estimating the variance of a normal population by utilizing the information in a sample from a...

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Estimating the variance of a normalpopulation by utilizing the informationin a sample from a second relatednormal populationMohammad Fraiwan Al-Saleh a & Hani M. Samawi ba Department of Statistics , Yarmouk University , Jordanb Department of Mathematics and Statistics , Sultan QaboosUniversity , P.O. Box 36, Al-Khod 123, Sultanate of Oman E-mail:Published online: 13 May 2010.

To cite this article: Mohammad Fraiwan Al-Saleh & Hani M. Samawi (2004) Estimating thevariance of a normal population by utilizing the information in a sample from a second relatednormal population , Journal of Statistical Computation and Simulation, 74:2, 79-90, DOI:10.1080/0094965031000104323

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Journal of Statistical Computation & Simulation

Vol. 74, No. 2, February 2004, pp. 79–90

ESTIMATING THE VARIANCE OF A NORMAL

POPULATION BY UTILIZING THE INFORMATIONIN A SAMPLE FROM A SECOND RELATED

NORMAL POPULATION

MOHAMMAD FRAIWAN AL-SALEHa,* and HANI M. SAMAWIb,y

aDepartment of Statistics, Yarmouk University, Jordan;bDepartment of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod 123,

Sultanate of Oman

(Received 24 August 2001; In final form 20 February 2003)

A class of estimators of the variance s21 of a normal population is introduced, by utilization the information in a

sample from a second normal population with different mean and variance s22, under the restriction that s2

1 � s22.

Simulation results indicate that some members of this class are more efficient than the usual minimum varianceunbiased estimator (MVUE) of s2

1, Stein estimator and Mehta and Gurland estimator. The case of known andunknown means are considered.

Keywords: Normal population; Prior; Mean squared error; Posterior; Bayesian estimation; Generalized Bayes; Steinestimator; Mehta and Gurland estimator

1 INTRODUCTION

Estimation of the variance of a normal population is a very well known problem. Extensive

research has been done on this problem. One very popular estimator is the sample variance,P(Xi � �XX )2=(n� 1), which is also the minimum variance unbiased estimator (MVUE).

However, this estimator is inadmissible and dominated byP

(Xi � �XX )2=(nþ 1), which is

the estimator that has lowest Mean squared error (MSE) among all estimators that are linear

inP

(Xi � �XX )2. Stein (1964) showed that the last estimator is also inadmissible and is domi-

nated by the estimator min(P

(Xi � �XX )2=(nþ 1),P

X 2i =(nþ 2)). However, these dominat-

ing estimators show little improvement over the usual MVUE.

Utilizing available information to estimate s21 has been considered by many authors. Pandy

(1979) obtained a double stage shrinkage estimator of s21 using an initial guess value of s2

1.

Al-Saleh (1993) obtained a similar estimator with a shrinkage factor that is a function of the

test statistic used to test the prior information. Mehta and Gurland (1969) considered the pro-

blem of estimating s21 by utilizing information from a second sample. To be more specific, let

* Corresponding author. E-mail: [email protected] E-mail: [email protected]

ISSN 0094-9655 print; ISSN 1563-5163 online # 2004 Taylor & Francis LtdDOI: 10.1080=0094965031000104323

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X1, X2, . . . , Xn be a random sample from a normal distribution with mean m1 and variance s21,

(N (m1, s21)). Let Y1, Y2, . . . , Yn be another independent random sample from N (m2, s2

2).

Mehta and Gurland (1969) considered the class of estimators of the form:

S ¼ S�21

a1 � a2Rþ a3R2

b1 � b2Rþ b3R2

, (1)

where

R ¼S�2

2

S�21

; S�21 ¼

P(Xi � �XX )2

n� 1; S�2

2 ¼

P(Yi � �YY )2

n� 1

a1 ¼ (n� 1)(1 � 2c) þ (nþ 1)c2; b1 ¼ (nþ 1)(1 þ c2) � 2(n� 1)c

a2 ¼ 2{(n� 1)d � (nþ 1)cd � 1}; b2 ¼ 2d{(n� 1) � (nþ 1)c}

a3 ¼ (nþ 1)d2; b3 ¼ (nþ 1)d2

and c and d are arbitrary constants. No condition was imposed on r ¼ s21=s

22. One disadvan-

tage of this estimator is that the choice of c and d depends on the sample size. For some other

related topics, readers may refer to Eden and Zidek (2001); Kubokawa and Saleh (1994) and

Kushary and Cohen (1989).

