03_07 - n be a random sample of size n from a population...

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STAT 410 Examples for 03/07/2008 Spring 2008 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = ± ² ³ - otherwise 0 1 0 1 1 x x 0 < θ < . a) Obtain the method of moments estimator of θ , ~ . b) Obtain the maximum likelihood estimator of θ , ˆ . c) Suppose n = 3, and x 1 = 0.2, x 2 = 0.3, x 3 = 0.5. Compute the values of the method of moments estimate and the maximum likelihood estimate for θ . Def An estimator ˆ is said to be unbiased for θ if E( ˆ ) = θ for all θ . d) Is ˆ unbiased for θ ? That is, does E( ˆ ) equal θ ? 2. Let X 1 , X 2 , … , X
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Unformatted text preview: n be a random sample of size n from a population with mean μ and variance σ 2 . Show that the sample mean X and the sample variance S 2 are unbiased for μ and σ 2 , respectively. 3. Let X 1 , X 2 , … , X n be a random sample of size n from N ( θ 1 , θ 2 ) , where Ω = { ( θ 1 , θ 2 ) : – ∞ < θ 1 < ∞ , 0 < θ 2 < ∞ } . That is, here we let θ 1 = μ and θ 2 = σ 2 . a) Obtain the maximum likelihood estimator of θ 1 , 1 & ˆ , and of θ 2 , 2 & ˆ . b) Obtain the method of moments estimator of θ 1 , 1 & ~ , and of θ 2 , 2 & ~ ....
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This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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