LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

YB 37

SECOND SEMESTER – April 2009

ST 2812 / 2809 – TESTING STATISTIACAL HYPOTHESIS

 

 

 

Date & Time: 22/04/2009 / 1:00 – 4:00       Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION A

Answer all questions.                                                                                    (10 x 2 = 20)

 

1. Define level and power of a test.
2. Let X be a random variable with pdf .

Obtain the Most Powerful Test of size for testing H₀: θ = 1 Vs H₁: θ = 2.

1. Give the general form of (k+1) parameter exponential family of distributions.
2. Define Uniformly Most Powerful Test.
3. Let. Consider the test function

for testing H₀: θ = 0.2 Vs H₁: θ > 0.2.Obtain the value of power function at

θ = 0.4.

1. What are the circumstances under which Locally Most Powerful test is used?
2. What is meant by shortest length confidence interval?
3. Define maximal invariant function.
4. What is meant by nuisance parameter? Give an example.
5. Define Likelihood Ratio Test.

 

 

SECTION B

Answer any FIVE questions.                                                                 (5 x 8 = 40)

 

1. Let  denote a random sample fromDerive a Most Powerful test of  level 0.05 for testing Vs. Also obtain the       cut-off point.
2. Show that the family of densities possesses MLR property.
3. Let denote a random sample of size n from. Consider the problem of testing Vs. Show that UMP test of  does not exist.
4. For (k+1) parameter exponential family of densities, derive an unconditional UMPUT of level for testing  Vs  clearly stating the assumptions.
5. State and prove the sufficient part of Generalized Neyman-Pearson lemma.
6. Show that any test having Neyman structure is similar. Also show that the converse is true under certain assumptions (to be stated).

 

 

1. Derive the Locally Most Powerful test for testing Vs based on a random sample of size n from, where  and  are known pdf’s.
2. Find maximal invariant function under the group of i.) Location transformations and ii.) Scale transformations.

 

SECTION C

Answer any TWO questions.                                                                 (2 x 20 = 40)

 

1. a.) Derive a UMP test of level  for testing  Vs  for the family of densities that possess MLR in T(x). Show that the power function of the above testing problem increases in

b.) Show that any UMP test is always UMPUT.                                          (16+4)

1. Consider a one parameter exponential family with density. Assume  is strictly increasing in. Derive a UMP test of level  for testing  Vs.
2. Let X and Y be independent Binomial variables with parameters and  respectively, where m and n are assumed to be known. Derive a conditional UMPUT of size  for testing  Vs.
3. Let anddenote independent random samples from  and respectively. Derive the Likelihood Ratio Test for testing Vs.

 

 

 

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