University Grants Commission

BAHADUR SHAH ZAFAR MARG NEW

DELHI – 110 002

E-Mail : ugc.@bol.net.in

website: www.ugc.ac.in

ESTIMATION OF PARAMETER OF RECTANGULAR

PROBABILITY DISTRIBUTION ON (0,θ) BASED ON TWO

SAMPLES AND PRELIMINARY TEST OF SIGNIFICANCE

Final Report

On

Minor Research Project

Submitted

To

University Grants Commission

Western Regional Office,

Ganeshkhind,Pune-411007

By

Dr. D.B. Jadhav

M.Sc., M. Phil., Ph.D.

Department of Statistics

Yashavantrao Chavan Institute of Science, Satara.

March 2015

Annexure -III UNIVERSITY GRANTS COMMISSION

BAHADUR SHAH ZAFAR MARG

NEW DELHI – 110 002. Annual/Final Report of the work done on the Minor Research Project.

(Report to be submitted within 6 weeks after completion of each year)

1. Project report No. 1st /2nd /3rd /Final Final

2. UGC Reference No. U.G.C.Letter No. F. No.47-1287/09(WRO)

Date.17 Nov. 2009.

3. Period of report: from: 1-12-2009 to 31-12-2011 4. Title of research project: Estimation of Parameter of Rectangular Probability Distribution on (0,θ) Based on Two Samples and Preliminary Test of Significance.

5. (a) Name of the Principal Investigator : Dr.D.B.Jadhav

(b) Deptt. and University/College where work has progressed: Dept. of

Statistics, Y.C.Inst. of Science, Satara 6. Effective date of starting of the project 20-12-2009 7. Grant approved and expenditure incurred during the period of the report:

a. Total amount approved Rs. 50,000/- b. Total expenditure Rs. 55,704/- c. Report of the work done: (Please attach a separate sheet)

i. Brief objective of the project: To propose a better estimator than available

by using two samples and preliminary test procedure. For this to develop a

test. To study properties of this test. To study the properties of the proposed

estimator. To find out situations where this estimator is better as compared

to the estimator in use today.

ii. Work done so far and results achieved and publications, if any,

resulting from the work (Give details of the papers and names of the

journals in which it has been published or accepted for publication):

The test is developed. It comes out to be a likelihood ratio test. Its

properties are studied. Estimator is proposed. Its properties are studied. It

estimates the unknown parameter more accurately than the estimator in use

today in a few situations. iii. Has the progress been according to original plan of work and towards

achieving the objective ? if not, state reasons:

Yes. The work was going on according to original plan. It was almost completed in time. But, the estimator I proposed is not that effective and handy

for practical use as the gain is less as compared to the efforts and computations

needed. Therefore, I was studying and thinking to still improve the results. During

this span I happened read about the attainable lower bound on the variance of

unbiased estimator of parameter of my interest in this project. Though, this is the

best bound its applications were limited. I tried to extend its utility and got some

results. This was quite encouraging fact due to which Dr. Ashok Shonubhogue of

Sardar Patel University, Vallabh-Vidyanagar invited me to work for my Ph.D. on

this topic. I registered, worked and received Ph.D. on Dec.20,2014. LAMBERT

Academic Publishing, Germany published my thesis in the form of a book. These

results were quite exiting as extension in utility of this bound was possible after

quite a long period after its inception in 1952. During this, Five research papers are

published in international journals. I presented Two of them in national

conferences one at North Maharashtra University, Jalgaon on Jan.30-31,2014 and

another at Sardar Patel University, V V nagar on Feb.21-22,2014. My Ph.D.

problem is motivated from the topic of project. As my retirement was nearing, I

wanted to finish up Ph.D. work first to enable me to apply for extension in service

without hampering my teaching work. Therefore, the submission is delayed. I am

extremely sorry for that. But, I want to express my gratitude towards UGC and all

the concerned persons for bearing with me and making it possible for me to add to

the knowledge of the subject, which provided me almost everything in life, on the

verge of the retirement.

iv. Please indicate the difficulties, if any, experienced in implementing

the project

Usual teaching and project work become heavy and exhausting.

