ON A CLASS OF ADMISSIBLE ESTIMATORS OF THE

COMMON MEAN OF TWO UNIVARIATE NORMAL

POPULATIONS

MOLOY DE

Tata Consultancy Services, Kolkata, India

ABSTRACT

The purpose of this article is to extend results of Sinha and Mouqadem

(Commun. Stat. Theo. Meth. II, 1982, 1603-1614), and present a class of

admissible estimators of the common mean of two univariate normal popu-

lations with unknown unequal variances.

Keywords and Phrases: Common Mean Estimation Problem, Bayes Esti-

mates, Admissible Estimates.

1. INTRODUCTION

The problem of estimation of common mean of two univariate normal

populations has attracted attention of researchers all over the world. The

topic has been thoroughly studied and documented in the literature (see

bibliography at the end).

Sinha and Mouqadem (1982) [abbreviated subsequently as SM (1982)]

confined their attention to certain subclasses of unbiased estimators of the

common mean and succeeded in producing admissible estimators. Our pur-

pose is to extend their results and provide a class of unbiased estimators.

Let

Πi = N(µ, σ2i ), i = 1, 2 (1.1)

be two univariate independent normal populations with unknown unequal

variances. Let n1 and n2 be the sizes of two random samples from Π1 and

Π2 respectively. The following notations are standard:

X =1

n1

n1∑i=1

Xi

Y =1

n2

n2∑i=1

Yi

s21 =

1

(n1 − 1)

n1∑i=1

(Xi − X2)

s22 =

1

(n2 − 1)

n2∑i=1

(Yi − Y 2)

D = Y − X (1.2)

Clearly (X, Y , s21, s

22) is sufficient for µ but it is not complete because

E(D) = 0.

2

Consider the following classes of unbiased estimators of µ:

C = {µ : µ = X + Dφ, 0 ≤ φ(s21 , s22 ,D

2 ) ≤ 1}

C0 = {µ : µ = X + Dφ, 0 ≤ φ(s22s21

) ≤ 1}

C1 = {µ : µ = X + Dφ, 0 ≤ φ(s21 , s22 ) ≤ 1}

C2 = {µ : µ = X + Dφ, 0 ≤ φ(s21D2

,s22D2

) ≤ 1} (1.3)

The restriction to 0 ≤ φ ≤ 1 is obvious as otherwise the estimators are

trivially inadmissible , see SM(1982).

Let us consider the estimator µ0 =Xs21/n1+Y s22/n2

s21/n1+s22/n2∈ C0 , as µ0 = X+Dφ0

where

φ0 =1

1 +n1s22n2s21

=1

1 + n1(n2−1)n2(n1−1)F

(1.4)

where F =s22/(n2−1)

s21/(n1−1). There are quite a few papers dealing with the proper-

ties of µ0 or its variations in relation to other estimators (say, µ0 beats the

individual sample means, see Cohen (1976)). However no exact optimum

property of µ0 regarding its admissibility or minimaxity is known.

2. PROPERTIES OF µ0

We begin with the following result.

Lemma 2.1. Let µ = X +Dφ ∈ C . Then

V ar(µ) =σ2

1σ22

n2σ21 + n1σ2

2

+ E{D2(φ− σ21

σ21 + n1

n2σ2

2

)2}. (2.1)

Proof.V ar(µ) =σ21n1

+2E[(X−µ)Dφ]+E[D2φ2]. Noting that X−µ and D

are jointly normally distributed, we get that E[X − µ|D] = − σ21/n1

σ21/n1+σ2

2/n2D.

Then the result follows trivially. QED.

3

We now proceed to prove the admissibility of µ0 in C0 . Using (2.1),

for µ0 ∈ C0 , we get,

V ar(µ) =ρσ2

n2(1 + n1n2ρ)

+σ2

n2(1 +

n1

n2ρ)E{φ(F )− 1

1 + n1n2ρ}2, (2.2)

where, σ2 = σ21, ρ =

σ22

σ21.

The following two lemmas are needed in sequel. The first is a version of

the well known theorem that a proper Bayes Estimator for prior distributions

assigning positive probability to any interval and risk continuous in θ is

admissible.

Lemma 2.2. Suppose there exists H0 on (0,∞) which is absolutely

continuous with respect to the Lebesgue Measure and satisfies H0{(a, b)} > 0

if 0 < a < b <∞ and∫ ∞0

(1 +n1

n2ρ)E{φ(F )− 1

1 + n1n2ρ}2H0(dρ) (2.3)

is finite and minimum at φ0. Then µ0 = X +Dφ0 is admissible in C0 .

Proof. Observing that E[φ(F )− 11+

n1n2ρ]2 is continuous in ρ the proof is

trivial using the method of contradiction. QED.

