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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
SECTION A (COMPULSORY)
QUESTION ONE (30 MARKS)
a) Define the following i. Parameter space. (1 Mark) ii. Loss function. (2 Marks) iii. Risk function (2 Marks) b) If X has binomial distribution with parameter n and p, show that the estimator ??+ ?? 2 ??+ ?? is a biased estimator of p unless = 1 2 . (3 Marks) c) Let ??1,.???? be iid Bernoulli (p) where p is the unknown parameter, 0 = = 1 ?????? = 0,1 = ?????? = (0,1). Consider the specific statistics ?? = ???? ?? ??=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 Marks) d) Given the likelihood function defined by = ??(????; ?? ??=1 ), . Show that a real valued statistic ?? = ??(??1,????) is sufficient for the unknown parameter iff the following factorization holds = ??(??1,???? ;)(??1,????),??1,???? ? Where the two functions .; ?????? (.) are both non-negative. (5 Marks)
e) Suppose that ??1,???? are iid Poisson () where 0 << 8 is the unknown parameter where = 0,1,2, and = R+. Find the maximum likelihood estimator (MLE) for . (4 Marks)
2
f) Given that??1,???? are iid Bernouli (p) where0 < < 1 is unknown, with ?? 2 Consider an estimator ?? = 1 2 (??1 + ??2) which is an unbiased estimator for p. i. Show that after going through the process of Rao-Blackwelization, one again ends up with ?? as the refined unbiased estimator of p. (3 Marks) ii. Show that when n=2 there is no improvement over T. (2 Marks) g) Suppose that 0~(370,400) and that |~(421,64). Compute (|??). (3 Marks)
SECTION B
QUESTION TWO (20 MARKS)
a) Let ??1,???? be iid random variable with pdf ?? .; ? R and consider the squared loss function, for an estimate ?? = ?? ??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 i.e ?? ??1,???? . (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 - 8 < < 8,0 < < 8,?? 2 where = R and = R × R +. Find MLE for . (5 Marks) b) State the invariance property of MLE. (2 Marks) c) From question three (a) above, considering the invariance property of MLE, what is the MLE of i. (1 Mark)
3
ii. + (1 Mark) iii. 2 2 (1 Mark) d) Consider a population described as (,2) where ? R is unknown but ? R + is known. Consider the following estimators of defined as
??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 of . (4 Marks) ii. Consider the unbiased estimators??2,??3 ??5 ?????? ??6. 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)
QUESTION FOUR (20 MARKS)
a) Suppose that T=T(X) is an unbiased estimator of a real valued parametric function ?? i.e ?? = ??() ? Let ?? ?? ??(), denoted by ??() exist and is finite ?. Show that (??) {??(}2 ?? ?? ?? log ?? 1; 2 (10 Marks) b) Define the following; i. Consistent estimator. (3 Marks) ii. Efficient estimator. (2 Marks) c) Let T be an unbiased estimator of a real valued parametric function ?? where the unknown parameter ? R?? R. 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)
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