a sequential procedure for estimating ratio of normal parameters
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A sequential procedure forestimating ratio of normalparametersC. Uno & E. IsogaiPublished online: 16 Feb 2011.
To cite this article: C. Uno & E. Isogai (1999) A sequential procedure forestimating ratio of normal parameters, Communications in Statistics - Theory andMethods, 28:1, 233-244
To link to this article: http://dx.doi.org/10.1080/03610929808832293
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COMMUN. STATIST .- THEO RY M ETH. , 28( 1), 233-244 (\ 999)
A SEQUENTIAL PROCEDURE FOR ESTIMATING RATIOOF NORMAL PARAMETERS
Chikara Uno
Department of MathematicsAkit a University
Akita 010-8502, Jap an
Eiichi Isogai
Department of MathematicsNiigata University
Niigata 950-2181, Japan
K ey Words: sequential est ima to r; second order approxima t ion; regret ; uniformintegrab ility.
ABSTRACT
Sequ ential point est imat ion of the ra tio J1. /a of normal parameters is conside red . We propose a class of sequential esti mators which includes estimatorswith th e smaller risk th an that of Sriram 's (1990) one.
1. INTRODUCTION
Let Xl , X 2 , • • • be independent observations from a normal population withmean J1. and variance a 2 (0 < a < 00), both unknown. Given a randomsample Xl, . . . , Xn of size n (~ 2) , one wishes to estimate ratio 0 = J1./a by
On = Xn/Sn, subject to the loss function
Ln = A(On - 0)2+ n ,
233
Copyr ight to 1999 by Marcel Dekker . Inc . www .de kker.com
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234 UNO AND ISOGAI
where X n = n - I L::?=I Xi , S~ = (n _ 1)- 1L::~ I(Xi - X n )2 and A is a knownpositive constant . The risk is given by
We want to find an appropriate sample size that will min imize the risk. This
problem was first considered by Sri ram (1990).
As shown in Sriram (1990), we have
Ro = E (L n ) = n- I A(1 + ~(P ) + n + O(n- ~) as n ~ 00.
If we ignore th e above ord er term O(n- ~ ) , t hen it can be seen th at th e riskRo is minimized at no ;:::: At -y (in pract ice, one of the two integers closestto this value) with Roo ;:::: 2At-y , where -y2 = (1 + !82) and -y > O. Since8 is unknown, however , t he bes t fixed sa mple size procedure no can not beused. Further, th ere is no fixed sample size procedure th at will attain therisk 2A t -y. Thus it is necessary to find a sequ ential sa mpling rule. For this
problem, Sri ram (1990 ) proposed the stopping rule
T = TA = inf{n ~ m : n ~ At(1 + ~B~)t} , (1.1 )
where m ~ 2 is th e st arting sample size. Estimating 8 by BT = XT/ST, the
risk of this sequ enti al pro cedure is given by
RT = E{A(BT - 8)2+ T} .
It is interest ing to know how close RT is to 2At -y. As a measure of closenesswe shall use th e difference RT - 2At -y called regret. Sriram (1990) gave anasymptotic expansion for the regret : as A ~ 00,
R - t - 1186+4484+7682+64
() (1.2)T 2A -y - 8(82 + 2)3 + 0 1 .
The result of (1.2) is slightly different from that of Theorem 2 in Sriram (1990) .We think that Sriram's (1990) result has a minor error.
In this paper, we consider a class of sequential estimators of 8 and showthat our procedure has the smaller risk than that of Sriram's (1990) procedureBT • In Section 2 we shall present main results, including the second orderapproximation to the risk of the proposed procedure. Proofs of the results ar e
given in the final section. In Section 3 we shall present simulation results. Asfor sequential interval estimation of ratio 8 of normal parameters, see Takada
(1997) .
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ESTIMATING RATIO OF NORMAL PARAMETERS
2. MAIN RESULTS
We consider a class of estima tors {OHk), -00 < k < oo} of (J , where
A (k) A(J;'(k) = 1 + T (JT ,
and define the risk by
RT(k) = E{A(O;'(k) - (J) 2+ T} .
