Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X₁,...,X, has the following probability density function (0%e-e x! 0, a) b) c) f(x; 8) = -, for x = 0,1,2,... elsewhere Show that for any &> 0 and S=1X₁, lim P (S₁ - 81 ≥ 8) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁ = *₁+2x;+2X₁-X and ₂ = (X₁ + X₂ + X₁ + X₁) be two unbiased estimators of 8. Which one of the two estimators is more efficient?
Suppose that a sequence of mutually independent and identically distributed discrete random variables X₁, X₂, X₁,...,X, has the following probability density function (0%e-e x! 0, a) b) c) f(x; 8) = -, for x = 0,1,2,... elsewhere Show that for any &> 0 and S=1X₁, lim P (S₁ - 81 ≥ 8) = 0. Show that a statistic S, in a) is the maximum likelihood estimator of the parameter 8. Let 8₁ = *₁+2x;+2X₁-X and ₂ = (X₁ + X₂ + X₁ + X₁) be two unbiased estimators of 8. Which one of the two estimators is more efficient?
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:Suppose that a sequence of mutually independent and identically distributed discrete random
variables X₁, X₂, X₁,...,X, has the following probability density function
(0%e-0
x!
a)
b)
c)
f(x; 0) =
0,
▸
for x = 0,1,2,...
elsewhere
n
Show that for any e > 0 and Sn = ₁X₁, lim P(|S₁ - 0 ≥ 8) = 0.
i=11
11-400
Show that a statistic S,, in a) is the maximum likelihood estimator of the parameter 8.
Let 0₁ = X₁+2X₂+2X-X₁ and ₂ = (X₁ + X₂ + X3 + X₁) be two unbiased estimators of 6. Which
one of the two estimators is more efficient?
d) What is the Cramer-Rao lower bound for the variance of the unbiased estimator of the
parameter 8?
e)
Use the one-parameter regular exponential family definition to find the functions,
h(x), c(0), w(0) and t(x).
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