Problem 1 Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where 0 > 0 is an unknown parameter. Recall that the pdf fof the U(0, 0) distribution is of the form 1 f(x) = { 0-¹ if 0 < x < 0 otherwise. Note that the information about contained in the random sample X₁, information about contained in the statistic T = max(X₁, ..., Xn equals the •, Xn). To understand why so, let's think of the random sample as being obtained in a sequential manner, that is, you obtain X₁ and pause before obtaining X₂. What does X₁ tell you about ? It tells you that > X₁. Once you have X₁ and the information that > X₁, obtain X₂. If X2 > X₁, then you know a bit more about 0, namely, 0 > X₂; however, if X2X₁, then it does not contribute anything, above and beyond what you already know from X₁, to your knowledge about 0. In other words, when you have obtained X₁ and X2, what you know about is that it is greater than the maximum of X₁ and X2. As such, any reasonable estimator of should be a function of T. (a) Carefully argue that T is the maximum likelihood estimator of 0. Recall that the likelihood function of 0, given a data sample x₁,, n, is the product of f(x₁), ..., f (x₂). (b) Construct an unbiased estimator of which is not a function of T and calculate its variance. To start with, you may like to calculate the expected value and the variance of the U(0,0) distribution. (c) Construct an unbiased estimator of which is a function of T and calculate its variance. To start with, you should calculate, in that order, the cdf, the pdf, the expected value, and the variance of T. Page 1 of 2 (d) Now consider a biased estimator of 0 of the form cT. Find c* that minimizes the MSE in the class of estimators of the form cT and explicitly verify that the MSE of c*T is less than that of the estimator constructed in part (c).