235
(2.1)
Then OT coincides with OHO) and hence RT == RT(O) . For given k, est imat ing(J by OHk), we obtain th e theorems conce rn ing the asymptotic expansions of
the bias and risk of OHk).
Theorem 1. As A --> 00 ,
E{O;'(k)} - (J = (k + i)(J'Y-1A-t + o(A- t ).
Theorem 2. As A --> 00 ,
Let 6. k = 'Y-2{ (J2k2+ 2k(1 + i(J2)}. For -~ < k < 0, 6.k < 0 for all (J .Hence from Theorem 2, the class of estimators {OHk), -~ < k < O} improves
Sriram's (1990) pro cedure OT in the risk . Specially for k = -i,from Theorem1, we get
E{O;'(-i)} = (J + o(A-t) as A --> 00,
that is, OH- i) = (1 - 4~ )OT is a second order asymptotically unbiased estimator of (J.
lf Jl = 0, then from Theorem 2 and (1.2), the second order approximationto the regret is the following:
RT(k) - 2At'Y RT(k) - RT + (RT - 2At'Y)
= 2k + 1 + 0(1), (2.2)
which surprisingly brings about large negative regret for sufficiently large neg
ative k , This phenomenon of large negative regret is observed in terms of
simulation in Section 3. Since Jl is unknown , however, we should not use the
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236 UNO AND ISOGAI
estimator OT(k) for large negative value of k, because it follows from Theorem2 that in the case of Jl i= 0, th e risk of th e proposed estimator OT(k) withk < -2(t + ~) is not small as compared with that of Sriram 's (1990) procedure. Therefore, for estimating unknown 0, we recommend our sequent ialestimator OT( - t) = (1 - 4~ )Or which has smaller risk than th at of Sriram 's(1990) procedure for any 0:
• ( 1) 02+ 8 ( )RT -4 - Rr = - 8(02 + 2) + 0 1 as A -> 00 .
3. SIMULATION RESULTS
In this sect ion , we shall present simulation results. We consid er here twocases N(O,I) with () = 0 and N(I ,I ) with 0 = 1. For each case, we takea = 50 and a = 100 (no ~ a, Rno ~ 2a) where a = A~ "y. Set th e pilotsample size m = A ~ . Our simulation resul ts are given by Tables I and H,in which each ent ry is based on 10,000 repetitions. It seems from the MonteCarlo simulation that our bias-corrected sequential procedure OT(-t) definedby (2.1) with k = -t is better than Sriram's (1990) pro cedure Or in terms ofthe estimate of () and the risk.
As we have seen in Section 2, for esti mating () = 0, a sequential pro cedureOT(k) defined by (2.1) with large negative k can reduce the risk . Table ill is asimulation result for estimating 0 = 0 by OT(k) with k = -15 when a = 150(A = 22500). The starting sample size is still m = A~ and each entry in Tableill is also based on 10,000 repetitions. Our simulation result given by Table illseems to justify the equation (2.2). Since 0 is unknown, however, OT(k) withlarge negative value of k is impractical.
4. PROOFS
In this section, we shall give th e proofs of Theorems 1 and 2. We will usethe notations in Sriram (1990). Let
Zi = (Xi - Jl)/u, D; = I:?=lZi , Zn = n -1Dn , a~ = S~/U2 ,
82 = Jl2+2u2, (}1=Jl2/(282), (}2=Jlu/82 and 03 = (Jl2+ u2)/(282).
