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1. Suppose that ri,.., n are a random sample having probability density function Here the paran neter θ > 0. (a) Determine the log-likelihood, (0), and a 1-dimensional sufficient statistio (b) Show that PX, b:0) =&+1 for f(z:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that Xi < 1/2. Let δί 1 or 0 according to whether 2a is observed or not and let d Σίδί denote the number of xi observed; thus n - d of the x, are only known to satisfy x; K 1/2. Determine the likelihood, L(0), 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.

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