M.Sc. (Semester II) (CBCS) Examination Oct/Nov-2017
Statistics
THEORY OF TESTING OF HYPOTHESES
Day Date: Wednesday, 22-11-2017 Max. Marks: 70
Time: 10:30 AM to 01.00 PM
Instructions: Attempt five questions.
Q. No. and Q. No are compulsory.
Attempt any three from Q. No. to Q. No.
Figures to the right indicate full marks.
Q.1 Choose correct alternative: 05
A test for testing against is called level test if:
Size of test does not exceed
Size of test is exactly equal to
Hypothesis of the test is simple hypothesis
The test is unbiased.
Let are iid with 1). Let and For any
0
There exists a UMP level test.
There does not exists a UMP level test.
There exists a test with one sided
None of these
Consider a one parameter exponential family θ with
probability function . When the
family has monotone likelihood ratio in
u is decreasing function of
is non-decreasing function of
is decreasing function of
is non- decreasing function of
The test with Neyman- structure is
Similar test
Not a subset of similar test
A subset of similar test
None of these
5)c For testing simple null against simple alternative hypothesis which of the
following statement is most appropriate?
UMP level test exists. UMPU level test exists.
UMP invariant test exists MP level test exists.
Fill in the blanks. 05
If hypothesis H is simple then probability of type I error of any test is
quantity.
When and are simple then the LRT will be same as test.
If is a likelihood ratio for testing against where
is scalar, then asymptotic distribution of is
UMAU confidence intervals are obtained from tests.
The symmetric kernel associated with the population variance is
Page 2 of 2
SLR-MS-646
State the following sentence are True or False: 04
If ∅ is MP test of level then probability of type I error exceed the level

Cauchy distributions with location parameter and scale parameter
unity possess the MLR property.
If ∅ is a test function, then ∅ is also a test function.
The test obtained by Neyman-Pearson lemma is an unbiased test.
Q.2 Define: 06
Size of test
Level of significance
Randomized test
Write short notes on the following: 08
Wilcoxon signed-rank test
Tests with Neyman structure
Q.3 Define most powerful test. Explain the method of obtaining MP test of
size for testing simple hypothesis against simple alternative.
07
Obtain a most powerful test of size for testing against
based on a random sample of size n from where is
known.
07
Q.4 When a family of densities is is said to have monotone likelihood ratio?
Show that the one-parameter exponential family of densities belongs to this
class of MLR densities.
07
Let … be a random sample drawn from uniform distribution
Find UMP size test for testing against .
07
Q.5 Define confidence set and UMA confidence set of level − Derive the
relationship between UMA confidence set and UMP test.
07
Let … be a random sample from exponential distribution with
. Consider the testing of hypothesis problem against
Find UMA(1 − level family of confidence sets corresponding
to size UMP test.
07
Q.6 Describe goodness of fit test based on Chi-square distribution. 07
Let … be a random sample from normal density with variance 1 and
unknown mean Find out likelihood ratio test of hypothesis 3
against the alternative 3 .
07
Q.7 State one sample and two sample U statistic theorems. 07
State and prove a necessary and sufficient condition for a similar test to
have Neyman structure.