In this paper, we assume that prior information suggests that s21 � s2

2 and derive a class of

estimators for s21 of the form

S�21b (k1, k2) ¼

(n1 � 1)S�21

n1 þ 2k1 � 4

�F(((n1 þ 2k1 � 4)=(n2 þ 2K2 � 2))(S�2

2 =S�21 ), n2 þ 2k2 � 2, n1 þ 2k1 � 4)

F(((n1 þ 2k1 � 2)=(n2 þ 2K2 � 2))(S�22 =S�2

1 ), n2 þ 2k2 � 2, n1 þ 2k1 � 2),

(2)

where F( � ; a, b) is the cumulative distribution function of an F random variable with a, bdegrees of freedom, n1(n2) is the sample size of the X (Y )-sample. The simulation results sug-

gest that S�21b (k1, k2) is substantially more efficient than S�2

1 , Stein and Mehta and Gurland

(M&G) estimators for some choices of k1 and k2.

The situation above can arise in different areas of applications. The following are some of

these applications:

1. The precision of a machine is usually measured by the reciprocal of the standard deviation

of a set of repeated measurements taken by the machine for the same subject. For example

the precision of the weighing machine can be estimated by using the machine to weigh a

subject many times. If we have two machines that can be used for the same purpose, then

we may confidently say that the precision of the newer machine is higher than that of the

old one, i.e. the variance of repeated measurements taken by the newer machine is smaller

than those taken by the older one. Frequently, a large amount of information is known

about the older machine that can be used to assess the precision of the new machine with

few new measurements. To ensure getting a random sample of observations, each machine

can be used to measure n different subjects that have a common nominal value.

2. If (X , W ) have a bivariate normal distribution then Var(X jw) < Var(X ). To be more

specific, suppose that the height and weight of a certain species have a bivariate normal

distribution, then a random sample of n heights of n species taken from the population can

be utilized (if available) to estimate the variability in the heights of all species that share a

common weight. Usually samples from the X-population are easier to obtain than samples

from the X jw-population.

80 M. F. AL-SALEH AND H. M. SAMAWI

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3. Taking a simple random sample from a population may be sometimes easier (cheaper to

obtain) than taking a sample of equivalent size from a specific portion (stratum) of the

population. Frequently, the variance of the elements in one stratum is small when com-

pared to the overall variance of the population. So a small sample from the stratum of

interest augmented with a larger simple random sample from the population can be used

to estimate the stratum variance.

The rest of the paper is organized as follows. In Section 2 we derive the class of estimators

for the cases of known and unknown means and discuss some of their properties. In Section 3

we discuss the behavior of the efficiency for some members of this class with respect to some

relevant estimators. The results of the simulation are reported in Section 4. Concluding

remarks are given in Section 5.

2 THE CLASS OF ESTIMATORS

Assume X1, X2, . . . , Xn1and Y1, Y2, . . . , Yn2

are two independent random samples from

N (m1, s21) and N (m2, s2

2), respectively. Suppose that prior information suggests that

s21 � s2

2. Estimation of s21 is considered utilizing the prior information in the Y-sample

which is available presumably at no extra cost. There are two cases to consider:

Case 1 Assume that m1 and m2 are known, so without loss of generality, we can assume that

m1 ¼ m2 ¼ 0. Let

S21 ¼

PX 2i

n1

; S22 ¼

PY 2i

n2

and R ¼ {(s21, s2

2): 0 < s21 � s2

2}:

S21 and S2

2 are the MVUE of s21 and s2

2, respectively.

Define the improper prior on the region R by

p(s21, s2

2) ¼1

s21

� �k1 1

s22

� �k2

(3)

and zero otherwise; k1 and k2 are non-negative real numbers. This is a type of diffuse (non-

informative) prior usually chosen if there is no prior information about the parameters to

construct a proper prior; it is a kind of ignorance prior. For more details about these types

of priors, and details about Bayesian analysis see Berger (1980).

The posterior distribution of (s21, s2

2) given x ¼ {x1, x2, . . . , xn} and y ¼ {y1, y2, . . . , yn} is

p(s21, s2

2jx, y) ¼(1=s2

1)k1þn1=2(1=s22)k2þn2=2e�(

PX 2i =(2s2

1)þP

y2i =(2s2

2))

Ð10

Ð s22

0 (1=s21)k1þn1=2(1=s2

2)k2þn2=2e�(P

X 2i=(2s2

1)þP

y2i=(2s2

2)) ds2

1 ds22

¼g(u1s2

1; ðk1 þ n1=2Þ � 1, 1)g(u2s22; ðk2 þ n2=2Þ � 1, 1)Ð1

0

Ð s22

0 g(u1s21; ðk1 þ n1=2Þ � 1, 1)g(u2s2

2; ðk2 þ n2=2Þ � 1, 1) ds21 ds2

2

, (4)

where g( � ; a, b) is the density of the inverse gamma random variable with parameters aand b, i.e.,

g(w; a, b) ¼1

G(a)ba1

waþ1e�1=(bw), for w > 0 and zero otherwise:

ESTIMATION OF THE VARIANCE 81

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Let u1 ¼ 2=P

x2i and u2 ¼ 2=

Py2i then (4) can be written as

p(s21, s2

2jx, y) ¼g(u1s2

1; (k1 þ n1=2) � 1, 1)g(u2s22; (k2 þ n2=2) � 1, 1)

(1=u1)Ð1

0G(u1s2

2; (k1 þ n1=2) � 1, 1)g(u2s22; (k2 þ n2=2) � 1, 1) ds2

2

,

(5)

where G is the cumulative distribution of g.

