2. Let X₁,..., Xn be a random sample from a distribution with the cumulative distribution func- tion (cdf) 0, x < 0, F(x

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2 Let X Xn Be A Random Sample From A Distribution With The Cumulative Distribution Func Tion Cdf 0 X 0 F X 1 (82.23 KiB) Viewed 15 times

2. Let X₁,..., Xn be a random sample from a distribution with the cumulative distribution func- tion (cdf) 0, x < 0, F(x; 0) =(), 0≤x <3, 1, x > 3, where > 0 is an unknown parameter. (a) Find the probability density function (pdf) for this distribution. (b) Write down the log-likelihood function and simplify it. (c) Find the maximum likelihood estimator of and check that your estimator gives the maximum of the likelihood function. (d) Find the Fisher Information and give the Cramér-Rao lower bound for the variance of unbiased estimators of 0. (e) Suppose n = 20 and the observations are: 2.698 1.758 2.706 2.203 1.162 1.973 2.442 1.993 2.201 2.899 2.855 0.917 2.871 2.130 1.570 0.962 2.475 2.588 2.293 2.161 with 20 20 ΣΧ; = 42.857, Σln(X;) = 14.251. i=1 i=1 i. Find the maximum likelihood estimate of 0. ii. Construct an approximate 95% confidence interval for 0. Some R. code output that may help: qnorm (c (0.5,0.9,0.975,0.99)) [1] 0.000000 1.281552 1.959964 2.326348 log (c(2,3,4,5)) [1] 0.6931472 1.0986123 1.3862944 1.6094379