Chapter 7

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a distribution that is \(N(0, \theta)\). Then \(Y=\sum X_{i}^{2}\) is a complete sufficient statistic for \(\theta\). Find the MVUE of \(\theta^{2}\).

Prove that the sum of the observations of a random sample of size \(n\) from a Poisson distribution having parameter \(\theta, 0<\theta<\infty\), is a sufficient statistic for \(\theta\).

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample from a normal distribution with mean zero and variance \(\theta, 0<\theta<\infty\). Show that \(\sum_{1}^{n} X_{i}^{2} / n\) is an unbiased estimator of \(\theta\) and has variance \(2 \theta^{2} / n\).

Let \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) represent a random sample from the discrete distribution having the pmf $$ f(x ; \theta)=\left\\{\begin{array}{ll} \theta^{x}(1-\theta)^{1-x} & x=0,1,0<\theta<1 \\ 0 & \text { elsewhere } \end{array}\right. $$ Show that \(Y_{1}=\sum_{1}^{n} X_{i}\) is a complete sufficient statistic for \(\theta .\) Find the unique function of \(Y_{1}\) that is the MVUE of \(\theta\). Hint: \(\quad\) Display \(E\left[u\left(Y_{1}\right)\right]=0\), show that the constant term \(u(0)\) is equal to zero, divide both members of the equation by \(\theta \neq 0\), and repeat the argument.

If \(X_{1}, X_{2}\) is a random sample of size 2 from a distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

Let \(Y_{1}

Let \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample of size \(n\) from a distribution with pdf \(f(x ; \theta)=\theta x^{\theta-1}, 00\). (a) Show that the geometric mean \(\left(X_{1} X_{2} \cdots X_{n}\right)^{1 / n}\) of the sample is a complete sufficient statistic for \(\theta\). (b) Find the maximum likelihood estimator of \(\theta\), and observe that it is a function of this geometric mean.

Show that the \(n\) th order statistic of a random sample of size \(n\) from the uniform distribution having pdf \(f(x ; \theta)=1 / \theta, 0