Question

. Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < (1) Here the parameter θ > 0. (a) Determine the log-likelihood, l(θ), and a 1-dimensional sufficient statistic. (b) Show that P(Xi ≤ b; θ) = b θ+1 for f(x; θ) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the xi are observed. For the rest of the observations, it is only known that xi ≤ 1/2. Let δi = 1 or 0 according to whether xi is observed or not and let d = P i δi denote the number of xi observed; thus n − d of the xi are only known to satisfy xi ≤ 1/2. Determine the likelihood, L(θ), and a 2-dimensional sufficient statistic. Note that d is a random quantity dependent on the data. You can use the result of (b) even if you were unable to show it.

Answer 1

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