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)
- Define a consistent estimator and give an example.
- Show that unbiased estimators do not always exist.
- Let X1, X2, …, Xn be iid b (1,p) random variables.
Show that T = is sufficient for p.
- State Lehmann – Scheffe theorem.
- State Bhattacharya Inequality.
- Write the test function associated with i) a non-randomized test ii) a randomized test.
- Define UMP test for testing a simple hypothesis against a composite hypothesis.
- Write any four applications of chi-square distribution.
- State the postulates for Poisson process.
- Define Markov chain and give an example.
SECTION – B
Answer any FIVE questions (5 ´ 8 = 40 marks)
- Let X1, X2,…, Xn be a random sample from N (m, s2). Show that and S2 are independent.
- Derive the pdf of F-distribution.
- Let X1, X2, …., Xn 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 £
- State and prove Factorization Criterion for determining sufficient statistics.
- State and prove Rao-Blackwell theorem.
- Let X1, X2, …, Xn be a random sample from Poisson distribution that has the mean q > 0. Show that is an efficient estimator of q.
- Let X1, X2,.., Xn be iid N (m, s2) random variables where both m and s2 are unknown. Obtain a MP test for testing Ho: m = mo ; s2 = against H1: m = m1; s2 = .
- 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)
- a) State and Prove Cramer-Rao Inequality. (10)
- b) Let X1, X2, …, Xn be iid N(m, s2) random variables. Obtain a confidence interval for m
when (i) s2 is known (ii) s2 is unknown. (10)
- a) State and prove Neyman-Pearson lemma. (10)
- b) Let X1, X2, …, Xn be a random sample for N(m, s2) where both m and s2 are unknown.
Derive the likelihood ratio test for testing Ho: m = mo against H1 : m ¹ mo. (10)
- a) Let X1, X2,…, XN be iid b (n, p) random variables, where n and p are unknown. Find
the method of moments estimator for (n,p). (7)
- b) Let X1, X2, …,Xn be a sample for U [q – , q + ]. Show that the maximum
likelihood estimator of q is not unique. (7)
- c) Explain normal test of significance for single mean and give an example. (6)
- a) Classify the stochastic processes with respect to time and state space. (2)
- b) State the characteristics of the Brownian motion process (4)
- c) Establish Chapman – Kolmogorov equaion and hence show that the m – step tpm is the
mth power of 1 – step tpm. (8)
- d) Explain chi-square test for goodness of fit and give an example. (6)
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