, 1994), Childs and Balakrishnan

(2006), Samuel and Thomas (1998), Cramer et al. (2009), Thomas and Sajeevkumar

(2002, 2003, 2005) and Sajeevkumar and Thomas (2005, 2006).

1.2 Concomitants of Order Statistics

Suppose (Xl, Yi), (X2 , 1'2), ... , (Xn , Yn ) are the observations of a random sampl of

size n drawn from a bivariate population. It is to be noted that for the bivariate

observations there is no unique way of ordering them as a natural generalization of

order statistics of a sample drawn from a univariate population. However, if we order

the Xi's in the above bivariate sample as X 1:n X 2:n, ... ,X n:n th n the random vari­

able which occur together as a Y component in the ordered pair with X compon nt

3

(1.2.1)

equal to X r :n (1 ::::; r ::::; n) is termed as the concomitant of the r th order statistic and

is usually denoted by Yir:nl (see David, 1973). Similarly, w can define the concomi­

tants of order statistics by ordering the Y observations as well. An earlier review of

literature on concomitants of order statistics is available in Bhattacharya (1984). For

a review of later studies on this topic, see David and Nagaraja (1998, 2003).

Let (Xl, Yi), (X2 , 1'2), ... , (Xn , Yn ) be a random sample of size n drawn from a

continuous bivariate population having joint pdf h(x, y) with marginal pdf's fx(x)

and fy (y) respectively. Let Fx (x) and Fy (y) denote the distribution functions cor­

responding to the pdf's fx(x) and fy(y) respectively. Then the pdf fY[r:nj(Y) of the

concomitant ofthe r th order statistic, Yir:nl for 1 ::::; r ::::; n is given by (see Yang, 1977)

f¥rr:nJ(Y) = I: fxr:Jx)fYlx(ylx)dx

100 ,

= (_ )~( _ ),[F(XW-1 [1-F(x)r-rh(x,Y)dx.-00 r 1. n r.

The joint pdf of the concomitants of the r th and 8 th order statistics, Yir:nJ and Yis:n],

for 1 ::::; r < 8 ::::; n is given by (see Yang, 1977)

f¥rr,s:nJ (Yl, Y2) = I:I: fXr,s:JXl' x2)fYIX(Yllxdfy/x(Y2Ix2)dxldx2

100 lx2 n'= -00 -00 (r - 1)'(8 _ r'- l)(n _ 8)! [F(Xl)r-

1[F(X2) - F(Xl)]s-r-l[l - F(X2)r s

x h(Xl, Yl)h(X2, Y2)dxldx2. (1.2.2)

Concomitants of order statistics are applied successfully to deal with statistical

inference problems associated with several real life situations.The most important

use of concomitants of order statistics arises in selection procedures, when k « n)

individuals are chosen on the basis of their X-values. In this case the corresponding

Y-values on a variable of primary interest may represent performance on an associ t d

characteristic which is hard to measure or can be observed only later. For exampl , if

the top k out of n rams, as judged by their g netic make-up, are selected for bre ding,

then Yin-k+l:n] ... , Yin:nJ might represent th quality of the wool of the first offspring

4

in each of the above crosses. As another example, if the top k out of n candidates are

judged by their performance in a preliminary screening test, then 1[n-k+l:n] , ... ,1[n:n]

might represent the performance of the candidates in a final test. Concomitants of

order statistics appear naturally in the estimation of correlation and r gression co­

efficients in a variety of situations, most importantly in censored bivariate sampl s

where the censoring is practised on the basis of X values. Kim and David (1990) giv s

a detailed account of the dependence structure of order statistics and concomitants

of order statistics. For a detailed study on the use of concomitants of order statis­

tics in estimation see, Chacko (2005, 2007) and Chacko and Thomas (2004, 2006).

Concomitants of order statistics of independent non-identically distributed random

variables was first introduced by Eryilmaz (2005).

The study of concomitants of order statisitics has also led to ranked set sampling

which is very effective and widely applicable because of its observational economy

consideration. Concomitants of order statistics also play an important role in doubl

sampling. Chen et al. (2004) have given a detailed account of rank d seL sampling

for a population involving joint distribution of an auxiliary variable X and a variabl

of primary interest Y.

