Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from uniform distribution on an interval \((0, \theta)\). Show that $\left(\prod_{i=1}^{n} X_{i}\right)^{1 / n}\( is consistent estimator of \)\theta e^{-1}$.

Prove that method of moment estimator is consistent for the estimation of \(r>0\) in the gamma family $$ f(x ; r)=\left\\{\begin{array}{ll} \frac{e^{-x} x^{r-1}}{\Gamma(r)} & 0

Consider repeated observation on a \(m\) -dimensional random variable with mean $E\left(X_{i}\right)=\mu, i=1,2, \ldots, m, \quad \operatorname{Var}\left(X_{i}\right)=\sigma^{2}, i=1,2, \ldots, m$ and \(\operatorname{Cov}\left(X_{i}, X_{j}\right)=\rho \sigma^{2}, i \neq j .\) Let the \(i\) th observation be \(\left(x_{1 i}, \ldots, x_{m i}\right)\) \(i=1,2, \ldots, n\). Define $$ \begin{array}{c} \bar{X}_{i}=\frac{1}{m} \sum_{j=1}^{m} X_{j i} \\ W_{i}=\sum_{j=1}^{m}\left(X_{j i}-\bar{X}_{i}\right)^{2}, \\ B=m \sum_{i=1}^{n}\left(\bar{X}_{i}-\bar{X}\right)^{2}, \\ W=W_{1}+\cdots+W_{n} . \end{array} $$ where \(B\) is sum of squares between and \(W\) is sum of squares within samples. 1\. Prove (i) \(\left.W \sim(1-\rho) \sigma^{2} \chi^{(} m n-n\right)\) and (ii) \(B \sim(1+(m-1) \rho) \sigma^{2} \chi^{2}(n-1)\). 2\. Suppose \(\frac{(1-\rho) B}{(1+(m-1) \rho) W} \sim F_{(n-1),(m n-n)} .\) Prove that when \(\rho=0, \frac{W}{W+B}\) follows beta distribution with parameters \(\frac{m n-n}{2}\) and \(\frac{n-1}{2}\).

Suppose that the random sample arises from a distribution with PDF $$ f(x ; \theta)=\left\\{\begin{array}{l} \theta x^{\theta-1}, \quad 0

If \(X_{1}, X_{2}\) and \(X_{3}\) are three independent random variables having the Poisson distribution with the parameter \(\lambda\), show that $$ \hat{\lambda_{1}}=\frac{X_{1}+2 X_{2}+3 X_{3}}{6} $$ is an unbiased estimator of \(\lambda\). Also, compare the efficiency of \(\hat{\lambda_{1}}\) with that of the alternate estimator. $$ \hat{\lambda_{2}}=\frac{X_{1}+X_{2}+X_{3}}{3} $$.

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from normal distributed population with mean \(\mu\) and variance \(\sigma^{2}\). Let \(s^{2}\) be the sample variance. Prove that \(\frac{(n-1) s^{2}}{\sigma^{2}}\) has \(\chi^{2}\) distribution with \(n-1\) degrees of freedom.