Suppose that X1, X2, . . . , Xn are i.i.d. random

Chapter , Problem 17

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Suppose that \(X_{1}, X_{2}, \ldots, X_{n}\) are i.i.d. random variables on the interval [0, 1] with the density function

\(f(x \mid \alpha)=\frac{\Gamma(2 \alpha)}{\Gamma(\alpha)^{2}}[x(1-x)]^{\alpha-1}\)

where \(\alpha>0\) is a parameter to be estimated from the sample. It can be shown that

\(\begin{aligned}E(X) & =\frac{1}{2} \\ \operatorname{Var}(X) & =\frac{1}{4(2 \alpha+1)}\end{aligned}\)

a. How does the shape of the density depend on \(\alpha\)?

b. How can the method of moments be used to estimate \(\alpha\)?

c. What equation does the mle of \(\alpha\) satisfy?

d. What is the asymptotic variance of the mle?

e. Find a sufficient statistic for \(\alpha\).

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