M.Sc. (Semester III) (CBCS) Examination Oct/Nov-2017
Statistics
ASYMPTOTIC INFERENCE
Day Date: Thursday, 16-11-2017 Max. Marks: 70
Time: 02.30 PM to 05.00 PM
Instructions: Attempt five questions
Q.1 and Q.2 are compulsory.
Attempt any three questions from Q. 3 to 7.
Figures to the right indicate full marks.
Q.1 Select the correct alternatives of the following questions: 05
For sufficiently large sample size, the likelihood equation admits
Unique consistent solution
Two consistent solutions
More than two consistent solutions
No consistent solution
The asymptotic distribution of Wald's statistics is
Normal t
Chi-square F
The variance stabilizing transformation for Poisson population is
Logarithmic Tanh-1
Sin-1 Square root
Let … be iid and
i
. Then which of the
following statement is true.
is unbiased for is consistent for
is CAN for is not MLE of
In a random sample of size n from Poisson distribution, MLE of
was reported to be 2. The variance of the asymptotic normal
distribution of − is estimated by



Q.1 Fill in the blanks: 05
For distribution belonging to one parameter exponential family, the
asymptotic variance of MLE of is
Test based on score functions was proposed by
For Cauchy distribution the consistent estimator of is
Exponential class of densities is of Cramer class.
Let … be iid CAN estimator of is
Q.1 Write whether the following statements True or False: 04
Consistent estimator is always unique.
Exponential distribution with location is not a member of Cramer
family.
Every consistent estimator is BAN.
Based on random sample … from is consistent
for
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SLR-MS-654
Q.2 Explain the following: 06
Weak Consistency
Strong consistency
Mean squared error consistency
Write short notes on the following: 08
CAN and BAN estimators in multiparameter case.
Goodness of fit test.
Q.3 Define marginal consistency and joint consistency for a vector parameter.
Show that joint consistency is equivalent to marginal consistency.
07
Let … be iid exponential with location Let be the smallest
order statistic. Show that is consistent but not CAN for .
07
Q.4 Under certain regularity conditions (to be stated) show that the likelihood
equation estimator is CAN for
07
Let … be iid Obtain CAN estimator of . 07
Q.5 In case of one parameter exponential family, show that the moment
estimator based on sufficient statistic is CAN for the parameter.
07
Let … be iid Pareto with p.d.f.

0. Show
that p.d.f. belongs to one parameter exponential family and moment
estimator of based on sufficient statistic is CAN for with asymptotic
variance equal to CRLB.
07
Q.6 Explain variance stabilizing transformation and illustrate its use in the
construction of large sample estimation and tests.
07
Let … be iid from . Obtain 100(1 confidence interval
for based on variance stabilizing transformation.
07
Q.7 Derive the asymptotic distribution of likelihood ratio test statistic. 07
Describe asymptotic confidence interval based on CAN estimator. Let be
… be iid from Obtain 100 asymptotic confidence
interval for based on CAN estimator of