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Math 344: Theory Of Estimation Question Paper
Math 344: Theory Of Estimation
Course:Bachelor Of Science (Economics & Statistics)
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
THIRD YEAR EXAMINATIONS FOR THE AWARD OF DEGREE OF BACHELOR OF SCIENCE (ECONOMICS & STATISTICS)
MATH 344: THEORY OF ESTIMATION
STREAMS: B.SC (ECON&STAT) Y3S1 TIME: 2 HOURS
DAY/DATE: MONDAY 22/4/2013 11.30 AM – 1.30 PM
INSTRUCTIONS:
Answer Question ONE and any other TWO Questions
All working must be Clearly shown
Statistical tables are required for use in this exam.
QUESTION ONE (30 MARKS)
(a) Define the following terms:
(i) A complete statistic
(ii) A sufficient statistic
(iii) An estimator. [6 Marks]
(b) Let x_1,x_2,…x_3 be a random sample f(x)= ?x^(?-1 0<x<)
,0<?<8
0 ,otherwise
(i) Find a maximum likelihood estimator of ?. [4 Marks]
(ii) Obtain a sufficient statistic for ?. [3 Marks]
(c) Let x_1,x_2,…,x_3 be a random sample form the distribution
f(x;?)= 1/? e^((-x)/? , 0<x<8)
? ,elsewhere
(i) Find the moments estimator for ?. [5 Marks]
(ii) Show that the statistic T(?X)= is a unbiased of ?. [5 Marks]
(d) A random sample of 100 students from Chuka University showed an average I.Q score of
112 with a standard deviation of 10. Find a 95% and 99% confidence internal estimate of the mean I.Q score of all students attending Chuka University. [6 Marks]
(e) Define mean squared error consistency for a sequence of estimators of T(?).
[3 Marks]
QUESTION TWO (20 MARKS)
(a) Briefly describe the method of Biyesian point estimation. [7 Marks]
(b) Let x_1,x_2,…x_n, denote a random sample from the Bernouli distribution.
f(x;?)= ?^x (1-?)^(1-x ),x=0,1
O ,elsewhere
Assume that the prior distribution of ? is given by g(?)= 1 ,0<?<1
0 ,elsewhere
Find the posterior bases estimator for Q. [13 Marks]
QUESTION THREE (20 MARKS)
(a) Define the following statistical terms
(i) A statistic
(ii) Parameter
(iii) Population
(iv) Prior distribution [4 Marks]
(b) Let X be a Poisson random variable with parameter ? . let S=?_(i=1)^n¦xi. Show that S is a
sufficient statistic. [6 Marks]
(c) State the craner-Rao inequality when regularity conditions are satisfied. [3 Marks]
(d) Let x_1,x_2,…x_n be a population from a density function f(x;?)=?e^(-?x)
x>0 find the Cramer-Rao lower bound for the unbiased estimator. [7 Marks]
QUESTION FOUR (20 MARKS)
(a) Define a uniformly minimum variance unbiased estimator (UMVUE) T of t(Q)
[4 Marks]
(b) Let x_1,x_2,…x_n be a random sample from a Poisson distribution with parameter ?. Using
the Cramar-Rao inequality condition, show that the mean x ¯ is the UMVUE of the population mean. [12 Marks]
(c) Let X be a binomial variate with parameter n & p. find the estimator of P by method of
moment. [4 Marks]
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