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14. For each of the following distributions, derive a general expression for the Maximum Likelihood Estimator (MLE). Carry out the second derivative test to make sure you really have a maximum. Then use the data to calculate a numerical estimate. (a) p(z) = θ(1-θ) forェ= 0, 1, , where 0 < θ < 1 . Data: 4, o, 1, o, 1, 3, (b) f(x)-гет forz > 1, where cr > 0. Data: 1.37, 2.89, 1.52, 1.77, 1.04, (c) f(z)=ア-e_f, for z real, where τ > 0. Data: 1.45, 0.47, -3.33, 0.82, (d)/(z) 1e-z/e forr > 0, where θ> 0. Data: 0.28, 1.72, 0.08, 1.22, 1.86, 2, 16, 3, 0, 4, 3, 6, 16, 0, 0, 1, 1, 6, 10. Answer: 0.2061856 2.71, 1.19, 1.13, 15.66, 1.43 Answer: 1.469102 -1.59, -0.37, -1.56, -0.20 Answer: 0.6451059 0.62, 2.44, 2.48, 2.96 Answer: 1.517778 2π

math   Statistics-And-Probability   
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