Let f(x; θ) = θxθ−1 for 0 < x < 1 and θ ∈ Ω =...

Let f(x; θ) = θxθ−1 for 0 < x < 1 and θ ∈ Ω = {θ : 0 < θ < ∞}. Let X1, . . . , Xn denote a random sample of size n from this distribution. (a) Sketch the pdf of X for (i) θ = 1/2, (ii) θ = 1 and (iii) θ = 2. (b) Show that ˆθ = −n/ ln (Qn i=1 Xi) is the maximum likelihood estimator of θ. (c) Determine the first theoretical moment (i.e. the mean) of this distribution. (d) For each of the following three sets of ten observations from the given distribuion, calculate the values of the maximum likelihood estimate and the method-of-moments estimate of θ: (i) 0.0256 0.3051 0.0278 0.8971 0.0739 0.3191 0.7379 0.3671 0.9763 0.0102 (ii) 0.9960 0.3125 0.4374 0.7464 0.8278 0.9518 0.9924 0.7112 0.2228 0.8609 (iii) 0.4698 0.3675 0.5991 0.9513 0.6049 0.9917 0.1551 0.0710 0.2110 0.2154

Expert Solution

(a). (i)

(ii).

(iii).

orchestra answered 2 years ago

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Let f(x; θ) = θxθ−1 for 0 < x < 1 and θ ∈ Ω =...

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