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Consider the sample s = (x1, . . . , xn), realization of a random sample X1, . . . , Xn from a uniform on [0, θ], with θ unknown. The goal is to estimate the unknown mean θ/2, based on s. To this end we consider two competing estimators: T1 = X(n)/2, where X(n) is defined as the maximum of (X1, . . . , Xn), ̄ a.1) find the cumulative distribution function (cdf) of T1, a.2) from the cdf obtain the probability density function (pdf) of T1, a.3) use the pdf to compute the expected value of T1, a.4) use the pdf to compute the variance of T1. b) Is T1 an unbiased estimator of θ/2? If the answer is no, study the limit of the bias when the sample size n goes to infinity. c) Find an expression for MSEθ(T1). d) Compute expected value and variance of T2. e) Is T2 an unbiased estimator of θ/2? If the answer is no, study the limit of the bias when the sample size n goes to infinity. f) Find an expression for MSEθ(T2).
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