Question

7.6.4. Let X1, X2,... , Xn be a random sample from a uniform (0,) distribution. Continuing with Example 7.6.2, find the MVUEs for the following functions of (a) g(0)-?2, i.e., the variance of the distribution (b) g(0)- , i.e., the pdf of the distribution C) or t real, g(9)- , î.?., the mgf of the distribution. Example 7.6.2. Suppose X1, X2,... , Xn are iid random variables with the com- mon uniform (0,0) distribution. Let Yn - max{X1, X2,... , Xn). In Example 7.4.2, we showed that Yn is a complete and sufficient statistic of ? and the pdf of Y, is given by (7.4.1). Let g(8) be any differentiable function of. Then the MVUE of g(0) is the statistic u(Yn), which satisfies the equation n-1 0 or equivalently, 0 Differentiating both sides of this equation with respect to ?, we obtain Solving for u(0), we obtain TL Therefore, the MVUE of g(0) is (7.6.2) For example, if g(0) 0, then which agrees with the result obtained in Example 7.4.2. Other examples are given in Exercise 7.6.4.

Answer 1

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