1.3 Record Values

Let Xl, X 2 , •.. be a sequence of independent observations arising from a populati n.

An observation X j will be called an upper record value (or simply a record) if its valu

exceeds that of all previous observations. Thus X j is a record if X j > Xi, 'Vi < j.

The first observation Xl is taken as the initial record RI . The next record R2 is the

observation following RI which is greater than RI and so on. The records R1, R2 , . ..

as defined above are sometimes referred to as the sequence of upp r records. Sim­

ilarly, an observation X j will be called a lower record value if its value is 1 ss than

that of all previous observations.

5

The hottest day ever, the longest winning streak in professional basket ball and

the lowest stock market figure are only some examples of records whi h v ryone

observes with curiosity. Record values are kept for almost every conceivable phe­

nomenon. Not only do we want to know who holds the record for running 100 metr s

in the Olympics and the record time, but we also want to predict the next record

breaking time. Here, we require statistical theory to predict the future using past

data. Disastrous floods and destructive earthquakes recur throughout history. Dam

construction has long focused on capturing and storing water to contain the so call d

100-year floods. Architects in California are particularly concerned with construction

designed to withstand the biggest earthquake. The designers of dams and skyscrap­

ers should be concerned with the distribution of upper record values from a possibly

dependent and non-identically distributed sequence of random variables. Observ­

ing record values also has a place in destructive stress testing and industrial quality

control experiments. The fact that: "a chain is no stronger than its weake~l link"

underlies much of the theory of strength of materials. Record value sampling in which

only the record values are measured is extremely useful when the measurement pro­

cess is costly, time consuming or destructive.

Let Xl, X 2 , .•. be a sequence of independent observations arising from a population

with absolutely continuous cumulative distribution function Fx(x) and pdf fx(x). If

we write Rn to denote the nth upper record value, then its pdf is given by (see Arnold

et al., 1998, p.10)

f ()= (-Zog[l - Fx(x )])n-l f ( )

Rn x (n _ I)! x x .

The joint pdf of the mth and nth upper record values Rm and Rn for m < n is given

by (see Arnold et al., 1998, p.ll)

6

! ( )_ [-log{l - FX(XI)}]m-1 [-log{l - FX(X2)} + log{l - FX(XI)}]n-m-1

Rm,n Xl, X2 - (m _ I)! x (n _ m - I)!

!X(XI)!X(X2) <x F ( ) , Xl X2·1- X Xl

If we write L n to denote the nth lower record value then its pdf is given by

! () = (-log[Fx(x)])n-l! ( )L n X (n _ I)! x X .

The joint pdf of the mth and nth lower record values Lm and Ln for m < n is given

by

! ( ) _ [-log{Fx(XI)}]m-1 [-log{Fx (x2)} + log{Fx(XI)}]n-m-1Lm,n Xl, X2 - (m _ I)! x (n _ m - I)!

!X(XI)!X(X2) <x FX(XI) , X2 Xl'

Chandler (1952) introduced the study of record values and documented many of

the basic properties of records. Resnick (1973) and Shorrock (1973) documented the

asymptotic theory of records. Glick (1978) provides a survey of the literature on

records. For a detailed discussion on the developments in the theory and applica­

tions of record values, see Arnold et.al. (1998), Nevzorov (1987), Nagaraja (1988),

Nevzorov and Balakrishnan (1998) and Ahsanullah (1995).

1.4 Concomitants of Record Values

The study of record concomitants was initiated by Houchens (1984). Suppose (Xl, YI ),

(X2 , Y2), .. , is a sequence of bivariate observations from a populaton. Let Rn denote

the nth upper record with respect to the X variable. Then the random variable which

occur together as a Y component in the ordered pair with X component equal to Rn

is termed as the concomitant of the nth upper record and may be denoted by Y[n].

