LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 34

SECOND SEMESTER – APRIL 2006

                                         ST 2809 – TESTING STATISTICAL HYPOTHESIS

(Also equivalent to ST 2807/2802)

 

 

Date & Time : 21-04-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

SECTION  A                           Answer all the questions                            10 x 2 = 20

1. Define test function and randomized test function.
2. Let X be B(1, q), q = 0.2,0.4,0.5. For testing H: q = 0.2,0.4 Vs K: q = 0.5, a test is given by

f(x)   =  0.3,    x = 0

=  0.6,    x =  1.

Find the size of the test.

3. Show that a UMP level a test is unbiased.
4. Define MLR property and give an example.
5. Show that a test with Neyman structure is similar.
6. Describe Type I and Type II right censoring.
7. Give two examples for multiparameter exponential family.
8. Define location family and give an example.
9. Describe likelihood ratio test.
10. Explain UMA and UMAU confidence sets.

 

SECTION B                            Answer any five questions                           5 x 8 = 40

11. Let X be DU{1,2,…, q }, q = 1,2. For testing H: q = 1 Vs K: q = 2,  find MP level a test using LP technique.
12. Give an example of a testing problem for which UMP test does not exist.
13. Given a random sample of size n from E(0, q ), q > 0, derive UMP level a test for testing H: q £ q ₀ Vs K: q > q ₀.Examine whether the test is consistent.
14. If the power function of an unbiased test is continuous, show that the test is similar.

15.Given a random sample of size n from P( q ), q > 0, derive UMPU level a test for testing H: q = q ₀ Vs K: q ¹ q _0.

16.Show that a statistic is invariant if and only if it is a function of a maximal invariant statistic.

17.Derive likelihood ratio test for testing H: q = q ₀ Vs K: q > q ₀ based on a random sample from E(0,q), q >0.

18.Explain shortest length confidence interval and illustrate with an example.

 

 

 

SECTION C                           Answer any two questions                         2 x 20 = 40

19 a).   State and establish the sufficient part of Neyman-Pearson lemma.

1. b) Let X₁,X₂,…X_n denote a random sample of size n from E(q ,1), q e Examine if there exists UMP level a test for testing H: q = q ₀ Vs K: q ¹ q ₀.

20 a)  In the case of one-parameter  exponential family show that there exists UMP level a  test for testing one-sided hypothesis against one-sided alternative. State your assumptions.

1. b) Derive UMPU level a test for testing H:  q₁ £ q £ q₂ Vs K: q < q₁  or q > q                               based on a random sample from N(q , 1), q e R. Explain the determination of the constants.Is the test unique?

21 a)   Discuss the relation between similar tests and tests with Neyman structure.

3. b) Let X₁,X₂,…X_n  be a random sample from P( ) and Y₁,Y₂,…Y_m be a random sample from an independent  Poisson population P( ).Derive UMPU level a test for testing H:l £ m  Vs K:l > m. Determine the constants when  n = 2 and m = 1, X₁ = 1, X₂ = 2 and Y₁ = 3.

22 a)   State and establish the asymptotic null distribution of the likelihood ratio statistic.

1. b) For testing H:(X₁ , X₂ ) is BVN(q, q ,1,1, 0.5) Vs K: (X₁ , X₂ ) is BVN(q, q,1, 4, 0.5), derive UMPI level a test with respect to location transformations.

 

 

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