4

Define,

Ii(x) =

∫ ∞0

ρifρ(x)H0(dρ), i = 0, 1, (2.4)

where fρ(x) = (ρν2ν1 )12ν2 Γ( 1

2(ν1+ν2))

Γ( 12ν1)Γ( 1

2ν2)

x12 ν2−1

(1+ν2x

ν1ρ)12 (ν1+ν2)

, x > 0. Note fρ(x) is the

density of F with degrees of freedom ν1, ν2.

Lemma 2.3. Let H0 satisfy Ii(x) <∞, i = 0, 1 for x > 0 and∫ ∞0

I0(x)I1(x)

I0(x) + n1n2I1(x)

dx <∞,∫ ∞

0

ρ

1 + n1n2ρH0(dρ) <∞.

Then φ0(x) = [1 + n1n2

I1(x)I0(x) ]−1 minimizes (2.3).

Proof. Using Fubini’s Theorem (2.3) can be written as∫ ∞0

[I0(x) +n1

n2I1(x)][φ(x)− 1

1 + n1n2

I1(x)I0(x)

]2dx

+n1

n2

∫ ∞0

I0(x)I1(x)

I0(x) + n1n2I1(x)

dx

−n1

n2

∫ ∞0

ρ

1 + n1n2ρH0(dρ).

QED.

Theorem 2.1. For n1 + n2 > 4, µ0 is admissible in C0 .

Proof. By (1.4) µ0 corresponds to φ0(x) = 1

1+n1n2

(n2−1)x

n1−1

. So for µ0 to be

admissible by lemma 2.3. we need to have I1(x)I0(x) = n2−1

n1−1x. Putting H0(dρ) =

ρp−1, ρ > 0, p ∈ R, we get,

Ii(x) =Γ(1

2ν1 + i+ p)Γ(12ν2 − i− p)

Γ(12ν1)Γ(1

2ν2)(ν2

ν1)i+pxi+p−1 (2.5)

where νi = ni − 1, i = 1, 2. Now I1(x)I0(x) = n1+2p−1

n2−2p−3n2−1n1−1x. Hence p = 1

4(n2 −n1 − 2). For I0(x), I1(x) to be defined we need −1

2(n1 − 1) < p < 12(n2 − 3),

which implies n1 + n2 > 4. QED.

5

Theorem 2.2. µ0 is extended admissible in C .

Proof. For µ ∈ C ,V ar(µ) =σ21σ

22

n2σ21+n1σ2

2+ E[D2(φ − 1

1+n1n2ρ)2]. Simple

calculations show,

E[D2(φ0 −1

1 + n1n2ρ

)2] ≤ (n1 − 1

n2 − 1)2 σ2

1

n1(1 + n1n2ρ)E(

1− n2−1n1−1F

∗

F ∗)2 (2.6)

where F ∗ = 1ρF ∼ χ2

n2−1,n1−1 and the expectation exists. Right hand side

of (2.6) can be made as small as possible making σ1 small and keeping ρ

constant. QED.

3. ADMISSIBLE ESTIMATORS IN C1

Theorem 3.1. Any estimator of the form µ = X + Dφ where φ =(s21+λ1)/n1

(s21+λ1)/n1+(s22+λ2)/n2, λ1 > 0, λ2 > 0 is admissible in C1 .

Proof. Using (2.1) for µ ∈ C1 ,

V ar(µ) =σ2

1σ22

n2σ21 + n1σ2

2

+1

n2(σ2

1 +n1

n2σ2

2)E(φ− σ21/n1

σ21/n1 + σ2

2/n2)2 (3.1)

Following arguments of SM(1982)[(3.1)], given a probability measure H for

(σ21, σ

22), µH = X +DφH is admissible in C1 where

φH(s21, s

22) =

E(σ21n1|s2

1, s22)

E(σ21n1

+σ22n2|s2

1, s22). (3.2)

Taking H as product of Inverted Gamma priors given by

H(dσ21, dσ

22) ∝

2∏i=1

e− λi

2σ2i (

1

σ2i

)p−1d(1

σ2i

), p > 0

we get the result. QED.

6

4. ADMISSIBLE ESTIMATORS IN C2

Lemma 4.1. (Zacks 1970) Let H be a prior assigning positive prob-

ability to any interval. Then µH = X + DφH is admissible in C2 where

φH(s21D2 ,

s22D2 ) = φ∗H(U, V ) =

∫∞0 ρ−

12

(n2−1)(1 + n1n2ρ)

12

(n1+n2)−1[1 + (1 + n1n2ρ)U + (1

ρ)(1 + n1n2ρ)V ]−

12

(n1+n2+1)H(dρ)∫∞0 ρ−

12

(n2−1)(1 + n1n2ρ)

12

(n1+n2)[1 + (1 + n1n2ρ)U + (1

ρ)(1 + n1n2ρ)V ]−

12

(n1+n2+1)H(dρ)

(4.1)

where U =s21

n1D2 , V =s22

n2D2 .