Let Ln denote the nth lower record with respect to the X variabl . Then the random

variable which occur together as a Y compon nt in the ord red pair with X compo­

nent equal to Ln is termed as the concomitant of the nth lower record and may b

denoted by l(n)' Let (Xl Yi) (X2 Y2), ... be as qu nc of observations drawn fr m

7

a continuous bivariate population with joint pdf h(x, y) and marginal pdf's fx (x) and

fy(y). Let Fx(x) and Fy(y) be the cumulative distribution functions orr sponding

to the pdf's fx(x) and fy(y) respectively. Suppose Rl, R2 ,.·. is the s qu nc of

upper record values obtained from the observed X values. Then the pdf of the con­

comitant of the nth upper record, Y[n], is given by (see Arnold et al., 1998, p.272)

j oo JOO (-log[l- Fx(x)])n-lfYr,nj(y) = -00 fRn(x)fYlx(ylx)dx = -00 (n -I)! h(x,y)dx.

Suppose L 1 , L 2 , ... is the sequence of lower record values obtained from the observ d

X values. Then the pdf of the concomitant of the nth lower record, Y(n), is given by

JOO joo (-log[Fx (x)])n-1fY(n) (y) = -00 hn (x)fYlx(ylx)dx = -00 (n _ I)! h(x, y)dx.

The joint pdf of concomitants of two upper records, Y[mj and Y[nJ, for m < n is giv n

by

(104.1)

The joint pdf of concomitants of two lower records, Y(m) and Y(n), for m < n is

given by

(1.4.2)

For some works relating to the distribution theory and properties of concomitants

of record values, see Ahsanullah (1994), Arnold et al. (1998) and Nevzorov and Ah­

sanullah (2000). For more details about the use of concomitants of record valu s in

estimation, see Chacko (2007). Suppose, in an experiment, the X observations ar

based on an inexpensive test and the Y observations are based on an xpensiv or

destructive test. Then with an objective of observational economy one may go ah ad

with measuring Y values of only those individuals whose X measurement br aks the

previous records. The resulting data output of such an experim nt is th s quenc

{Yin]} of concomitants of upper record values. Similarly in som oth r situati ns th

8

data of interest is only the sequence {}(n)} of concomitants of lower r cord valu .

1.5 Rank of Concomitants of Order Statistics

The concomitant of the r th order statistic Y[r:n] , may take any rank from 1 to n

on ranking the Y observations. Thus the rank of the concomitant of the r th order

statistic is a discrete random variable for each r, 1 :::; r :::; n, and is usually denoted by

qr:n. The probability that qr:n takes the value s is given by (see David et 81., 1977),

7fr,s:n = P[qr:n = s]00 00 min(r-1,s-1)

= n11 L Ck(r, s : n)e~(1;-1-k(13-1-k(1~-r-s+l+kh(x, y)dxdy,-00 -00 k=max(O,r+s-n-1)

where (11 = P[X < x, Y < y],(12 = P[X < x, Y > y],

(13 = P[X > x, Y < y], (14 = P[X > x, Y > y] and

(n - I)!Ck(r, s : n) = ( ) ( ) ( ) .k! r - 1 - k ! s - 1 - k ! n - r - s + 1 + k !

For more details on the theory of the ranks of concomitants of ord r statistics and

applications, see David et al. (1977). David and Galambos (1974) discuss d the

asymptotic properties of the ranks of the concomitants of order statistics. FUrther

works on the ranks of concomitants of order statistics were done by Yang (1977), He

and Nagaraja (2009) and so on.

1.6 Generalized Morgenstern Family of Bivariate Distribu-

tions

Morgenstern(1956) introduced a family 9J1 of bivariate distribution functions F(x, y)

possessing a representation of the form,

Fx,y(x, y) = Fx (x)Fy (y){l + k[l - Fx (x)][l - Fy(y)]},

9

(1.6.1)

where Fx(x) and Fy(y) are two univariate distribution functions and the association

parameter k is constrained to lie in the interval [-1,1]. When Fx(x) and Fy(y) are

absolutely continuous with corresponding pdf s fx(x) and fy(y) respectively, th

joint pdf corresponding to Fx,y(x, y) is given by

fx,Y(x, y) = fx(x)fy(y){l + k[l - 2Fx (x)][l - 2Fy(y)]}. (1.6.2)

The family 9J1 of bivariate distributions with distribu"tion function Fx,y(x, y) as given

in (1.6.1) is also called in the literature as Farlie-Gumbel-Morgenstern family of bivari­

ate distributions. For a detailed discussion on the distribution theory of concomitants

of order statistics arising from the Morgenstern family of bivariate distributions, see

Chacko (2007). Several applications of concomitants of order statistics in estimating

the parameters of Morgenstern family of distributions are also elucidated in Chacko

(2007).

