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Sma2110:Real Analysis Question Paper

Sma2110:Real Analysis

Course:Bachelor Of Computer Science

Institution: Meru University Of Science And Technology question papers

Exam Year:2013

QUESTION ONE (30 MARKS)
a) Define the properties of estimators. (5 Marks) b) Let ??1,…???? be iid Poisson (?) where (?< 0) is the unknown parameter. i. Find the maximum likelihood estimator for ?. (4 Marks) ii. Find UMVUE (uniformly minimum variance unbiased estimator) for ?. (2 Marks) c) Let ??1,…???? be iid Bernoulli (p) where 0 < < 1 is the unknown parameter. Consider T(p)=p i. Show that the cramer Rao lower bound of p is the same as var(??) and thus ?? is the UMVUE of p. (4 Marks) ii. Consider the specific statistic = ?? ?? =1 for ?? 0,1,2,….??. Verify that T is sufficient for P by showing that the conditional distribution of (??1,…????) given = ?? does not involve p(5). iii. With ?? = 2 consider an estimator = 1 2 (??1 + ??2) which is a biased estimator for p. show that through Rao-Blackwelization process, one again ends up with a refined unbiased estimator of P i.e ??. (3 Marks) Similarly for n=2 there is no improvement over T. (2 Marks) iv. Show that the estimator = ?? ?? =1 is minimal sufficient for p by Lehmanscheffe theorem. (3 Marks) v. Let = ??1??2 + ??3 is u a sufficient statistic for p? (2 Marks)
2
QUESTION TWO (20 MARKS)
a) Let ??1,…???? be iid random variable with pdf .; and consider the squared loss function, for an estimator = ??1,…???? , ; = ;(??1,…????) = - (??1,…????) 2 Let be a random variable with prior pdf ?. Determine T so that it is a Bayes estimate of . (8 Marks)
b) Let ??1,…???? be iid random variable from Bernoulli (), = (0,1). Choose ? to be the Beta density with parameter and , that is
? = ( +)
-1 1 - -1
0 ????????????
(0,1)
Find the Bayes estimate of (8 Marks) c) Suppose there is a prior pdf ? on such that for the Bayes estimate T defined by
??1,…???? =
f x1; ….f xn; ? d f x1; …f xn; ? d

The risk ; is independent of . Show that T is minimax. (4 Marks)
QUESTION THREE (20 MARKS)
a) Suppose that ??1,…???? are iid (,2) where and 2 are both unknown, = ,2 - < < ,0 < < ,?? = 2 where = and = × +. Find MLE for . (5 Marks) b) State the invariance property of MLE. (2 Marks) c) Consider a population described as (,2) where is unknown but + is known. Consider the following estimators of
1 = ??1 + ??2, 2 =
1 2
??1 + ??3 3 = ?? , 4 =
1 3
??1 + ??3
5 = ??1 + 2 - ??4 6 =
1 10
?? 4 =1 i. Show that 1 and 4 are both biased estimators of but 2,3 5 ?????? 6 are unbiased estimators. (4 Marks) ii. Consider the unbiased estimators, Compute the mean squared error (MSE) for each, and hence determine the best unbiased estimator amongst them. (4 Marks) iii. Compute the MSE for the biased estimators 1 and 4. (2 Marks)
3
iv. Show that any statistic which is one to one function of a minimal sufficient statistic is itself minimal sufficient. (3 Marks)
QUESTION FOUR (20 MARKS)
a) Let ??1,…….???? be iid random (,2) where = ,2 and both and are unknown. Where ?= and = × +. Find the minimal sufficient statistic for . (4 Marks) b) Suppose that T=T(X) is an unbiased estimator of a real valued parametric function a. such that its derivative exists b. Show that c. () = {(}2 ???? ?? ?? log ??1; 2 (8 Marks) c) Let T be an unbiased estimator of a real valued parametric function where the unknown parameter . Suppose that U is a jointly sufficient statistic for . Given that the domain space of U is defined as = ?? = , for ?? show that. (2 Marks) i. If = , then W is an unbiased estimator of (). (2 Marks) ii. () = () , with the equality holding iff T is the same as W. (3 Marks)
d) Define i. Prior distribution (1 Mark) ii. Posterior distribution. (2 Marks)

Sma2110:Real Analysis Question Paper

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