07_13 - likelihood estimate for θ Def An estimator θ ˆ...

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STAT 410 Examples for 07/13/2009 Summer 2009 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 ) = - otherwise 0 1 0 θ 1 θ θ 1 x x 0 < θ < . a) Obtain the method of moments estimator of θ , θ ~ . b) Obtain the maximum likelihood estimator of θ , 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
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Unformatted text preview: likelihood estimate for θ . Def An estimator θ ˆ is said to be unbiased for θ if E( θ ˆ ) = θ for all θ . d) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? 2. Let X 1 , X 2 , … , X 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....
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