In the subsequent chapters of this work, we repeatedly use a generalized cla: s ~

of bivariate distributions which can be viewed as an extension of the Morgenst rn

family of bivariate distributions and is defined as the class of all distributions with

distribution function H(x, y) having the following form:

t

H(x, y) = Fx(x)Fy(y) + L ki{Fx (x)[l - Fx(x)]}mi{Fy~y)[l - Fy(y)]}Pi , (1.6.3)i=l

where Fx(x) and Fy(y) are univariate distribution functions. Clearly Fx(x) and

Fy(y) are the marginal distribution functions of H(x, V). The constants mi and Pi

involved in (1.6.3) are reals such that mi 2: l,Pi 2: 1 and k/s are real constants

constrained to lie in intervals about zero. If both Fx(x) and Fy(y) involved in

(1.6.3) are absolutely continuous then the joint probability density function (pdf)

corresponding to H(x, y) may be denoted by h(x, y) and is given by

t

h(x, y) = fx(x)fy(y) +L kimiPi{Fx(x) [1 - Fx (x)]}m i -l[l - 2Fx (x)]fx(x)i=l

10

Clearly, the constants ki , mi, Pi; i = 1,2, ... t determine the xtent of orrelation

that exists between X and Y involved in (1.6.4). In chapter 2 we have illustrat d

that it include members of bivariate distributions possessing correlation coeffici nt

which exceeds the maximum correlation allowed for the distributions belonging to

the Morgenstern family. This illustrates that the family ~ is more useful than 9J1

in terms of its model flexibility. The distributions belonging to ~ have considerabl

analytical appeal as each H E ~ is determined by the respective univariate marginal

distributions.

1.7 Generalized Gumbel's Family of Bivariate Distributions

The joint pdf f(x, y) of the well known Gumbel's bivariate exponential distributi n

is given by

-(x + Y + Bxy)f(x, y) = [(1 + Bx)(1 + By) - B]e ,0< B < 1, x> 0, Y > 0. (1.7.1)

Clearly the marginals of f (x, y) are standard exponential densities.

In this work, we also consider a family Qj of bivariate distributions with pdf g(x, y)

given by

g(x, y) = {(I - Blog[1 - Gx(x)]) (1 + fJy) - fJ}gx(x)e-y [1 - Gx(x)ty

, (1.7.2)

°< B< 1, y > 0, x E R, where gx(x) and Gx(x) are the pdf and distribution function

respectively of a random variable X. We have

1 [1 - Gx(X)]l+OY] 00

x g(x, y)dx = e-Y (1 + By - B) -(1 + By) -00

[1 G (x)]l+OY] 00

- B(1 + By)e-Ylog[l - Gx(x)] -=-(t+ fJy) -00

+ B(1 + By)e-Y1 -gx(x) [1 - Gx(x)]l+OY dxx [1 - Gx(x)] -(1 + By)

. 1 + By - fJ 1= e-Y + B(1 + By)e-Y-,-------:-=-

1+ By (1 + By)2

11

and

1.g(x,y)dy ~ gx(x) { (1 Hy - 0) (11- C:X(X)]') y log [[1\'")1']10

1() ([1- GX(X)]O)Y 1 d }- Y e log [ll-G:(XW] Y

~Olog(1 - Gx(x)]gx(x) { (1 + Oy) (11 - C:X(X)]') Y log [[1-~:'")1'] 1~

1() ([1- GX(X)]O)Y 1 d }- Y e log [[I-G:(XW] Y

{

- (1 - ()) () [ll-G:(x)]Br 1OO}= gx(x) [()log[1 - Gx(x)] - 1] - [()log[1 - Gx(x)] - 1]2 0

1+()log[1 - Gx(x)]gx(x) [()log[1 _ Gx(x)] - 1]

2 [ll-G:(X)]Sr 100

+() log[1 - Gx(x)]gx(x) [()log[1 - Gx(x)] - 1]2 0

= gx(x).