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ESTIMATING RATIO OF NORMAL PARAMETERS
TABLE
237
N (O,l)() =O
E(T)E(8T)E(8T(- t ))
RTRj.(-t )
a = 50
( A = 2500)
51.055 1
- 0.002088
- 0.002077
101. 757152
101.264 427
a = 100
(A = 10000 )
101.0524
0.000817
0.000 815
201.4 85893
200.990879
N( l, l)
() = 1
E(T)E(8T)E(8T(-t ))RTRj.(-l)
TABLE n
a = 50
(A = 2500/1.5)
50.8104
1.003265
0.998361
101.441268
101.095399
TABLE ill
a = 100
(A = 10000/1.5)
100.8550
1.002696
1.000219
201.0 80460
200 .701734
a = 150
(A = 22500 )
N(O,l)
() =o
E(T)E(8T)E(8T(-15.0))
RTRj.(-15.0)
151.0500
-0.000340
-0.000306
298 .262064
270.520534
Further , let K n = Li':l(Zl - 1) and a = At )'. From (1.1 ), we get t hatp eT < 00) = 1 for each A > 0 and T ---; 00 a.s. as A ---; 00 . The sto pping
rule T in (1.1) can be rewritten as
T=inf{n?:m: n+(}lKn-(}2Dn+(n?:a} ,
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238
where
UNO AND ISOGAI
and AI ,n and A2,n are intermediate random variables such that IAI ,n - 11 <Io-~ - 11 and IA2,n - 11 < 1(2S~ + X~)D-2 - 11- Sri ram (1990) showed thefollowing lemma.
Lemma. (i) T']a --> 1 a .s. as A --> oo. Further, {(1'/a) -P, A > o} and{(1'/ a)P, A > o} ar e uniformly integrable for all p > o.(ii) 1'-~~T --t 0 in probability as A --> oo.
(iii) For every q > 1,
E {SUP(o-~) -q} < (~rE(o-~)-q < oo if m > 2q + 1.n:2:m
(iv) As A --> oo,- d
UA == T + (}lKT - (}2DT + ~T - a --t H
for some random variable H . Here ,~ , means convergence in distribution.
(v) As A --> (Xl,
1'-~(DT' KT) .s: ((1>(2) ,
where ((I, (2) is a bi-variate normal random vector with mean vector (0,0)and variance-covariance matrix (~ ~) .
(vi) For all T > 0, {la-~DTlr, A > u} and {la-~KTlr, A > O} are uniformlyintegrable.
(vii) For all s > 0, {1(1' - a)/VCi!S, A > o} is uniformly integrable.
Since T ~ A! from (1.1), we may choose the starting sample size m ~ ALSince the results of this paper are valid only for large A, given any q > 1 wecan choose A large so that m > 2q + 1. We are now in the position to provethe results in Section 2.
Proof of Theorem 1. Let k be any fixed constant. Then OHk) - ()OT - () + ~OT ' Since OT - () = ZTo-T I + (}(o-T I - 1),
E(OT)-(}=E(ZTo-TI)+(}E(o-TI -1)= I + II , say . (4.1)
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ESTIMATING RATIO OF NORMAL PARAMETERS 239
From t he results (3.25) and (3.26) of Sriram (1990) , it follows th at E{Zr(aTI
I)} = o(a- I ) . Hence by Wald 's lemm a, as A -+ 00,
I E{Zr (aTI - I)} + EZr
a-IE{ (f -1) Dr} +o(a- I). (4.2)
From Lemma (ii) , (iv) and (v) , we have
(f -1) Dr = - [T- 4{UA + B2Dr - BIKr - (r}T - 4Dr ]
and as A -+ 00 ,
[T- 4{UA + B2Dr - BIKr - ~r}T- 4Drl ~ (B2( 1 - BI(2)(1'
Since, by Schwarz's inequality, for Q' > 1,
E !(f - 1)DrlQ~ {E (a/T )2Q}t{EI(T - !l)/vaI4Q}t{E la- tDr I4Q} t ,
it follows from Lemma (i), (vi) and (vii) that {I(f - 1) Drl' A > o] is uniforml y integrable , which yields
-E{(B2( 1 - BI(2)(d + 0(1)
-B2 + 0(1 ). (4.3)
To obtain such a un iform integrability, in this paper, Schwar z's inequalityargumen ts similar to tha t above will be used several times in th e proofs ofTheorems 1 and 2, and omitted. From (4.2) and (4.3) , we get, as A -+ 00,
(4.4)
By Taylor 's theorem ,5
aTI -1 = - t(a} -1) + ~f3;:2(a~ -I?, (4.5)
where f3r is a random variable such that 1f3r - 11 < lu} - 11, for which , it
follows from (A.15) of Sriram (1990) that for all q > 0,
{1f3rl- Q, A> o} is uniformly integrable. (4.6)
From (4.5),
II = Ba- IE{ - ~ a (u} -1) + ~af3;:! (u} _1 )2}. (4.7)
From the equation
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240 UNO AND ISOGAI
-2 T -1K 1 ;-Z2 1 D2aT - 1 = T + T(T _ 1) (;;: i - T(T _ 1) T,
Lemma and Wald's lemma, we have, as A ...... 00,
(4.8)
{T }1 - I 1 2 1 2
- "2 aE T KT + T(T _ 1) ~ Zi - T(T _ 1) DT
- t aE {T - 1Kr} + 0(1)
(4.9)
It follows from Lemma th at as A ...... 00 ,
(1 - -f) KT = T -! {VA + 02DT - (JIKT - ~r}T-!KT
.s: ((J2(1 - (JI( 2)(2 .