v. If project has not been completed, please indicate the approximate

time by which it is likely to be completed. A summary of the work done for

the period (Annual basis) may please be sent to the Commission on a

separate sheet : ----.

vi. If the project has been completed, please enclose a summary of the

findings of the study. Two bound copies of the final report of work done

may also be sent to the Commission: Enclosed. ___________________________________________________________ ___________________________________________________________

vii. Any other information which would help in evaluation of work done on the project. At the completion of the project, the first report should indicate the

output, such as (a) Manpower trained (b) Ph. D. awarded (c) Publication of results (d) other impact, if any

I learned to apply some ideas in this problem in a non regular set up. Also I learned some computer operations etc. (a) I could learn to apply some ideas

in this problem in a non regular set up. Also I learned some

computer basics. (b) The uniformly minimum variance unbiased estimator

(UMVUE) of the parameter of interest in project and Kiefer bound giving lower

bound on the variance of its unbiased estimator motivated me to extend the utility

of Kiefer bound and receive Ph.D. in Dec.2014. (c) I could publish 5 (Five)

research papers all in international journals. (d) This made me eligible for

extension of Two more years in my service. So, I can motivate and help some more

students to learn and come up.

I express my gratitude to all the concern for making it possible for

me to add something to the knowledge of our subject

SIGNATURE OF THE PRINCIPAL REGISTRAR/PRINCIPAL INVESTIGATOR

Annexure – VIII

UNIVERSITY GRANTS COMMISSION BAHADUR SHAH ZAFAR MARG

NEW DELHI – 110 002 PROFORMA FOR SUBMISSION OF INFORMATION AT THE TIME OF SENDING THE

FINAL REPORT OF THE WORK DONE ON THE PROJECT 1. NAME AND ADDRESS OF THE PRINCIPAL INVESTIGATOR : Dr.D.B.Jadhav

Yashwantrao Chavan Institute of Science, Satara. 2. NAME AND ADDRESS OF THE INSTITUTION: Yashwantrao Chavan Institute of Science, Satara.

3. UGC APPROVAL NO. AND DATE: F. 47-1287/09(WRO) dated 17 Nov. 2009

4. DATE OF IMPLEMENTATION: 20-12-2009

5. TENURE OF THE PROJECT 20-12-2009 to 31-12-2011 6. TOTAL GRANT ALLOCATED: Rs. 50,000/- 7. TOTAL GRANT RECEIVED:Rs. 40,000/- 8. FINAL EXPENDITURE : Rs. 55704/-

9. TITLE OF THE PROJECT: Estimation of Parameter of Rectangular Probability

Distribution on (0,θ) Based on Two Samples and Preliminary Samples and

Preliminary Test of Significance.

10. OBJECTIVES OF THE PROJECT :To propose a better estimator than available by

using two samples and preliminary test procedure. For this to develop a test. To

study properties of this test. To study the properties of the proposed estimator. To

find out situations where this estimator is better as compared to that in use today. 11. WHETHER OBJECTIVES WERE ACHIEVED : Yes.

12.ACHIEVEMENTS FROM THE PROJECT : The work was going on according to original plan. It was almost completed in time. But, the estimator I proposed is not

that effective and handy for practical use as the gain is less as compared to the efforts

and computations needed. Therefore, I was studying and thinking to still improve the

results. During this span I happened to read about the attainable lower bound on the

variance of unbiased estimator of parameter of my interest in this project provided by

Jack Kiefer in 1952. Though, this is the best bound its applications were limited. I tried

to extend its utility and got some results. This was quite encouraging fact due to which

Dr.Ashok Shonubhogue of Sardar Patel University, Vallabh-Vidyanagar,Anand,Gujrat

invited me to work for my Ph.D. on this topic. I got registered on14-9-2011, worked

hard and received Ph.D. on Dec.20,2014. LAMBERT Academic Publishing, Germany

published my thesis in the form of a book. These results were quite exiting as

extension in utility of this bound was possible after quite a long period after its

inception in 1952. During this, Five research papers are published in international

journals. I presented Two of them in national conferences one at North Maharashtra