Now, consider the integral

h(u1, u2) ¼

ð10

g(u2s22; a2, 1)G(u1s2

2; a1, 1) ds22: (6)

Taking the partial derivative of h with respect to u1 we have

qh(u1, u2)

qu1

¼

ð10

g(u2s22; a2, 1)g(u1s2

2; a1, 1)s22 ds2

2

¼G(a1 þ a2)

G(a1)G(a2)

ua1þa2

3

ua1þ11 ua2þ1

2

, (7)

where u3 ¼ u1u2=(u1 þ u2).

Let v1 ¼ 2a2 and v2 ¼ 2a1 then

qh(u1, u2)

qu1

¼G((v1 þ v2)=2)

G(v1=2)G(v2=2)

v(v1þv2)=22

uv1=2þ12

uv1=2�11

(v2 þ v2(u1=u2))(v1þv2)=2, (8)

Hence

h(u1, u2) ¼G((v1 þ v2)=2)

G(v1=2)G(v2=2)

v(v1þv2)=22

uv1=2þ12

ðu1

0

uv1=2�1

(v2 þ v2(u=u2))(v1þv2)=2du: (9)

Using the transformation v1t ¼ v2(u=u2), we have

h(u1, u2) ¼G((v1 þ v2)=2)

G(v1=2)G(v2=2)vv1=21 v

v2=22

1

u2

ðu1

0

tv1=2�1

(v2 þ v1t)(v1þv2)=2

dt

¼1

u2

Fa1u1

a2u2

, 2a2, 2a1

� �, (10)

where F( � , 2a2, 2a1) is the cumulative distribution function of the F distribution with 2a2,

2a1 degrees of freedom. Using (10) above, the denominator of (5) is equal to

1

u1u2

Fnþ 2k1 � 2

nþ 2k2 � 2

u1

u2

, nþ 2k2 � 2, nþ 2k1 � 2

� �: (11)

Note: As noted by the referee, the above results can be obtained by using the transfor-

mation: u ¼ s21=s

22 and v ¼ 1=s2

2.

The Generalized Bayes estimator of s21 with respect to the squared error loss function

L(y, d) ¼ (y� d)2 is the posterior mean given by

E(s21jX, Y) ¼

ð10

ðs22

0

s21p(s2

1, s22jX, Y) ds2

1 ds22: (12)


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Now

ð10

ðs22

0

s21g u1s2

1; k1 þn1

2� 1, 1

� �g u2s2

2; k2 þn2

2� 1, 1

� �ds2

1 ds22

¼2

(n1 þ 2k1 � 4)u1

ð10

ðs22

0

s21g u1s2

1; k1 þn1

2� 2, 1

� �g u2s2

2; k2 þn2

2� 2, 1

� �ds2

1 ds22

¼2

(n1 þ 2k1 � 4)u21

ð10

G u1s22; k1 þ

n1

2� 2, 1

� �g u2s2

2; k2 þn2

2� 2, 1

� �ds2

2

¼2

(n1 þ 2k1 � 4)u21u2

1

u2u2

Fn1 þ 2k1 � 4

n2 þ 2k2 � 2

u1

u2

, n2 þ 2k2 � 2, n1 þ 2k1 � 4

� �: (13)

Note that (13) is obtained by using (10) again. Thus, the generalized Bayes estimator of s21 is

S21b(k1, k2) ¼

2

(n1 þ 2k1 � 4)u1

�F(((n1 þ 2k1 � 4)=(n2 þ 2k2 � 2))(u1=u2), n2 þ 2k2 � 2, n1 þ 2k1 � 4)

F(((n1 þ 2k1 � 2)=(n2 þ 2k2 � 2))(u1=u2), n2 þ 2k2 � 2, n1 þ 2k1 � 2),

which can be written in terms of S21 and S2

2 as

S21b(k1, k2) ¼

n1S21

(n1 þ 2k1 � 4)

�F(((n1 þ 2k1 � 4)=(n2 þ 2k2 � 2))(S2

2=S21 ), n2 þ 2k2 � 2, n1 þ 2k1 � 4)

F(((n1 þ 2k1 � 2)=(n1 þ 2k2 � 2))(S22=S

21 ), n2 þ 2k2 � 2, n1 þ 2k1 � 2)

:

(14)

Note that S21b(k1, k2) can be regarded as a class of estimators, i.e. different values of (k1, k2)

give different estimators. The following observations about this class can be easily seen:

1. If S22 is very large (relative to S2

1 ) then S21b(k1, k2) behaves as nS2

1=(nþ 2k1 � 4) (i.e. as S21

for k1 ¼ 2). This is a reasonable property because large values of S22 relative to S2

1 indicate

that s22 is so large compared to s2

1 and hence the information in the Y-sample is irrelevant

to the problem of estimating s21. In this case the Bayes formula ignores the information in

the Y-sample and estimate s21 using the information in the X-sample only.

