Let X1,, X, be a random sample from a population with pdf (0 + 1)r", if 0

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
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Let X1,,Xn be a random sample from a population with pdf
(0 +1)r", if 0<x<1
fo(x) =
0,
otherwise,
where -1 < 0 < o is an unknown parameter.
(a) Prove that E(X) = That is, the population mean is .
(b) Derive a method of moments estimator of 0. Denote it by ÔMOM-
(c) Derive the maximum likelihood estimator, denoted Omle, of 0.
(d) Do the two estimators ÔMOM and êmte always produce estimates that are in the
0+2
0+2
parameter space? In other words, can you assure that (i) Ômte > -1 and (ii) ÔMOM > -1?
Transcribed Image Text:Let X1,,Xn be a random sample from a population with pdf (0 +1)r", if 0<x<1 fo(x) = 0, otherwise, where -1 < 0 < o is an unknown parameter. (a) Prove that E(X) = That is, the population mean is . (b) Derive a method of moments estimator of 0. Denote it by ÔMOM- (c) Derive the maximum likelihood estimator, denoted Omle, of 0. (d) Do the two estimators ÔMOM and êmte always produce estimates that are in the 0+2 0+2 parameter space? In other words, can you assure that (i) Ômte > -1 and (ii) ÔMOM > -1?
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