University, Jalgaon on Jan.30-31,2014 and another at Sardar Patel University, V V

nagar on Feb.21-22,2014. My Ph.D. problem is motivated from the topic of project. As

my retirement was nearing, I wanted to finish up Ph.D. work first to enable me to

apply for extension in service without hampering my teaching work. Therefore, the

submission is delayed. I am extremely sorry for that. But, I want to express my

gratitude towards UGC and all the concerned persons for bearing with me and making

it possible for me to add to the knowledge of the subject, which provided me almost

everything in life, on the verge of the retirement. 13. SUMMARY OF THE FINDINGS : ( IN 500 WORDS ):

Let X1 ,X2, … ,Xn and Y1 ,Y2 , … ,Ym be two independent random samples

from continuous uniform probability distributions on (0, ) and (0, )

respectively. Let X and Y be the maximum values in these two samples

respectively. The problem is to estimate the value of . It is suspected but

not known for sure that .Therefore, we use the two samples to

test the hypothesis that Ho: . A likelihood ratio test is developed to

test this hypothesis against the alternative that H1: . The test is

Ф(t) = 1, if t < c1/n for 0< t ≤ 1 and if t > c-1/m for t>1.

=0,otherwise.

The properties of this test are studied. The probability distribution of the

test statistic T= X/Y is obtained. It comes out to be

(t)= (nm/(n +m))( )n t n-1, 0< t ≤ ( )

= (nm/(n +m))( ) m(1/t m+1) , ( ) < t < ∞.

If the hypothesis Ho is accepted, we use both the samples to estimate ;

otherwise, use just the first sample. The necessary estimator in case of

1 < 2 is found to be

If >

The properties this estimator are studied. The expressions of this estimator,

its bias and mean square error (MSE)s are computed. The values of the MSE

are compared with the variance of uniformly minimum variance unbiased

estimator (UMVUE) based on the single sample. It is found that for small

samples when ,the proposed estimator gives MSE less than the

variance of UMVU estimator. Some numerical values of various quantities

are provided in the form of tables and graphs to have the idea at a glance.

This provides the idea about the regions where the proposed estimator can

be profitably used instead of usual UMVU estimator. Careful computations

are needed as the estimator and the concerned quantities depend upon

the sample sizes, values of the parameters and the level of significance etc.

of the test. Of course, these things are common in preliminary test

estimation.

14.CONTRIBUTION TO THE SOCIETY ( GIVE DETAILS ): A test to test the hypothesis is developed. Needy persons would apply it. The parameter of interest is provided a better estimator in some situations. The statisticians would use it and find accurate results. The results are useful when one has to deal with boundary value. 15.WHETHER ANY PH.D. ENROLLED/PRODUCED OUT OF THE PROJECT: Yes. The UMVU estimator and attainable lower bound on the variance of the estimator of parameter of interest in the project as introduced by Jack Kiefer motivated me to extend

the Kiefer bound to various situations. I registered for Ph.D. and received it on Dec.20, 2014 from Sardar Patel University, Vallabh- Vidyanagar, Anand, Gujrat for my thesis entitled “Some Contributions to Kiefer Bound on Variance ”. The thesis is published in the form of a book by LAMBERT Academic Publishing, Heinrich-Bocking-Str.6-8,66121 Saarbrucken,Germany. Its ISBNumber is ISBN 978-3-659-67188-3. 16.NO. OF PUBLICATIONS OUT OF THE PROJECT ( PLEASE ATTACH RE-PRINTS): The publications are based on the results motivated from the project work, but do not include the exact results of the project. Therefore, I provide here the details of the publications instead of the reprints of the publications.

1. Jadhav,D.B. and Shanubhogue,A.(2014). Kiefer Bound in Truncated

Distributions. International Journal of Mathematics and Computer

Research, 2(6), 469-483.

2. Shanubhogue,A. and Jadhav,D.B.(2014). Attainable Kiefer Bounds Using

Censored Samples from Left Truncated Family of Distributions.

International Journal of Applied Mathematics and Statistical Sciences.