2. If S22 is very small (relative to S2

1 ) then S21b(k1, k2) behaves as (S2

1 þ S22)=2 (for k1 ¼ 2

and n1 ¼ n2). This is also a reasonable property because small values of S22 relative to S2

1

support the near equality of s22 and s2

1 which makes (S21 þ S2

2 )=2 a reasonable estimator ofs21.

3. The efficiency of S21b(k1, k2) with respect to the usual S2

1 , measured by

MSE(S21 )=MSE(S2

1b(k1, k2)) depends on s22 and s2

1 through the ratio r ¼ s22=s

21. Here

MSE(d) ¼ E(d� E(d))2, where d is an estimator.

4. The first factor of the estimator above, namely n1S21=(n1 þ 2k1 � 4) is the generalized

Bayes estimator of s21 with respect to the prior (1=s2

1)k1, without using the information in

the second sample. The second factor can be written as F(Z1, a, b)=F(Z2, a, bþ 2), where

Z1 < Z2. Now, F(Z1, a, b) < F(Z2, a, b) � F(Z2, a, bþ 2), (for moderate values of b).

Hence F(Z1, a, b)=F(Z2, a, bþ 2), can be regarded as a shrinkage factor of n1S21=

(n1þ 2k1 � 4).

Case 2 In practice, it is most likely that m1 and m2 are unknown. In case of the absence of

any prior information about (m1, m2), the extended uniform prior (non-informative prior)

can be used, i.e. p�(m1, m2) ¼ 1 for (m1, m2) 2 R2. Assuming that (m1, m2) and (s21, s2

2) are


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independent, then the joint posterior distribution of (s21, s2

2, m1, m2) given (x, y) is

proportional to

1

s21

� �k1þn1=21

s22

� �k2þn2=2

e�(P

(Xi��xx)2=(2s21)þP

(yi��yy)2=(2s22))�n1=(2s2

1)(�xx�m1)2�n2=(2s2

2)(�y�m2)2

:

By integrating (m1, m2) out, we can obtain the joint posterior distribution of (s21, s2

2) given

(x, y) as

p(s21, s2

2jx, y)

¼(1=s2

1)k1þn=2(1=s22)k2þn=2e(

P(Xi��xx)2=(2s2

1)þP

(yi��yy)2=(2s22))

Ð10

Ð s22

0 (1=s21)k1þn1=2(1=s2

2)k2þn2=2e�(P

(Xi��xx)2=(2s21)þP

(yi��yy)2=(2s22))ds2

1 ds22

, (15)

which the same as in case 1, except thatP

x2i is replaced by

P(xi � �xx)2 and

Py2i is replaced

byP

( yi � �yy)2. All the steps lead to the generalized Bayes estimators of s21 in the previous

case are valid to this case. Thus, the generalized Bayes estimator of s21 is

S�21b (k1, k2) ¼

(n1 � 1)S�21

(n1 þ 2k1 � 4)

�F(((n1 þ 2k1 � 4)=(n2 þ 2k2 � 2))(S�2

2 =S�21 ), n2 þ 2k2 � 2, n1 þ 2k1 � 4)

F(((n1 þ 2k1 � 3)=(n2 þ 2k2 � 3))(S�22 =S�2

1 ), n2 þ 2k2 � 2, n1 þ 2k1 � 2),

(16)

where,

S�21 ¼

P(Xi � �XX )2

n1 � 1; S�2

2 ¼

P(Yi � �YY )2

n2 � 1:

All of the above comments about the class of estimators in case 1, are valid for this case

as well.

It can be seen that the closed form of the suggested class of estimators is a complicated one

and it is not possible to study its property analytically; so we will rely on simulation to get an

idea about the performance of these estimators.

3 THE BEHAVIOUR OF THE EFFICIENCY WITH RESPECT TO

SOME RELEVANT ESTIMATORS

It can be seen from the form of S�21b (k1, k2) that each member of the class is a modification of the

generalized Bayes estimator of s21 based on the first sample, i.e. with n1 ¼ n2 ¼ n,

S��1b (k1) ¼ (n� 1)S�21 =(nþ 2k1 � 4). Hence, naturally S�2

1b (k1, k2) should be compared to

S��1b (k1). If k1 ¼ 1:5 then S��1b (1:5) ¼ S�21 , which the MVUE of s2

1 based on the first sample. If

k1 ¼ 2 then S��1b (2) ¼ (n� 1)S�21 =n, which the MLE of s2

1 based on the first sample, and finally

if k1 ¼ 2:5 then S��1b (2:5) ¼P

(Xi � �XX )2=(nþ 1), which is the estimator that has lowest mean

squared error (MSE) among all estimators that are linear inP

(Xi � �XX )2. The MSE of each of

these three estimators are very closed to each other; so we will only compare S�21b (1:5, k2) to

S��1b (1:5) ¼ S�21 . Also S�2

1b (1:5, k2) will be compared to Stein estimator as well as to Mehta

and Gurland estimator. These two last estimators are derived based on unknown means.

