4.14 Let X1, X2, ..., X, be a random sample from a Pareto population with probability density function f(x) = (1+x)0+1 where 0 is a positive unknown parameter. x> 0, (a) Find the Cramér-Rao lower bound on the population variance of any unbiased estima- tor of 0. (b) Find an exact two-sided 100(1 – a)% confidence interval for 0 from a single (n = 1) observation X. (c) A single observation X is used to test Ho : 0= 2 versus H1:0 < 2. Find the critical value at a = 0.01.
4.14 Let X1, X2, ..., X, be a random sample from a Pareto population with probability density function f(x) = (1+x)0+1 where 0 is a positive unknown parameter. x> 0, (a) Find the Cramér-Rao lower bound on the population variance of any unbiased estima- tor of 0. (b) Find an exact two-sided 100(1 – a)% confidence interval for 0 from a single (n = 1) observation X. (c) A single observation X is used to test Ho : 0= 2 versus H1:0 < 2. Find the critical value at a = 0.01.
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
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A and b please

Transcribed Image Text:4.14
Let X1, X2, ..., X, be a random sample from a Pareto population with probability density
function
f(x) =
x> 0,
(1+x)®+1
where 0 is a positive unknown parameter.
(a) Find the Cramér–Rao lower bound on the population variance of any unbiased estima-
tor of 0.
(b) Find an exact two-sided 100(1 – a)% confidence interval for 0 from a single (n = 1)
observation X.
(c) A single observation X is used to test
Ho : 0 = 2
versus
H1 :0 <2.
Find the critical value at a=0.01.
(d) Plot the power function for the hypothesis test in part (c).
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