LOYOLA COLLEGE (AUTONOMOUS), CHENNAI –600 034

M.Sc., DEGREE EXAMINATION – STATISTICS

  FOURTH SEMESTER – APRIL 2004

ST 4953 – MATHEMATICAL STATISTICS – II

12.04.2004                                                                                                           Max:100 marks

1.00 – 4.00

SECTION – A

 

Answer ALL questions                                                                                (10 ´ 2 = 20 marks)

 

1. Define a consistent estimator and give an example.
2. Show that unbiased estimators do not always exist.
3. Let X₁, X₂, …, X_n be iid b (1,p) random variables.

Show that T =  is sufficient for p.

4. State Lehmann – Scheffe theorem.
5. State Bhattacharya Inequality.
6. Write the test function associated with i) a non-randomized test ii) a randomized test.
7. Define UMP test for testing a simple hypothesis against a composite hypothesis.
8. Write any four applications of chi-square distribution.
9. State the postulates for Poisson process.
10. Define Markov chain and give an example.

 

SECTION – B

 

Answer any FIVE questions                                                                        (5 ´ 8 = 40 marks)

 

11. Let X₁, X₂,…, X_n be a random sample from N (m, s²). Show that  and S² are independent.
12. Derive the pdf of F-distribution.
13. Let X₁, X₂, …., X_n be a random sample from a distribution of continuous type with pdf f(x; q). Derive the joint pdf of jth and kth order statistics, 1 £ j < k £
14. State and prove Factorization Criterion for determining sufficient statistics.
15. State and prove Rao-Blackwell theorem.
16. Let X₁, X₂, …, X_n be a random sample from Poisson distribution that has the mean q > 0.  Show that  is an efficient estimator of q.
17. Let X₁, X₂,.., X_n be iid N (m, s²) random variables where both m and s² are unknown. Obtain a MP test for testing H_o­: m = m_o­ ; s² =  against H₁: m = m₁; s² = .
18. Show that if {N(t)} is a Poisson process, then for s< t, the conditional distribution of N(s) given N(t) = n is binomial b (n, ).

 

 

SECTION – C

 

Answer any TWO questions                                                                        (2 ´ 20 = 40 marks)

 

19. a) State and Prove Cramer-Rao Inequality.            (10)

 

1. b) Let X₁, X₂, …, X_n be iid N(m, s²) random variables. Obtain a confidence interval for m

when (i) s² is known (ii) s² is unknown.                                                                    (10)

 

20. a) State and prove Neyman-Pearson lemma.            (10)

 

1. b) Let X₁, X₂, …, X_n be a random sample for N(m, s²) where both m and s² are unknown.

Derive the likelihood ratio test for testing H_o: m = m_o against H₁ : m ¹ m_o.                  (10)

 

21. a) Let X₁, X₂,…, X_N be iid b (n, p) random variables, where n and p are unknown. Find

the method of moments estimator for (n,p).                                                                 (7)

 

1. b) Let X₁, X₂, …,X_n be a sample for U [q – , q + ]. Show that the maximum

likelihood estimator of q is not unique.                                                                        (7)

 

1. c) Explain normal test of significance for single mean and give an example. (6)

 

22. a) Classify the stochastic processes with respect to time and state space. (2)

 

1. b) State the characteristics of the Brownian motion process (4)

 

1. c) Establish Chapman – Kolmogorov equaion and hence show that the m – step tpm is the

mth power of 1 – step tpm.                                                                                           (8)

 

1. d) Explain chi-square test for goodness of fit and give an example. (6)

 

 

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