Likewise, when the means are known then S21b(2, k2) is compared to S2

1 .


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For a given n1, n2, the MSE of S�21b (1:5, k2) depends on k2 and r ¼ s2

2=s21, but not on

(m1, m2). The distribution of S�21b (k1, k2) is free of (m1, m2). Hence, without loss of generality

we may take (m1, m2) ¼ (0, 0) in the process of calculating the MSE. Thus, the efficiency of

this estimator with respect to S�2i and M&G estimator doesn’t depend on (m1, m2). However,

for Stein estimator, the MSE does depend on m1. Stein estimator was derived based on a pre-

test of the hypothesis H0: m1 ¼ 0. The estimator isP

X 2i =(n1 þ 2) if the H0 is accepted and it

isP

(Xi � �XX )2=(nþ 1) if H0 is rejected. Hence Stein tends to have the smallest MSE when

m1 ¼ 0. The efficiency of S�21b (1:5, k2) with respect to Stein estimator tends to be minimal

when (m1, m2) ¼ (0, 0). The efficiency of an estimator d1 with respect to an estimator d2 is

given by

Eff (d1; d2) ¼MSE(d1)

MSE(d2)

Before we report the result of simulation, several point have to be emphasized. As we men-

tioned above, the efficiency of S�21b (1:5, k2) with respect to the relevant estimators depends on

r and k2 (except for Stein, it depends also on m1). The value of r is an indication of the use-

fulness of the second sample in estimating s21; the larger the value of r the less useful is the

second sample. The value of k2 serves as a weight factor of the information in the second

sample in the proposed estimator. Using a small value of k2 means that the information in

the second sample is highly utilized, while using a large value of k2 means that the informa-

tion in the second sample is minimally utilized. It is intuitively clear that the efficiency as a

function of r for a given k2, may not be independent of k2, i.e. there is an interaction effect of

(r, k2) on the efficiency. Thus, it is expected that for small values of (r, k2) and for large

values of (r, k2), the efficiency tends to be high. Likewise the efficiency should tend to be

smaller for the other two cases. The efficiency should eventually approaches 1 as r or k2

gets very large.

4 NUMERICAL COMPARISONS

4.1 Plan of the Simulation

The normal distribution is used in the simulation to compare S�21b (1:5, k2) with the usual

sample mean estimator S�21 (and S2

1b(2, k2) w.r.t S21 ), Stein and M&G estimators. Two

equal sample size combinations n1 ¼ n2 ¼ 7 and 11 and one unequal samples sizes n1 ¼

5 and n2 ¼ 7 were investigated for combinations of values of k2 and r. Without loss of

generality, we assume throughout the simulations that s21 ¼ 1 and (m1, m2) ¼ (0, 0). For

each of the combinations of sample sizes, k2 and r, 5000 data sets were generated for

the simulation study. The efficiency of S�21b (1:5, k2), which is the MSE of the relevant esti-

mator divided by the MSE of S�21b (1:5, k2) is approximated for each of the combinations.

The efficiency of S�21b (1:5, k2) w.r.t M&G is obtained based on tables provided by M&G

(1969); The values of n and r are chosen here to match their values. Tables IA, IB and IC

(n1 ¼ 5 and n2 ¼ 7) contain the efficiency of S21b(2, k2) w.r.t S2

1 . Tables IIA, IIB and IIC

(n1 ¼ 5 and n2 ¼ 7) contain the efficiency of S�21b (1:5, k2) w.r.t S�2

1 and Stein estimator;

Table III contains the efficiency of S�21b (1:5, k2) w.r.t G&M. We also report the case of

S�21b (2, k2); the case of k1 ¼ k2 ¼ 2 corresponds to the noninformative priors of s1 and

s2. Table IId (n1 ¼ 5 and n2 ¼ 7) contains the efficiency of S�21b (2, k2) w.r.t S�2

1 . Other

values of k1 are also investigated; some of them yield high efficiency of S�21b (k1, k2),

but will not be report here.


