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 Â > 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 Â satisfy?
d What is the asymptotic variance of the maximum likelihood estimate?
e Find a sufficient statistic for .