LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034.

M.Sc. DEGREE EXAMINATION – STATISTICS

FourTH SEMESTER – APRIL 2003

St 4953 / s 1072  –  mathematical statistics -ii

 

26.04.2003

1.00 – 4.00                                                                                                      Max : 100 Marks

 

                                                                section – A                                (10´ 2=20 marks)

      Answer ALL the questions. Each carries two marks.

 

1. Define empirical distribution function.
2. For the sample central moment compute E (b₂).
3. Let X₁, X_2, …, X_n be a sample from a Compute the .
4. Define t and F statistics.
5. Let X₁, X₂, X₃, X₄ be iid random variables with pdf f. Find the marginal pdf of the order statistic X₍₂₎ .
6. Let X₁, X₂, … be iid N random variables. Show that is an unbiased and consistent estimator of  .
7. Let X ~ N (0,q). Check if the family of pdf’s is complete.
8. State Neyman-Pearson fundamental lemma.
9. Define a likelihood ratio test.
10. Define a process with independent increments.

 

 

 

                                                         section – B                                         (5´ 8=40 marks)

      Answer any FIVE questions. Each carries EIGHT marks.

 

11. Let X₍₁₎, X₍₂₎, X₍₃₎ be the order statistics of iid random variables X₁, X₂, X₃ with common pdf

Let Y₁ = X₍₃₎ – X₍₂₎ and Y₂ = X₍₂₎ . Show that Y₁ and Y₂ are independent.

12. State and prove factorization criterion for determining sufficient statistics.
13. State and prove Rao-Blackwell theorem.
14. Let X₁, X₂, …,X_n be iid random variables, where both n and p are unknown. Find the estimates of p and n by the method of moments.
15. Let X₁, X₂, …,X_n be a sample from U [q -, q + ]. Show that the statistic T(X₁,X₂,…,X_n) such that max X_i – T (X₁, X₂, …, X_n) min X_i +

 

is an MLE of q .

 

16. Let X₁, X₂,…,X_n ~ U[0, q], q>0. Show that the family of uniform densities on [0,q] has
    an MLR in max

 

17. (a)  Explain Chi-square test of goodness of fit.

• Explain normal test for single proportion.

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

• State the characteristics of the Brownian motion process.

 

 

 

                                                        section – C                                         (2´20=40 marks)

Answer any TWO questions. Each carries TWENTY marks.

 

19. (a) Derive the distribution of   in sampling from a normal population.

• Derive the pdf of Chi-square distribution.

20. (a) State and prove Cramer-Rao inequality.

• Let X ~ P (). Find the UMVUE of based on a sample of size one.

21. (a)  Let X ~ N . Obtain a 100 confidence interval for based on a
    random sample of size n.

• Let X ~ N (0, 1) under H₀ and X ~ C (1, 0) under H₁. Find an MP size  test of H₀ against H₁, and obtain the power of the test.
• Explain F – test for the equality of population variances.

22. (a) Establish Chapman – Kolmogorov equation and hence show that the   is
    the power of 1-step tpm.

• State the postulates for Poisson process.
• Show that if {N (t)} is a Poisson process, then for , the conditional distribution of N (s) given N (t) = n is binomial .

 

 

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