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4.2 Results of the Simulation

In this subsection, we will comment on the results of the efficiency of S�21b (1:5, k2); similar

comments can be said about S21b(2, k2). Our Simulation results demonstrate that the estimator

S�21b (k1, k2) of s2

1 is substantially more efficient than S�21 for some choices of k2. Furthermore,

S�21b (k1, k2) is almost always dominating Stein and M&G estimators. The results are in a

relatively strong agreement with our insight to the properties of the efficiency outlined in

the previous section. Based on Table II we may observe the following:

1. S�21b (1:5, k2) is substantially more efficient than S�2

1 for a combination of small values of r

and k2 and also for a combination of large values of them. The efficiency is moderate for

the other two cases. The suggested estimator is also more efficient than Stein estimator for

TABLE IA The Efficiency of S21b(2, k2) w.r.t S2

1 for n1 ¼ n2 ¼ 7.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.61 1.62 1.54 1.51 1.48 1.38 1.28 1.24 1.010.5 1.63 1.63 1.64 1.62 1.54 1.45 1.39 1.31 1.021.0 1.56 1.58 1.62 1.62 1.61 1.51 1.43 1.36 1.021.5 1.52 1.60 1.74 1.69 1.65 1.63 1.54 1.40 1.032.0 1.46 1.55 1.55 1.68 1.69 1.74 1.60 1.49 1.032.5 1.42 1.50 1.67 1.65 1.64 1.65 1.62 1.58 1.043.0 1.35 1.48 1.55 1.56 1.65 1.72 1.65 1.61 1.074.0 1.19 1.34 1.40 1.56 1.52 1.65 1.74 1.68 1.07

TABLE IB The Efficiency of S21b(2, k2) w.r.t S2

1 for n1 ¼ n2 ¼ 11.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.53 1.52 1.50 1.45 1.42 1.30 1.24 1.15 1.000.5 1.53 1.55 1.52 1.49 1.44 1.37 1.25 1.19 1.001.0 1.46 1.56 1.54 1.54 1.49 1.40 1.30 1.23 1.001.5 1.42 1.46 1.49 1.51 1.53 1.47 1.35 1.25 1.002.0 1.34 1.44 1.49 1.52 1.56 1.46 1.41 1.31 1.002.5 1.35 1.41 1.48 1.54 1.57 1.53 1.42 1.33 1.003.0 1.22 1.43 1.50 1.56 1.59 1.57 1.50 1.36 1.014.0 1.09 1.25 1.40 1.55 1.52 1.55 1.50 1.40 1.00

TABLE IC The Efficiency of S21b(2, k2) w.r.t S2

1 for n1¼ 5, n2¼ 7.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.40 1.48 1.59 1.59 1.57 1.59 1.45 1.44 1.350.5 1.54 1.45 1.45 1.43 1.51 1.48 1.53 1.54 1.451.0 1.41 1.51 1.46 1.51 1.54 1.46 1.44 1.51 1.411.5 1.43 1.38 1.33 1.63 1.47 1.59 1.61 1.58 1.452.0 1.20 1.40 1.47 1.42 1.46 1.35 1.43 1.64 1.332.5 1.16 1.36 1.39 1.45 1.40 1.56 1.66 1.54 1.343.0 1.15 1.25 1.26 1.28 1.55 1.48 1.32 1.58 1.404.0 1.13 1.12 1.18 1.35 1.41 1.45 1.58 1.38 1.44


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most of the combination. The reported efficiency with respect to Stein is the minimal

possible one. (see the note in Section 3 regarding this efficiency).

2. For small to moderate values of r, the efficiency is slightly increasing. We believe that the

cause of increase is due to the value of k2 for small k2, the increase is very slight and may

be due to simulation error while the increase is significant for large values of k2. For large

values of r, the efficiency is decreasing and goes to 1 as r gets very large regardless of the

value of k2. As we move from n ¼ 7 to n ¼ 11, the efficiency w.r.t S�21 decreases by 10 to

20%. This decrease is justifiable, because the more sample information one have from the

TABLE IIA The Efficiency of S�21b (1:5, k2) w.r.t S�2

1 and Stein Estimator (Bold) for n1¼ n2¼ 7.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.64 1.65 1.64 1.61 1.58 1.49 1.45 1.36 1.031.24 1.25 1.24 1.22 1.18 1.13 1.07 1.02 0.78

0.5 1.59 1.74 1.76 1.64 1.66 1.57 1.53 1.45 1.041.22 1.28 1.28 1.25 1.23 1.21 1.16 1.08 0.80

1.0 1.63 1.69 1.61 1.70 1.70 1.62 1.62 1.53 1.051.22 1.26 1.25 1.28 1.28 1.24 1.19 1.15 0.78

1.5 1.50 1.58 1.69 1.66 1.75 1.67 1.62 1.64 1.061.14 1.21 1.26 1.26 1.29 1.30 1.22 1.18 0.81

2.0 1.47 1.57 1.63 1.66 1.72 1.75 1.68 1.73 1.071.10 1.17 1.21 1.25 1.28 1.30 1.28 1.23 0.80

2.5 1.46 1.50 1.56 1.60 1.68 1.75 1.74 1.71 1.111.06 1.11 1.16 1.20 1.26 1.30 1.31 1.27 0.81

3.0 1.23 1.36 1.49 1.56 1.65 1.73 1.76 1.68 1.100.96 1.04 1.12 1.17 1.23 1.30 1.33 1.28 0.85