3(6), 57-61

3. Shanubhogue,A. and Jadhav,D.B.(2014). Lower Bounds on Variance of

Estimators: Past , Present and Future. International Journal of Science

and Research .3(11),1296-1298.

4. Shanubhogue,A. and Jadhav,D.B.(2014). Attainable Kiefer Bound Using

Censored Samples From Doubly Truncated Distributions (2014)

International Journal of Mathematics and Computer

Research,3(1),850-860.

5. Shanubhogue,A. and Jadhav,D.B.(2015). Attainable Kiefer Bounds Using

Censored Samples from Right Truncated Family of Distributions. Far

East Journal of Theoretical Statistics,50(1),21-30.

( PRINCIPAL INVESTIGATOR ) (REGISTRAR/PRINCIPAL)

INDEX

Section Topic Page

Number

1 Introduction 1

2 Likelihood Ratio Test To Test Ho: 1

3 Estimator To Estimate

4 Case – I 1 < 2

5 Case- II: >

6 Cases I and II Together

7 Comparison of Estimators

8 References

ESTIMATION OF PARAMETER OF RECTANGULAR (UNIFORM) PROBABILITY

DISTRIBUTION ON (0, ) BASED ON PRELIMINARY TEST AND TWO

SAMPLES

1.INTRODUCTION: Let X1 ,X2, … ,Xn and Y1 ,Y2 , … ,Ym be two independent

random samples from continuous uniform probability distributions on (0,

) and (0, ) respectively. Let X and Y be the maximum values in these

two samples respectively. The problem is to estimate the value of . It is

suspected but not known for sure that Therefore, we use the two

samples to test the hypothesis that . If the hypothesis is accepted,

we use both the samples to estimate ; otherwise, use just the first

sample. We provide here the necessary test and the estimator in this

situation. We study the properties of the test and the estimator. This

estimator is compared with the usual unbiased estimator based on

maximum likelihood estimator based on single sample.

2. LIKELIHOOD RATIO TEST TO TEST Ho: : The pdf’s of X and Y are

given by

(x)= (n/ n)xn-1 , 0<x< , >0 (2.1)

and (y)= (m/ m)ym-1 , 0<x< , >0. (2.2)

The joint pdf of (x,y) is

(x,y)=[mn/(( n)( m)) ]xn-1 ym-1 (x) (y) (2.3)

The parameter space is,

To test Ho: against H1: ,we have,

and

If and both are unknown, x and y are maximum likelihood

estimators (mles) of and respectively. Therefore,

Sup ( (x,y) = (nm/( xn ym)) xn-1 ym-1 = nm/(xy).

If , say, the mle of is max(x, y). Therefore,

Sup ( ) { (x,y) } = [nm xn-1 ym-1 /{ (max(x,y))m+n}]

Therefore, the likelihood ratio

L.R.(x ,y) =

=[[nm xn-1 ym-1 /{ (max(x,y))m+n}]/[nm/xy]]

= (y/x)m, if x ≥ y (2.4)

Putting T= X/Y, we have,

L.R.(x ,y)=tn, if t ≤ 1

= t-m, if t> 1.

Thus, the likelihood ratio test to test Ho against H1 is: Reject Ho iff L.R.(x

,y) < c , where, c is some constant so chosen that size of the test becomes

α. The test is equivalent to: Reject Ho iff L.R.(x,y)= LR(t) < c, i.e.; iff tn < c

for 0< t ≤ 1 and (1/tm) < c for t> 1.That is, reject Ho if t < c1/n for 0< t ≤ 1

and if t > c-1/m for t>1.

Thus, the test is

Ф(t) = 1, if t < c1/n for 0< t ≤ 1 and if t > c-1/m for t>1.

=0,otherwise. (2.5)

Applying equal tail criterion,

P Ho{ t < c1/n} = P Ho{ t > c-1/m} = α/2 (2.6)

To calculate probabilities in (2.6) we have to know the probability

distribution of T. For this, let T = X/Y and U = Y. If 0< t ≤ ( ), 0< u < .If

( ) < t < ∞, 0 < u < ( /t). X = TU. The Jacobian of transformation

J = (2.7)

The joint pdf of T and U, using (2.3) and (2.7) is

(u,t) = (n/ n)( m/ m)t n-1u n+m-1, 0 < t < ∞,0 < u < , 0 < tu <

(2.8)

= 0 , otherwise.

