STAT 400 Lecture AL1 Examples for 6.4, Part 2 Spring 2015 Dalpiaz 4. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; ) = otherwise010θ1 θ θ1xx 0 < < . Recall: Maximum likelihood estimator of is niin1 X 1 lnθˆ. Method of moments estimator of is 1X1 XX1 θ~. E ( X ) = θ11. Def An estimator θˆ is said to be unbiased for if E(θˆ) = for all . d) Is θˆ unbiased for ? That is, does E(θˆ) equal ? Jensen’s Inequality: If g is convex on an open interval I and X is a random variable whose support is contained in I and has finite expectation, then E [ g ( X ) ] g [ E ( X ) ]. If g is strictly convex then the inequality is strict, unless X is a constant random variable.e) Is θ~ unbiased for ? That is, does E(θ~) equal ? _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ sample mean sample variance nn X...XXX21 S 2 = 2 X X 11 in 6. Let X 1 , X 2 , … , X n be a random sample of size n from a population with mean and variance 2. Show that the sample mean X and the sample variance S 2 are unbiased for and 2, respectively. For an estimator θˆ of , define the Mean Squared Error of θˆ by MSE ( θˆ ) = E [ ( θˆ – ) 2 ]. E [ ( θˆ – ) 2 ] = ( E ( θˆ ) – ) 2 + Var ( θˆ ) = ( bias ( θˆ ) ) 2 + Var ( θˆ
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