Suppose that X1, . . . , Xn form a random sample from a uniform distribution with the following p.d.f.: f(x|θ) = 1/θ * I (θ ≤ x ≤ 2θ),where θ > 0 is the unknown parameter of interest. (a) What’s the support of ∑ i=1 Xi? (b) Let Z = min{X1, . . . , Xn} be the minimum of X1, . . . , Xn. Find the CDF of Z, i.e., P(Z ≤ z). (Hint: consider P(Z > z), and don’t forget to include therange of values that z could take) (c) What’s the MLE of θ? (d) Here’s the observed data, x 1 = 4 (only one data point been collected).What’s the maximum likelihood estimate of θ? (e) Find an estimator for θ using method of moments. (f) What’s the mode of f(x|θ)? (g) What’s the expectation of X1? (h) Apply WLLN on the sample mean X¯n. State the result. (i) Let θ → 0 as n → ∞. What’s the limiting distribution of f? (Hint: check the support)
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