4.0 1.17 1.26 1.34 1.58 1.52 1.72 1.72 1.69 1.130.88 0.93 1.01 1.08 1.13 1.27 1.29 1.30 0.84

TABLE IIB The Efficiency of S�21b (1:5, k2) w.r.t S�2

1 and Stein Estimator (Bold) for n1¼ n2¼ 11.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.48 1.52 1.59 1.49 1.51 1.37 1.28 1.20 1.001.26 1.28 1.31 1.27 1.23 1.16 1.08 1.02 0.85

0.5 1.45 1.58 1.56 1.58 1.52 1.43 1.34 1.23 1.001.22 1.30 1.30 1.30 1.28 1.20 1.11 1.03 0.85

1.0 1.44 1.53 1.59 1.54 1.58 1.49 1.40 1.27 1.001.20 1.27 1.31 1.30 1.31 1.24 1.14 1.07 0.84

1.5 1.39 1.47 1.55 1.58 1.56 1.51 1.42 1.31 1.011.16 1.24 1.29 1.32 1.30 1.29 1.20 1.10 0.84

2.0 1.29 1.47 1.52 1.59 1.57 1.57 1.46 1.38 1.011.08 1.22 1.27 1.32 1.32 1.28 1.24 1.14 0.85

2.5 1.30 1.39 1.42 1.60 1.60 1.63 1.45 1.40 1.011.06 1.15 1.25 1.30 1.32 1.32 1.26 1.15 0.83

3.0 1.15 1.37 1.49 1.52 1.52 1.62 1.49 1.42 1.010.97 1.12 1.19 1.27 1.30 1.35 1.27 1.20 0.85

4.0 1.08 1.23 1.38 1.42 1.51 1.58 1.56 1.50 1.020.90 1.01 1.12 1.19 1.26 1.32 1.31 1.26 0.85


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target population the more is the tendency to ignore other related data. The efficiency with

respect to Stein estimator is slightly higher for n ¼ 11 than for n ¼ 7. (Again see the note

in Section 3 regarding this efficiency)

Based on the above, if we feel strongly that the second sample is very relevant to our esti-

mation problem, i.e. r is small, then we may gain efficiency by using small values of k2,

0 � k2 � 1, say. In the other extreme, one may use large values of k2, 2:5 � k2 � 4, say.

Also, the suggested estimator is more useful with small sample size. This property is actually

TABLE IIC The Efficiency of S�21b (1:5, k2) w.r.t S�2

1 and Stein Estimator (Bold) for n1¼ 5, n2¼ 7.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.65 1.59 1.49 1.56 1.47 1.49 1.51 1.63 1.571.02 1.03 1.03 1.05 1.04 1.04 1.05 1.05 0.98

0.5 1.62 1.52 1.43 1.53 1.62 1.57 1.61 1.70 1.610.99 1.00 1.01 1.02 1.05 1.05 1.06 1.07 0.99

1.0 1.42 1.46 1.48 1.50 1.58 1.51 1.56 1.60 1.520.94 0.95 0.99 1.01 1.04 1.04 1.52 1.06 1.01

1.5 1.29 1.40 1.40 1.50 1.55 1.58 1.63 1.74 1.480.89 0.94 0.94 0.97 1.01 1.03 1.05 1.08 1.02

2.0 1.34 1.29 1.25 1.48 1.48 1.63 1.54 1.65 1.560.86 0.88 0.88 0.95 0.97 1.02 1.02 1.06 1.00

2.5 1.26 1.30 1.31 1.48 1.55 1.65 1.34 1.60 1.580.81 0.83 0.89 0.94 0.96 1.01 0.99 1.05 1.01

3.0 1.11 1.29 1.13 1.46 1.38 1.52 1.47 1.55 1.500.75 0.83 0.83 0.89 0.90 0.96 0.98 1.02 1.02

4.0 1.13 1.11 1.18 1.28 1.27 1.24 1.55 1.46 1.620.72 0.74 0.78 0.82 0.84 0.88 0.97 0.99 1.04

TABLE IID The Efficiency of S�21b (2, k2) w.r.t S�2

1 and Stein Estimator (Bold) for n1¼ 5, n2¼ 7.

r

k2 1.0 1.1 1.2 1.3 1.4 1.6 1.8 2.0 5.0

0.0 1.34 1.42 1.49 1.53 1.51 1.56 1.43 1.41 1.390.91 0.92 0.95 0.97 0.95 0.96 0.95 0.95 0.94

0.5 1.38 1.37 1.33 1.34 1.44 1.41 1.47 1.52 1.470.90 0.89 0.90 0.92 0.93 0.94 0.96 0.97 0.95

1.0 1.33 1.45 1.31 1.37 1.41 1.40 1.41 1.48 1.480.85 0.90 0.88 0.90 0.92 0.94 0.94 0.96 0.94