Integrating (2.8) w.r.t. u, we get, the marginal pdf of T as

(t)= (nm/(n +m))( )n t n-1, 0< t ≤ ( )

= (nm/(n +m))( ) m(1/t m+1) , ( ) < t < ∞. (2.9)

If , i.e.; under Ho, we have,

(t) = (nm/(n +m))t n-1, 0< t ≤ 1

= (nm/(n +m))(1/t m+1) , 1< t < ∞. (2.10)

Note that this is independent of .

If the samples are of equal size, i.e.; if n = m, (2.9) becomes,

(t)= (n/2)( )n t n-1, 0< t ≤ ( )

= (n/2)( ) m(1/t m+1) , ( ) < t < ∞. (2.11)

If , as well as n = m, this reduces to

h(t) = (n/2) t n-1, 0 < t ≤ 1

= (n/2) (1/ t n+1), 1< t < ∞. (2.12)

From (2.5), we have,

(nm/(n +m)) = (nm/(n +m)) = α/2.This gives

the critical region to be

Rα = {(0,( α(m +n)/(2m))(1/n))U((2n/ α(n+m))(1/m), ∞)}

= {(0, U( , ∞)} (2.13)

If m =n, the critical region reduces to, Rα = {(0, ) U ( , ∞ )}.(2.14)

The power of the test (2.5) is given by ,

βф(ѳ1,ѳ2)= E ѳ1,ѳ2(Ф(T))= P[ ( t < c1/n )U (t > c-1/m)]

= +

=

= α/2[( ѳ2/ ѳ1)n + ( ѳ1/ ѳ2) m ] = (2.15)

= , if n=m. ( 2.16)

3. ESTIMATOR TO ESTIMATE :

Consider the following estimator

T*= (3.1)

where Z= max(X,Y)

The pdf of Z is given by,

f(zIθ1,θ2)= ,θ1<θ2 (3.2)

And

f(zIθ1,θ2)= ,θ2≤ θ1 (3.3)

The expected value of Z is,

E(Z)= (3.4)

and

E(Z2)=

If θ1= θ2,

1. E(Z)= (3.5)

Which is expectation of maximum of (n+ m) observations.

2. E(Z2)=

Which is expectation of square of maximum of (m+ n) observations.

We have the p.d.f.s,

fX(x I θ1)=

fY(y I θ2)=

hT(t I θ1, θ2)=

hY,T(y,t I θ1,

θ2)= (3.6)

If and if 0< t <

h(yI t, θ1, θ2) = (3.7)

Let us put Y =

Jacobean of transformation is,

=

Therefore,

h(x,v I θ1, θ2) =

In our notation, V=T= , therefore,

h(x,t I θ1, θ2) =

(3.8)

0< 0 < x < also 0 < x < , therefore 0 < x < min { θ1, θ2}

If 0 < t <

If

Therefore, the conditioned density of X given T is given by,

h(x I t, θ1, θ2) = (3.9)

let A1 = (0, , A2 = ( and A= A1 U A2.

Consider the estimator T* defined earlier,

T* =

Thus, T* = X

Therefore,

E(T*) = E (X ) + E(

On A1, 0 < T < < , therefore,

E(X I t 1) =

E(X ) =

= (3.10)

E(X2 I t 1) = t2

E(X2 ) =

= (3.11)

In general,

E(Xr ) =

On A2 , t > and

E(X I t 2) =

E(X ) =

= (3.12)

E(X2 I t 2)=

E( X2 (t) ) = (3.13)

In general,

E( Xr (t) ) = , r > 0

But T*= X if Tє A1 or Tє A2 , i.e. if Tє A

Therefore actually we want,

E (X ) = E[X + X ]

= (3.14)

E(Xr (t) ) = E(Xr ) + E(Xr )

=

Consider,

E[(X – θ1)2] = E[X2 - 2θ1 X + ]

= E[X2 ]- 2 θ1E[X ] +

=

(3.15)

