LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034                    B.Sc. DEGREE EXAMINATION – STATISTICS

AB 13

FIFTH SEMESTER – NOV 2006

ST 5500 – ESTIMATION THEORY

(Also equivalent to STA 505)

 

 

Date & Time : 25-10-2006/9.00-12.00         Dept. No.                                                       Max. : 100 Marks

 

 

 

Part A

Answer all the questions.                                                                        10 X 2 = 20

 

 

1. Define bias of an estimator in estimating the parametric function.
2. Explain efficiency of an estimator.
3. Explain Uniformly Minimum Variance Unbiased Estimator (UMVUE).
4. What is Cramer-Rao lower bound?
5. Define bounded completeness.
6. State Bhattacharyya inequality.
7. Let X_1, X₂ denote a random sample of size 2 from B (1, θ), 0 < θ < 1. Show that    X₁ + 3X₂ is sufficient for θ.
8. Describe sufficient statistic.
9. Explain Bayes estimation.
10. What is BLUE?

Part B

Answer any five questions.                                                                           5 X 8 = 40

 

1. Let X_1, X₂, … , X_n denote a random sample of size n from B(1, p), 0 < p < 1. Suggest an unbiased estimator of (i)  p and  (ii).  p (1- p).
2. If T_n asymptotically unbiased with variance approaching zero as n ® ¥ then show that T_n is consistent.
3. State and establish Factorization Theorem in the discrete case.
4. Show that the family of Bernoulli distributions { B(1,p), 0< p < 1} is complete.
5. State and establish Lehmann-Scheffe theorem.
6. Let X₁, X₂, … , X_n denote a random sample of size n from a distribution with p.d.f.                                  e^{– (x-q)}, x ³q, q Î Â

f(x; q) =     0      , otherwise.

 

Obtain UMVUE of q.

1. Give an example where MLE is not unique.
2. Explain Gauss-Markov model.

 

 

 

 

 

Part C

Answer any two questions.                                                                         2 X 20 = 40

 

1. a). Let X₁, X₂, … ,.X_n  denote a random sample of size n from P(q), q >0. Suggest

an unbiased estimator of i)  q        ii) 5q + 7.

b). If T_n is consistent estimator for and g is continuous then show that g(T_n)

is consistent for g().                                                                        (10 +10)

 

1. a). Show that UMVUE is essentially unique.

b). Obtain CRLB for estimating q in case of

1

f(x ; q) =                                   , – µ < x < µ and – µ < q < µ,

• [1 + (x – q)² ]

 

based on a random sample of size n.                                                            (10 +10)

1. a). State and establish Chapman – Robbins inequality.

b). Describe the method of moments with an illustration.                            (12 + 8)

1. a). Let X_1, X₂, … , X_n denote a random sample of size n from N (m, s²). Obtain

MLE of q = (m, s²).

b). Illustrate the method of moments with the help of G (a, p).                   (12 + 8)

 

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