Thus the marginal pdf's of g(x, y) are gx(x) and the standard exponential density

function

{

-ygy(y) = e ,Y >. a .

o otherWIse

If we further assume that

{

-x9 (x) = e l x> ax 0 otherwise '

then the corresponding g(x, y) E Q:) is the well known Gumbel's bivariat expon ntial

distribution. Hence we call the class Q:) of distributions as th g neraliz d Cumb l'

12

family of bivariate distributions. This family is di cussed in s veral plac in the

subsequent chapters.

1.8 Objectives of the Study

Order statistics finds immense applications m statstical inference problems. Also

certain properties of order statistics of a random sample arising from univariate dis­

tributions are used for characterizing the corresponding parent distributions. For a

bivariate data comprising of the observations (Xl, Yi), (X2 , Y2), ... , (Xn , Yn ), order­

ing of the values recorded on the X variable will give rise to concomitants of ord r

statistics with respect to the associated random variable Y. Similarly if we construct

the record values (either upper or lower) of the marginal observations Xl, X 2 ,· .. , X n

then those ordered random variables induce another set of associated random variables

called concomitants of record values. The rank of a concomitant of order statistic is

another discrete random variable of interest in statistical inference problems. Math­

ematical statistics associated with the study of bivariate random variables attains its

completion only when the theory relating to the above mentioned random variables

associated with the ordered random variables such as order statistics and record val­

ues are developed to the extent it is required. This motivates the authors to take up

the study 'On Concomitants of Some Ordered Random Variables'.

It is interesting to note that concomitants of order statistics are applied suc­

cessfully to deal with statistical inference problems associated with several r al life

situations. However no work is seen carried out till now to utilize the properti s

of concomitants of order statistics to characterize the parent bivariate distribution.

If some results characterizing bivariate distributions ar generat d, then those r ­

suIts would enormously help in the construction of a suitabl bivariate model to the

population from which the bivariate data sets are available. H nce one of th main

objectives of this work is to derive properties of concomitants of ord r statisti s whi h

13

characterize some families of bivariate distributions.

Extensive literature is available on the distribution theory and other d velopm nts

of order statistics of inid random variables. For example, see Vaughan and Ven­

ables (1972), Balakrishnan et al. (1992), Samuel and Thomas (1998), Thomas and

Sajeevkumar (2002, 2003, 2005), Sajeevkumar and Thomas (2005, 2006) and so on.

However only very recently Eryilmaz (2005) derived the expression for distribution

function of concomitants of order statistics arising from independent non-identically

distributed (inid) random variables. Not any other distribution theory and oth r

associated problems are seen attempted on concomitants of order statistics arising

from inid random variables. Hence another objective of this work is to derive the pdf,

joint pdf, moments and recurrence relations on moments of concomitants of order

statistics arising from inid random variables. Though Vaughan and Venables (1972)

have outlined the distribution theory of order statistics of inid random variables, only

very recently those order statistics have been used for inference problems. For some

applications of order statistics of inid random variables in inference problems, see

Sajeevkumar and Thomas (2005, 2006) and Thomas and Sajeevkumar (2002, 2003,

2005). But till now concomitants of order statistics of inid random variabl s ar not

seen used to deal with any inference problem. Hence another objective of this work

is to utilize the concomitants of order statistics of inid random variables to deal with

certain inference problems in statistics.

An outlier in a data set is usually viewed as an observation which appears to b

inconsistent with its remaining observations. A discordancy test for an outli I' is a

statistical test to examine whether the suspected observation is not only an extreme,

but also statistically unreasonable even when viewed as an extrem . In the available

literature, one can see that most of the discordancy tests for univariate outliers ar

based on the theory of order statistics. But if we have to check whether ther ar

14

discordant outliers in a bivariate data, the problem is complicated. We kn w that

bivariate normal distribution finds appli atioll in many real life ituations. H 'nee

there arise situations, where in one requires an ascertainment that all observations in

the data could be regarded as from the same statistical population, before switching

over to use the data for drawing inferences. Hence another objective of this study

is to find a possible way of detecting discordant outliers when the assurn d parent

model is bivariate normal.

