2. Let be a random sample from an exponential distribution with pdf: , Please derive The maximum likelihood estimator for . θ Is the above MLE unbiased for ? θ Please derive the distribution of the first order statistic and further show whether is an unbiased estimator of or not. θ Please calculate the MSE (mean squared error) of Y, and the MSE of the MLE for , which one is θ smaller? Is there a best estimator (UMVUE) for Please show the entire derivation. θ? Is there an efficient estimator for Please show the entire derivation. θ? Hint: Cramér-Rao Inequality: Let be unbiased for, where is a random sample from a population with pdf satisfying all regularity conditions. Then Solution: The likelihood function is The log likelihood function is Solving We obtain the MLE for Since We know the MLE is an unbiased estimator for Now we derive the general formula for the pdf of the first order statistic as follows: Therefore we have Differentiating with respect to and then multiplying by (-1) at both sides leads to: x, Now we can derive the pdf of the first order statistic for a random sample from the given exponential family. First, the population cdf is: Now plugging in the population pdf and cdf we have:
2. Let be a random sample from an exponential distribution with pdf: , Please derive The maximum likelihood estimator for θ . Is the above MLE unbiased for θ ? Please derive the distribution of the first order statistic and further show whether is an unbiased estimator of θ or not. Please calculate the MSE (mean squared error) of Y, and the MSE of the MLE for θ , which one is smaller? Is there a best estimator (UMVUE) for θ? Please show the entire derivation. Is there an efficient estimator for θ? Please show the entire derivation. Hint: Cramér-Rao Inequality: Let be unbiased for, where is a random sample from a population with pdf satisfying all regularity conditions. Then Solution: The likelihood function is The log likelihood function is Solving We obtain the MLE for Since We know the MLE is an unbiased estimator for Now we derive the general formula for the pdf of the first order statistic as follows: Therefore we have Differentiating with respect to x, and then multiplying by (-1) at both sides leads to: Now we can derive the pdf of the first order statistic for a random sample from the given exponential family. First, the population cdf is: Now plugging in the population pdf and cdf we have:
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