solution

Let X1, … Xn be i.i.d. random variables with the density function

a Find the method of moments estimate of .

b Find the mle of .

c Find the asymptotic variance of the mie.

d Find a sufficient statistic for .

Let X1, .., Xn be i.i.d. uniform on [0, 0].

a Find the method of moments estimate of  and its mean and variance.

b Find the mie of .

c Find the probability density of the mie, and calculate its mean and variance. Compare the variance, the bias, and the mean squared error to those of the method of moments estimate.

d Find a modification of the mie that renders it unbiased.

 
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