Review - Review 1 Review 1.1 Statistics Estimation Maximum...

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ReviewDecember 12, 20161Review1.1StatisticsEstimation:Maximum likelihood estimation (MLE); large sample properties of theMLE; Information matrix; method of moments.Consistency of estimators; Mean squared error and its decomposition; unbiased esti-mation; minimum variance unbiased estimator; sufficiency.Bayes estimators: prior distribution; posterior distribution.Sampling distribution of an estimator; sampling from a normal distribution;t-distribution.Exercise: Suppose thatX1, . . . , Xnform a random sample from a normal distribution with mean0 and unknown varianceσ2.Determine the asymptotic distribution of the statisticT=(1nni=1X2i)-1.
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Exercise: Consider i.i.d observationsX1, . . . , Xnwhere eachXifollows a normal distributionwith mean and variance both equal to 1, whereθ >0. Thus,fθ(x) =θ2πexp-(x-θ-1)22θ-1.Show that the MLE is one of the solutions to the equation:θ2W-θ-1 = 0,whereW=n-1ni=1X2i.Determine which root it is and compute its approximatevariance in large samples.

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