Z= max (T,1) , Ac= (c1,c2)

Note that , c1 < 1 < c2

4. CASE – I 1 < 2

Thus,

T*= X { (0,c1)(t) + (c2,∞)(t) } + X (1,c2)(t)+ Y (t) + Y (t)

(4.1)

Therefore, E(T*) = a+b+c+d;

Where, a = E[X { (0,c1)(t) + (c2,∞)(t) }], b = E[X (1,c2)(t)],

c = E[Y (t)], d = E[Y (t)]

a= E[X { (0,c1)(t) + (c2,∞)(t) }]

=

E[X2 { + }]=

b= E[X ] = E[ E[X I t є (1,c2)]]

=

=

E[X2 (t)] = .

Now, we consider

c= E[Y (t)] = E [ (t) E[Y I t є ]

= =

E[Y2 (t)]=

d= E[Y (t)] = E[ (t) E[Y I t є ]

=

=

E(Y2 (t))=

From a,b,c and d

E(T*)= +

+ +

Thus,

E(T*)= + +

(4.2)

In estimating by T* the bias is,

= E(T*) -

= + + -

(4.3)

We also have,

E(T*2) = E[ X2 { (t) + (t) }] + E [X2 (t) ] + E [ Y2

(t) ]

+E [ Y2 (t) ]

=

+

E(T*2) = + + (4.4)

The MSE (mean squared error) of T* in estimating is given by,

E(T*- 1)2= E(T*)2 - 2 1 E(T*) + 12=

–

(4.5)

If m=n,

= (4.6)

where, .

5.CASE- II: >

In this situation,

Thus,

= X + Y ( ,1)(t) + X (t) + X (t) (5.1)

= (I) + (II) + (III) + (IV)

E(I) = E[X (0, )(t)]

= E[ (0, )(t) E{X I t є (0, )} ]

= , =

=

E(I2) = E[X2 (0, )(t)]

=

E(II) = E[ ( ,1)(t) E{Y I t є ( ,1)} ]

=

=

=

=

E(II2) = E [Y2 ( ,1)(t) ]

=

E(III) = E [ (t) E{X I t є (1, )}]

=

=

E(III2)= E[X2 (t)]

=

E(IV)= E [X (t)]

= E [ (t) E{X I t є } ]

=

=

E(IV)=

E(IV2)= E [X2 (t)]

=

=

Thus,

E( = + +

+ .

=

E( =

(5.2)

E( = + +

+

=

E( =

(5.3)

In this situation, i.e., when > the bias of in estimating is

given by,

= E(T*) -

= + - (5.4)

Also

E(T*- )2= E(T*)2 - 2 E(T*) + 2

= -

= –

(5.5)

6. CASES I AND II TOGETHER.

Thus, from the cases I and II above we conclude that,

Its bias is given by,

(6.2)

The MSE of is given by,

E(T*- )2

(6.3)

Where

b2=

(6.4)

If n=m, c1= , c2= and

(6.5)

If α=0, gives always pooled estimator and c1=0, c2=∞. This

gives

(6.6)

If n=m, this becomes,

(6.7)

In this putting , we have

: the bias of the maximum of a sample of size 2n.

If m=n the MSE of is given by,

E(T*-

)2 (6.8)

Where

If α=0, in this, then, =0 and

E(T*- )2 (6.9)

(6.10)

(6.10)is the MSE of the maximum in the 2n observations, if .

If α=1, m=n we have c1=c2=1 and becomes never pool estimator. In this

case

(6.11)

If , this becomes,

. (6.12)

(6.12) gives bias of the maximum in n observations in single sample.

If m=n, α=1,

b2=

If , this becomes,

b2=

If n=m, α=1,

E(T*- )2 : MSE of maximum of sample of size n (i.e., of X).

Note that,

, V n≥1 and >0.

That is, MSE of proposed estimator T* is smaller than that of the sample

maximum of size n V n≥1 and >0.

