LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

SECOND SEMESTER – APRIL 2012

ST 2812 – TESTING STATISTICAL HYPOTHESES

 

 

Date : 19-04-2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION – A

 

Answer  ALL  questions.  Each  carries TWO  marks:                                                 (10 x 2 =  20 marks)

 

1. How do the Loss and Risk functions quantify the consequences of decisions?

 

2. Specify the three elements required for solving a decision problem.

 

3. Describe a situation where the decision rule remains invariant or symmetric.

 

4. Define Bayes Rule and Bayes Risk.

 

5. Illustrate that the consequences of Type I error and Type II error are quite different.

 

6. Define Most Powerful Test of level α.

 

7. Write UMPT for one parameter exponential family for testing

(i)  H: θ ≤ θ₀ versus K: θ > θ₀ when Q (θ) is increasing

(ii) H: θ ≥ θ₀ versus K: θ < θ₀ when Q(θ) is decreasing.

 

8. When do we say that a test  has Neyman Structure?

 

9. State any two asymptotic results regarding likelihood equation solution.

 

10. What is an invariant test?

 

SECTION – B

 

Answer any FIVE questions.  Each carries EIGHT marks:                                              (5 x 8 = 40 marks)

 

11. Distinguish between randomized and non-randomized tests and give an example for

each test.

 

12. Let ‘N’ be the size of a lot containing ‘D’ defectives, where ‘D’ is unknown. Suppose a

sample of size ‘n’ is drawn and the number of defectives ‘X’ in the sample is observed.

Obtain UMPT of level α for testing H: D ≤ D₀ versus K: D > D₀.

 

13. Let ‘X’ denote the number of events observed during a time interval of length ‘τ’ in a

Poisson process with rate ‘λ’.  When τ = 1, at 5% level, find the power at λ = 1.5 of the

UMPT for testing H: λ ≤ 0.5 versus K: λ > 0.5.

 

14. Obtain the UMPUT for H: p = p₀ versus K: p ≠ p₀ in the case Binomial distribution

with known ‘n’ and deduce the ‘side conditions’ that are required to be satisfied.

 

15. State and prove a necessary and sufficient condition for similar tests to have Neyman

structure.

 

16. If X ~ P (λ₁) and Y ~ P (λ₂) and are independent, then compare the two Poisson populations

through UMPUT for H: λ₁ ≤ λ₂ versus K: λ₁ > λ₂, by taking random sample from P (λ₁) and

P (λ₂) of sizes ‘m’ and ‘n’ respectively.

 

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

 

18. Using a random sample of size ‘n’ from N(μ, 1), derive the likelihood ratio test of level α

for testing H: μ = 0 against K: μ ≠ 0.

 

SECTION – C

 

Answer  any TWO  questions.  Each  carries TWENTY  marks:                               ( 2 x 20 =  40 marks)

 

19. State and prove the existence, necessary and sufficiency parts of Neyman-Pearson

Fundamental Lemma.

 

20(a) For a two decision problem, with zero loss for a correct decision, prove that every

minimax procedure is unbiased.                                                                        (10)

 

(b) Prove that an unbiased procedure is minimax if P_θ(A) is a continuous function of θ

for every event ‘A’ and there is a common boundary point of Θ₀ and Θ₁.        (10)

 

21. Let X₁, … , X_n be a random sample from E(a, b), where ‘a’ is unknown and ‘b’ is

known.  Using the UMPT for testing H: θ = θ₀ versus K: θ ≠ θ₀ in U(0, θ), obtain the

UMPT for testing H: a = a₀ versus K: a ≠ a₀ and find its power function.

 

22(a) Derive the conditional UMPUT of  level α for testing the independence of  attributes

in a 2 x 2 contingency table.                                                                               (16)

 

(b) Discuss the criteria for choosing the value of significance level α.                     (4)

 

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