Proof. For µ ∈ C2 , µ = X + Dφ(V1, V2), where Vi =s2iD2 , i = 1, 2.

Following the arguments of lemma 2.1 and using E[D2|V1, V2] = σ2(1 −n2−1n1

)(1 + n1n2ρ)[1 + 1

n1(1 + n1

n2ρ)V1 + 1

n2ρ(1 + n1

n2ρ)V2]−1 we get V ar(µ) =

σ2

n1+σ2(1+

n2 − 1

n1)(1+

n1

n2ρ)E[{φ2−2(1+

n1

n2ρ)−1φ}{1+

1

n1(1+

n1

n2ρ)V1+

1

n1ρ(1+

n1

n2ρ)V2}−1]

where σ2 = σ21, ρ =

σ22

σ21. Given the prior H one calculates the prior risk

function. Applying Fubini’s Theorem, µ is admissible if the corresponding

φ minimizes∫ ∞0

H(dρ|V1, V2)[φ− (1 +n1

n2ρ)−1]2[(1 +

n1

n2ρ)−1 +

1

n1V1 +

1

n1ρV2]−1

where H(ρ|V1, V2) designates the posterior distribution of ρ given V1, V2, see

Zacks(1970). The minimizer is given by

φ(V1, V2) =Eρ|V1,V2 [1 + 1

n1(1 + n1

n2ρ)V1 + 1

n1ρ(1 + n1

n2ρ)V2]−1

Eρ|V1,V2(1 + n1n2ρ)[1 + 1

n1(1 + n1

n2ρ)V1 + 1

n1ρ(1 + n1

n2ρ)V2]−1

(4.2)

where Eρ|V1,V2 designates the posterior expectation given (V1, V2).

7

Calculating the joint density of (U,V), we note that H(dρ|V1, V2) ∝H(dρ|U, V ) ∝

ρ−12

(n2−1)(1+n1

n2ρ)

12

(n1+n2−2)[1+(1+n1

n2ρ)U+

1

ρ(1+

n1

n2ρ)V ]−

12

(n1+n2−1)H(dρ)

(4.3)

Using (4.3) in (4.2) we get the result. QED.

We now obtain the final form of the class of admissible estimators in C2

by the following steps:

I =

∫ ∞0

dx

(ax2 + 2bx+ c)32

=1√

c(√ac+ b)

(4.4)

where a ≥ 0, c > 0,√ac+ b > 0. Vide, Gradstein and Ryshik (1980).

Iq =

∫ ∞0

xn+qdx

(ax2 + 2bx+ c)n+ 32

=(−1)n2q

(2n+ 1)!!

∂n

∂aq∂bn−q(I) (4.5)

where (2n+ 1)!! = 1.3.5 · · · (2n+ 1), q = 0, 1, 2, · · · , n+ 1. Expression (4.6)

is obtained by taking derivatives of (4.5).

∂q

∂bq(I) = (−1)p!

1√c(√ac+ b)q+1

(4.6)

q = 0, 1, 2, · · · .I0 =

n!

(2n+ 1)!!

1√c(√ac+ b)n+1

(4.7)

I1 =n!

(2n+ 1)!!

1√a(√ac+ b)n+1

(4.8)

I2 =n!

(2n+ 1)!!

√ac+ (

√ac+ b)n−1

a32 (√ac+ b)n+1

(4.9)

8

Theorem 4.1. µ = X+Dφλ, λ ≥ 0, is an admissible class of estimators

in C2 where

φλ = [1 +n1

n2

I1 + λI2

I0 + λI1]−1 (4.10)

where n = n1+n22 − 1, a = n1

n2U, 2b = 1 + U + n1

n2V, c = V, U =

s21n1D2 , and

V =s22

n2D2 .

Proof. Let us consider the following family priors:

(1 + λρ)ρn22−2(1 +

n1

n2ρ)−

n1+n22

+1, λ ≥ 0 (4.11)

It is a family of proper priors as the Fν1,ν2 distribution has all the r moments,

r = 0, 1, · · · , [ν22 ], ν2 > 4. Vide for example, Goon et al (1988). Using this

family of priors in (4.1) we get the result. QED.

ACKNOWLEDGEMENT

I thank Professor Bikas K. Sinha of Indian Statistical Institute, Kolkata,

for suggesting the problem and for his encouragement and help.

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9

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