For univariate distributions, the distributional properties of record valu shave

been extensively used for characterizing the parent distributions. For example, s e

Ahsanullah (2004), Arnold et al. (1998), Kirmani and Beg (1984), Nevzorov (1992)

and so on. But till now properties of concomitants of record values which charact riz

a bivariate distribution are not seen identified. So another objectiv of the study is

to find the possibility of characterizing bivariate distributions using th distributional

properties of concomitans of record values.

1.9 Summary of the Present Work

In chapter 1, brief reviews on the developments in the theory of order statistics, con­

comitants of order statistics, record values and concomitants of record values and

rank of concomitants of order statistics are recited. An introduction to the gener­

alized Morgenstern family of bivariate distributions, generalized Gumbel's family of

bivariate distributions, objectives of the study and summary of the present work ar

also displayed in this chapter.

In chapter 2, we have dealt with the problem of characterization of bivariat dis­

tributions using properties of concomitants of order statistics. In section 2.1, we hav

considered the generalized family ~ of bivariate distributions which includ s the Mor­

genstern family as its subclass and have derived the admissible rang of COlT lation

15

coefficient for a special case of this family. In section 2.2, we have obtained solution

of an integral equation which is helpful in proving the th orems consid red in th

subsequent sections. In section 2.3, we have derived some distributional prop rti s

of concomitants of order statistics which help to characterize the family .J of bivari­

ate distributions. We have derived in section 2.4 a characterization property of the

Morgenstern family of bivariate distributions using concomitants of order statistics.

Application of the characterization result is also discussed in this section. In section

2.5, we have established the role of concomitants of order statistics in uniquely iden­

tifying the parent bivariate distribution. A characterization property of a bivariat

Pareto distribution and that of a generalized Gumbel's bivariate exponential family

of distributions are also identified in this section. Most of the results estabished in

this chapter are available in Thomas and Veena (2011a), Veena (2008) and Veena and

Thomas (2008b).

In chapter 3 we have dealt with the theory of concomitants of order statisticB of

inid random variables. In section 3.2, the pdf of a single concomitant of order statis­

tics and the joint pdf of two concomitants of order statistics have been derived. In

sections 3.3 and 3.4, some recurrence relations on the single moments and product

moments respectively of concomitants of order statistics of inid random variables have

been derived. Most of the results established in this chapter are published in Veena

and Thomas (2008a).

In chapter 4 we have described the application of concomitants of order statistics

of inid random variables in estimation of common location and scale parameters of

several distributions. In section 4.2, we have illustrat d the application of concomi­

tants of order statistics in finding the Best Linear Unbiased Estimate (BLUE) of a

common parameter involved in several bivariate Pareto distributions. In section 4.3,

we have derived BLUE of certain paramet rs involv d in inid bivariat normal dis-

16

tributions. The results generated in our research and form the conten of chapter -!

are available in eena and Thomas (2011a 2011b).

In chapter 5 we have proposed a discordancy test for bivariate normal outlier.

This discordancy test makes use of the joint distribution of ranks of order stati tics

and the ranks of the corresponding concomitant of order tatistics. In ection 5.2

we have outlined the theory and have also illustrated tables which provide the critical

region of the test statistic and power of the test for specific alternati e hypothe e .

iost of the results established in this chapter are available in eena and Thomas

(2011c).

In chapter 6 we have dealt with the problem of characterization of bivariate dis­

tributions using properties of concomitants of record values. In section 6.2 we ha e

identified a property of concomitant of record values as a characteristic property of

the generalized forgenstern family J. In section 6.3 ",e have establi h d h p '­

sibility of characterizing bivariate distributions using concomitants of second upper

and second lower record values and illustrated the application of this result in charac­

terizing the forgenstern family of bivariate distributions. Characterizing properti s

of bivariate Pareto and generalized Gumbel s family of distributions have been ab­

lished in sections 6.4 and 6.5 respecti ely. The results generated in our research and

form the contents of chapter 6 are available in Thomas and Veena (2011b).

17