ood estimator of 8 for this data set. 4.89 Let X₁, X2,..., X₁, be a random sample from a population with probability den

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Ood Estimator Of 8 For This Data Set 4 89 Let X X2 X Be A Random Sample From A Population With Probability Den 1 (52.63 KiB) Viewed 35 times

ood estimator of 8 for this data set. 4.89 Let X₁, X2,..., X₁, be a random sample from a population with probability density function 0 f(x)= x>0, (1+x)0+1 where is a positive unknown parameter. (a) Find the maximum likelihood estimator ê. (b) Show that the maximum likelihood estimator ê is a biased estimator of 8. Show that ê is asymptotically unbiased. Find an unbiasing constant c. which is a function of n. such that E[ce] =0. (c) Determine whether the unbiased estimate cê from part (c) is efficient. Cn (d) Calculate the maximum likelihood estimator and give an exact two-sided 93% con- fidence interval for e for the wooden toy price data set (in pounds, from Hand, et al., 1994, Small Data Sets, Chapman and Hall, page 48): 4.20 1.12 1.39 2.00 3.99 2.15 1.74 5.81 1.70 2.85 0.50 0.99 11.50 5.12 0.90 1.99 6.24 2.60 3.00 12.20 7.36 4.75 11.59 8.69 9.80 1.85 1.99 1.35 10.00 0.65 1.45. (e) Find the p-value associated with the test Ho: 0 = 1 versus H₁ : 0 1 using the likelihood ratio statistic for the wooden toy prices given in part (e). Use Wilks's theorem to arrive at the p-value.