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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034    M.Sc. DEGREE EXAMINATION – STATISTICSSECOND SEMESTER – NOVEMBER 2012ST 2812 – TESTING STATISTICAL HYPOTHESES
Date : 03/11/2012 Dept. No.   Max. : 100 Marks    Time : 1:00 – 4:00
SECTION – A
Answer  ALL  questions.  Each  carries TWO  marks:      (10 x 2 =  20 marks)
1.   Discuss the objective of Statistical Inference.
2.   Explain Interval Estimation as an infinite decision problem with many possible correct       decisions.
3.  Give the mathematical formulation of symmetry or invariance.
4.   Describe minimax procedure.
5.   Distinguish between size and level of significance of a test.
6.   Show that one parameter exponential family possesses MLR property.
7.   Define Similar test.
8.   State Generalized Neyman-Pearson Theorem.
9.   When do we say function is Maximal Invariant?
10.  Define likelihood ratio test statistic and state the test criterion.
SECTION – B
Answer any FIVE questions.  Each carries EIGHT marks:      (5 x 8 = 40 marks)
11.  Let β denote the power of a most powerful test of level α for testing simple hypothesis H       against simple alternative K.  Prove that  (i) β  ≥  α   (ii) α < β unless p0 = p1.
12.  Let X1, … , Xn be a random sample from N(μ, σ2), where σ2 is known.  Derive UMPT       of  level α for testing H: μ ≤ μ0 versus K: μ > μ0.
13.  Let ‘T’ denote the time required to get ‘r’ events in an inverse Poisson sampling with        process average rate λ.  Show that the minimum number of events to be observed is       r = 14 in order to get a power greater than 0.8 at λ = 1.5 in a UMPT of level α = 0.05       for testing H: λ ≤ 0.5 versus K: λ > 0.5.

14.  Obtain the UMPUT for H: λ = λ0 versus K: λ ≠ λ0 in the case of Poisson distribution        and deduce the ‘side conditions’ that are required to be satisfied.
15.  If X ~ B(m, p1) and Y ~ B(n, p2) and are independent,  then compare the two Binomial       populations using UMPUT for H: p1 ≤ p2 versus K: p1 > p2.
16.  If the power function of every test function is continuous in θ, then show that any unbiased        test  is similar on the boundary.
17.  Define unbiased test and UMP test.  Hence if there exists UMPT of level α for testing a            composite H against a composite K, then show that it is unbiased.
18.  Explain locally most powerful unbiased level α test with an example.
SECTION – C
Answer  any TWO  questions.  Each  carries TWENTY  marks.     ( 2 x 20 =  40 marks)
19.  Let X ~ Pθ with MLR in T(x).  Establish the existence of UMPT for H: θ ≤ θ0 versus       K: θ > θ0 and obtain its form. β  (θ)  Show that its power β  (θ) strictly increases for all θ       for which o <  β  (θ) < 1.  Also show that the test minimizes β  (θ)  θ < θ0.
20.  Let X1, … , Xn be a random sample from U(0, θ), θ > 0.  For testing H: θ ≤ θ0 versus       K: θ > θ0, show that there exists more than one UMPT.  In the same problem, obtain       UMPT for H: θ ≥ θ0 versus K: θ < θ0.
21.  Let X1, … , Xn be a random sample from E(a, b) where both ‘a’ and ‘b’  are unknown.       Using the test statistic T = U / V, where  U = X(1)  –  a0 and  V =  ,        obtain the power function of the level α  test                                   = {█(1  when T  ≤C_1@ 0  otherwise       )┤
for testing  H: a = a0 versus KL: a < a0.  Also obtain the power function of the level α test
= {█(1  when T  ≥C_2@ 0  otherwise       )┤
for testing  H: a = a0 versus KR: a > a0.
22(a) Let X1, … , Xn be a random sample from N(μ, σ2), with both parameters unknown.         Derive the LRT of level α for testing H: σ2 = σ_0^2 versus K: σ2 ≠ σ_0^2.              (10)        (b) Consider the test for H: θ = θ0 versus K: θ ≠ θ0 based on a random sample of  size ‘n’          from a distribution in the Cramer family.  Derive the  asymptotic null distribution of           the  LRT statistic.

 

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