Suppose that two independent binomial random variables X₁ and X2 are observed where X₁ has a Binomial(n; p) distribution and X2 has a Binomial(2n; p) distribution, where n > 0 and 0 < p < 1. You may assume that n is known, whereas p is an unknown parameter. Define two possible estimators of p, 1 1 P₁ = 2n X1+-X2+ 2 1 10 1 and P2: == (2X1+3X2). bn where b is a constant. (a) bias of P₁? Show that P₁ is not an unbiased estimator of p. What is the value of the (b) Find constant b such that estimator P2 is unbiased. (c) Find Var(P1) and Var(P2). (d) Show that both estimators are consistent estimators of p. (e) Derive the relative efficiency of the estimator P2 relative to P₁. Which estimator is preferred? Hint: Your final expression Eff(P2, P₁) may consist of n and p terms. Assume some values for those to choose the preferred estimator.
Suppose that two independent binomial random variables X₁ and X2 are observed where X₁ has a Binomial(n; p) distribution and X2 has a Binomial(2n; p) distribution, where n > 0 and 0 < p < 1. You may assume that n is known, whereas p is an unknown parameter. Define two possible estimators of p, 1 1 P₁ = 2n X1+-X2+ 2 1 10 1 and P2: == (2X1+3X2). bn where b is a constant. (a) bias of P₁? Show that P₁ is not an unbiased estimator of p. What is the value of the (b) Find constant b such that estimator P2 is unbiased. (c) Find Var(P1) and Var(P2). (d) Show that both estimators are consistent estimators of p. (e) Derive the relative efficiency of the estimator P2 relative to P₁. Which estimator is preferred? Hint: Your final expression Eff(P2, P₁) may consist of n and p terms. Assume some values for those to choose the preferred estimator.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Transcribed Image Text:Suppose that two independent binomial random variables X₁ and X2 are observed where X₁
has a Binomial(n; p) distribution and X2 has a Binomial(2n; p) distribution, where n > 0
and 0 < p < 1. You may assume that n is known, whereas p is an unknown parameter.
Define two possible estimators of p,
1
1
P₁ =
2n
X1+-X2+
2
1
10
1
and P2: ==
(2X1+3X2).
bn
where b is a constant.
(a)
bias of P₁?
Show that P₁ is not an unbiased estimator of p. What is the value of the
(b)
Find constant b such that estimator P2 is unbiased.
(c)
Find Var(P1) and Var(P2).
(d)
Show that both estimators are consistent estimators of p.
(e)
Derive the relative efficiency of the estimator P2 relative to P₁. Which
estimator is preferred?
Hint: Your final expression Eff(P2, P₁) may consist of n and p terms. Assume some
values for those to choose the preferred estimator.
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