To compare the values of MSE of the proposed estimator T* and the

variance of UMVU Estimator [(n+1)/n]X

7. COMPARISON OF ESTIMATORS

1. θ2=1

θ1 n m mse var

0.4 3 8 0.155928 0.010667

0.5 3 8 0.141231 0.016667

0.6 3 8 0.097822 0.024

0.7 3 8 0.058256 0.032667

0.8 3 8 0.032224 0.042667

0.9 3 8 0.022331 0.054

1 3 8 0.027665 0.066667

1.1 3 8 0.181955 0.080667

1.2 3 8 0.183311 0.096

1.3 3 8 0.193911 0.112667

2. θ2=1

θ1 n m mse var

0.6 5 7 0.065936 0.010286

0.7 5 7 0.047346 0.014

0.8 5 7 0.026049 0.018286

0.9 5 7 0.016461 0.023143

1 5 7 0.019571 0.028571

1.1 5 7 0.091949 0.034571

1.2 5 7 0.088216 0.041143

1.3 5 7 0.091916 0.048286

1.4 5 7 0.100039 0.056

3. θ2=1

θ1 n m mse var

0.8 8 3 0.015295 0.008

0.9 8 3 0.013654 0.010125

1 8 3 0.016591 0.0125

1.1 8 3 0.036653 0.015125

1.2 8 3 0.036585 0.018

1.3 8 3 0.039823 0.021125

1.4 8 3 0.044726 0.0245

1.5 8 3 0.050627 0.028125

1.6 8 3 0.057235 0.032

1.7 8 3 0.064418 0.036125

1.8 8 3 0.072113 0.0405

1.9 8 3 0.080288 0.045125

2 8 3 0.088928 0.05

2.1 8 3 0.098023 0.055125

2.2 8 3 0.107569 0.0605

2.3 8 3 0.117564 0.066125

2.4 8 3 0.128005 0.072

2.5 8 3 0.138891 0.078125

2.6 8 3 0.150224 0.0845

2.7 8 3 0.162001 0.091125

2.8 8 3 0.174222 0.098

2.9 8 3 0.186889 0.105125

1. θ2=1

θ1 n=m mse var

0.4 3 0.084078 0.010667

0.5 3 0.084268 0.016667

0.6 3 0.061419 0.024

0.7 3 0.042279 0.032667

0.8 3 0.033758 0.042667

0.9 3 0.036705 0.054

1 3 0.049507 0.066667

1.1 3 0.157842 0.080667

1.2 3 0.167972 0.096

1.3 3 0.18439 0.112667

1.4 3 0.205548 0.130667

1.5 3 0.230507 0.15

1.6 3 0.258679 0.170667

2. θ2=1

θ1 n=m mse var

0.6 5 0.053385 0.010286

0.7 5 0.040361 0.014

0.8 5 0.024444 0.018286

0.9 5 0.018647 0.023143

1 5 0.023456 0.028571

1.1 5 0.086482 0.034571

1.2 5 0.085198 0.041143

1.3 5 0.090243 0.048286

1.4 5 0.099125 0.056

1.5 5 0.110572 0.064286

1.6 5 0.123901 0.073143

1.7 5 0.138733 0.082571

3. θ2=1

θ1 n=m mse var

0.7 8 0.027597 0.006125

0.8 8 0.020979 0.008

0.9 8 0.010685 0.010125

1 8 0.011014 0.0125

1.1 8 0.044467 0.015125

1.2 8 0.039973 0.018

1.3 8 0.041341 0.021125

4. θ2=1

θ1 n=m mse var

0.9 15 0.006613 0.003176

1 15 0.003685 0.003922

1.1 15 0.014804 0.004745

1.2 15 0.012066 0.005647

References

Bancroft,T.A. and Han,Chaen-Pai(1977). Inferences based on Conditional

Specification: a Note and a Bibliography, International Statistical

Review,Vol.45,pp-117-127.

Hogg,R.V.(1956). On the distribution of the likelihood ratio.Annals of

Mathematical Statistics,Vol.27,pp-529-532.

Hogg,R.V.(1961). On the resolution of statistical hypotheses. Journal of

American Statistical Association,Vol.56,pp-978-989.

Kiefer,J.(1952).On minimum variance estimators. Annals of Mathematical

Statistics,23,627-629.