Further , we can show by Lemma that {1(1 - ~)KTI, A > o} is uniformlyintegrable, which yields
E{((J2(1 - (Jl(2)(2} + 0(1 )
- 201 + 0(1). (4.10)
From (4.9) and (4.10),
E{ -ta(iT} - I)} = -(JI + 0(1).
According to (4.8) and Lemma, we get that as A ...... 00 ,
T!(iT} - 1) .s; ( 2,
and
(4.11)
(4.12)
{IT!(iT} - 1)1" , A > O} is uniformly integrable for all s > O. (4.13)
From (4.6) , (4.12), (4.13) and Lemma (i), we have
EHa.B;:~(iT} -I?} = ~E[(a/T).B;:~{T!(iT} - 1)}2]
~E((~) + 0(1)
~ + 0(1),
which , together with (4.7) and (4.11), yields
II = (Ja -I{
- (JI + ~ + 0(1)} . (4.14)
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ESTIMATING RATIO OF NORMAL PARAMETERS
Hence, combining (4.1) , (4.4) and (4.14) , we have
E(BT ) - () a- l{ -(}2 - (}(}l + ~(}} + o(a- l)
~(}a -l + o(a- l) ,
from which , it follows that
241
E{Br(k)} - () = E(BT) - () + kE(T-1BT)
= ~(}a -l + ka- l E (fBT ) + o(a- l). (4.15)
As shown in Sriram (1990), BT --+ () a .s. as A --+ 00, so from Lemma (i) ,(a/T)BT --+ () a.s. as A --+ 00 . By Schwarz's inequality and Doob's maximalinequality, for Q' > 1,
EI(a/T)BTIOI
s :0 {E(frOr {EIXT I4o}t{Elu}I-2o}t
I I
~ :0 {E(f)20}' (4~~lr{EIXd40}t [E{~~~(U})-20}]~(4.16)
from which , together with Lemma (i) and (iii) , {I(a/T)BTI, A > o} is uniformly integrable. Hence, we get, as A --+ 00,
E (fBT) = ()+ 0(1),
which , together with (4.15) , concludes the theorem. 0
Proof of Theorem 2. Let k be any fixed constant. Since a = At'Y,
RT(k) - RT
{• k : } { k : }2= 2AE ((}T - (})T(}T + AE T(}T
=2k'Y-2a2E{T-I(BT - (}?} + 2k(}'Y-2a2E{T-I(BT - ())}
+k2'Y-2 E (;: O} )
I + II + III, say. (4.17)
By virtue of the equation OT - () = ZTUr l + (}(ur l- 1), we get
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242 UNO AND [SOGA[
a2E{T- 1(OT - B)2}
= a2E[T-IfZ~ur2 + B2(ur l - 1)2+ 2BZTUrl(Url - In] . (4.18)
It follows from Lemma that
E {(a2/T2)T- 1D}ur2}
E«(i) + 0(1)
1 + 0(1). (4.19)
In terms of (4.5), (4.6) , (4.12), (4.13) and Lemma (i) , we can show that
a282E{T-l(url _ 1)2}•
= 82E[(a2/T2)TH(u} - 1)2 - ~f3~2(U} - 1)3 + -bf3r5(u} _ 1)4}]
= t82E«(D + 0(1)
= ~82 + 0(1) (4.20)
and
a2E[T-l{ZTUrl(url - I)}]
