STAT 400 Lecture AL1 4 Spring 2015 Dalpiaz Examples for 6 4 Part 2 Let X 1 X 2 X n be a random sample of size n from the distribution with probability density function 1 1 x f x 0 0 x 1 otherwise 0 Recall Maximum likelihood estimator of is Method of moments estimator of is Def d 1 n ln X i n i 1 1 X 1 1 X X E X 1 1 An estimator is said to be unbiased for if E for all Is unbiased for That is does E 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 unbiased for That is does E equal sample mean X 6 sample variance X 1 X 2 X n n S2 1 n 1 X i X 2 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 of define the Mean Squared Error of by MSE E 2 E 2 E 2 Var bias 2 Var
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