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2 is the above mle unbiased for 3 the method of moment

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Unformatted text preview: hod of moment estimator (MOME) for θ . (4) Please derive the distribution of the first order statistic where further show whether is an unbiased estimator of θ or not. = min( and Solution: (1) The likelihood function is The log likelihood function is Solving We obtain the MLE for (2) Since 3 We know the MLE is an unbiased estimator for (3) Now we derive the method of moment estimator (MOME) for θ. Since we have only one parameter, θ, to estimate, the equation of the first population moment and sample moment will suffice. That is, solving: , we obtain the MOME: (4) 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 pop...
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