1.5 1.29 1.25 1.20 1.52 1.35 1.42 1.54 1.51 1.490.83 0.83 0.84 0.91 0.90 0.93 0.97 0.96 0.95

2.0 1.10 1.24 1.32 1.24 1.29 1.25 1.24 1.55 1.340.78 0.82 0.83 0.85 0.87 0.88 0.90 0.96 0.95

2.5 1.03 1.27 1.19 1.31 1.30 1.41 1.54 1.38 1.300.75 0.79 0.79 0.83 0.87 0.90 0.94 0.94 0.95

3.0 1.06 1.16 1.11 1.12 1.37 1.36 1.28 1.43 1.400.71 0.76 0.78 0.78 0.84 0.88 0.89 0.92 0.94

4.0 1.06 1.05 1.08 1.22 1.23 1.31 1.40 1.23 1.510.68 0.70 0.73 0.78 0.80 0.81 0.87 0.87 0.96


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inherited in the Bayesian formula. If our budget allows for large sample size then we don’t

need other related information which may not be very accurate, while if it is very costly to

sample from the population of interest then we may use some other related available or easily

and cheaply obtained information.

Based on Table III, we observe the following:

1. Except for few cases, S�21b (1:5, k2) is more efficient than M&G estimator. The efficiency

gets larger as we move from n ¼ 7 to n ¼ 11 for a combination of small values of r and k2.

2. If the second sample is irrelevant to the estimation problem (very large r), the suggested

estimator is safer to use.

Based on the above, and given the fact that the calculation of the values of M&G estimator

depends solely on the two constants c and d that should be specified for each choice of sam-

ple size, the suggested estimator S�21b (1:5, k2) is more reasonable to be used whenever

applicable.

Similar statements can be said about the other results of the other tables.

5 CONCLUSIONS

It appears that some members of the suggested class of estimators of s1, whenever applicable

are serious competitors to some of the well-known available estimators. In Particular,

S�21b (1:5, k2) for some carefully selected value of k2 is substantially more efficient than the

widely used sample variance, S�21 . This estimator appears to be also more efficient that the

Stein and Mehta and Gurland estimators. We recommend using some members of

S�21b (k1, k2) when the second sample is believed to be from a population with slightly higher

variance than the target population.

TABLE III The Efficiency of S�21b (1:5, k2) w.r.t Mehta and Gurland

Estimator when (c, d)¼ (�0.25, 3.5) for n1¼ n2¼ 7 and (c, d)¼ (1.76,1.76) for n1¼ n2¼ 11 (Bold).

r

k2 1.0 2.0 3.0 4.0 5.0 10.0

0.0 1.11 1.14 1.08 1.03 1.01 1.081.40 1.18 1.05 1.22 1.02 1.07

0.5 1.10 1.23 1.10 1.05 1.02 1.081.35 1.24 1.05 1.25 1.02 1.07

1.0 1.00 1.32 1.12 1.09 1.02 1.081.34 1.26 1.08 1.28 1.05 1.07

1.5 0.98 1.35 1.17 1.09 1.02 1.091.24 1.31 1.09 1.26 1.02 1.07

2.0 1.47 1.40 1.19 1.09 1.05 1.081.25 1.35 1.09 1.25 1.04 1.07

2.5 1.46 1.44 1.27 1.14 1.06 1.081.19 1.40 1.12 1.20 1.04 1.07

3.0 0.91 1.49 1.28 1.19 1.08 1.091.06 1.43 1.13 1.17 1.05 1.07

4.0 0.79 1.52 1.39 1.20 1.12 1.091.03 1.45 1.16 1.08 1.07 1.07


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Acknowledgements

The authors would like to thank Professor John Grego for his careful review and many

comments that greatly improved the manuscript. Special thanks to the referee for his useful

comments.

References

Al-Saleh, M. F. (1993). Two stage estimation of the variance of normal population. Information and OptimizationSciences, 15, 17–125.

Berger, J. (1980). Statistical Decision Theory. Springer-Verlag, New York.Eden, V. and Zidek, J. V. (2001). Estimating one of two normal means when their difference is bounded. Statistics

and Probability Letters, 51, 277–284.Kubokawa, T. and Saleh, A. K. Md. E. (1994). Estimation of location and scale parameters under order restrictions.

Journal of Statistics Research, 28, 41–51.Kushary, D. and Cohen, A. (1989). Estimation ordered location and scale parameters. Statistical Decisions, 7,

201–213.Mehta, J. S. and Gurland, J. (1969). On utilizing information from a second sample in estimating variance.

Biometrika, 56, 527–532.Pandy, B. N. (1979). Double stage estimation of population variance. Annals of the Institute of Statistical

Mathematics, 31, 225–233.Stein, C. (1964). Inadmissibility of the usual estimator of the variance of a normal distribution with unknown mean.

Annals of the Institute of Statistical Mathematics, 16, 155–160.


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estimating the variance of a normal population by utilizing the information in a sample from a...

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