= E[(a2/T2)T-! DTurIT! {-Hu} - 1) + ~ f3~~ (u} - 1)2}]
= -~E«(1(2) + 0(1)
= 0(1). (4.21)
Thus, (4.18)-(4.21) yield
1= 2k-y-2(1 + ~82) + 0(1). (4.22)
Next, as (4.18), we have
a2E[T-1{ZTUr
l + 8(url - In]
a 2 E{T-1ZT(UT1 - In + a 2E(T-1ZT)
+8a2E{T-1(UT1 - In. (4.23)
By an argument similar to (4.21), we obtain
a2E{T-1ZT(Ur l - In = - ~E{(a2 /T2)T-!DTT !(u} - Ins
+~E{(a2 /T2)T- 1DTf3~2T(u} - 1)2}
- ~E«(1 (2 ) + 0(1)
0(1). (4.24)
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EST IM AT ING RATIO O F NORMAL PARAMETERS
It follows from Wald 's lemma and Lemm a that
243
E { (;: - 1) DT }
-E [(j; + 1) T- ~ {VA+ B2DT - B1KT - ~r}T- ~ DT]
-2E {(B2(1 - Bt(2)(d + 0(1)
- 2B2+ 0(1). (4.25)
As (4.20) , we get
Ba2E{T- 1(ai l- I )}
= -~Ba2E {T- l (a} - I )} + ~Ba2E{T-l f3; ~(a} - 1)2}
=-~Ba2E{T- l (a} - I)} + ~ BE((i) + 0(1)
= -~BE{a2T-l (a} - I)} + ~ B + 0(1). (4.26)
According to Lemma, (4.8) and Wald 's lemma, we have
E {a2T - 1(a} - I )}
[a2 { 1 T I}]= E T2 J(T + T _ 1 t; z;- T _ 1D}
=E[(;: - 1) J(T] +1 -E(a)+o(1 )
= - E [(j; + 1) T-~ {VA+ B2DT - B1J(T - ~T }T- ~ J(T] + 0(1)
= -2E{(B2(1 - B1(2)(2} + 0(1)
=4B1 + 0(1),
which, together with (4.26), yields
Ba2E{T-1(ai 1 - I)} = - 2BB1+ ~B + 0(1). (4.27)
Hence, (4.23) -(4.25) and (4.27) give
II 2kB-y-2(-2B2 - 2BB1+ ~ B) + 0(1)
2kB-y-2(- ~B) + 0(1). (4.28)
Fin ally, by an argument similar to (4.16), we get
III = k2-y- 2E (;:O}) = e -y- 2()2 + 0(1). (4.29)
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244 UNO AND ISOGAI
Thus, combining (4.17), (4.22), (4.28) and (4.29) , we obtain
RT(k ) - RT
as desired . 0
2k,-2 (1 + ~ (2) + 2kO,- 2(- ~O) + e02, - 2+ 0( 1)
, -2{02e + 2k (1 + ~ (2) } + 0( 1),
ACKNOWLEDGEMENTS
The aut hors ar e grateful to the referees for th eir comments .
BIBLIOGRAPHY
Sriram , T . N. (1990) . "Sequent ial est imat ion of ratio of norm al paramet ers ,"
J. Statist . Plann. Inference, 26 , 305-324 .
Takada , Y. (1997). "Fixed-width confidence intervals for a function of normalpar ameters," Sequent ial Anal. , 16, 107-117.
Received April, 1998, Revised August; 1998.
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