\\\textup{Let}\, X_{1}, X_{2}, ..., X_{n}\, \, \textup{be iid exponential random variables with mean}\,\, \beta.\, \textup{Find the...

\\\textup{Let}\, X_{1}, X_{2}, ..., X_{n}\, \, \textup{be iid exponential random variables with mean}\,\, \beta.\, \textup{Find the following for}\, \beta .
\\\\\textup{a. The method of moments estimator}
\\\textup{b. The maximum likelihood estimator}
\\\textup{c. Determine whether the maximum likelihood is unbiased}
\\\textup{d. Find the mean squared error of the maximum likelihood}
\\\textup{e. Find the Cramer-Rao lower bound for the variances of the unbiased estimators}
\\\textup{f. What is the UMVUE (uniformly minimum variance unbiased estimator)? State the reasoning.}
\\\textup{g. Find the asymptotic distribution of the maximum likelihood as}\, n\rightarrow \infty
\\\\\textup{Useful information}
\\X\sim Exponential(\beta) : \frac{1}{\beta }e^{\frac{-x}{b}}
\\E[X]=\beta \; \; \;\; \; \; \; \; Var[X]=\beta ^2

Expert Solution

orchestra answered 1 year ago

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