solution

Consider an i.i.d. sample of random variables with density function

a Find the method of moments estimate of .

b Find the maximum likelihood estimate of .

c Find the asymptotic variance of the mie.

d Find a sufficient statistic for .

Suppose that X1, X2,. . . , Xnare i.i.d. random variables on the interval [0, 1] with the density function

Where  > 0 is a parameter to be estimated from the sample. It can be shown that

a How does the shape of the density depend on ?

b How can the method of moments be used to estimate ?

c What equation does the maximum likelihood estimate of  satisfy?

d What is the asymptotic variance of the maximum likelihood estimate?

e Find a sufficient statistic for .

 
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