Assignment 5

[This assignment covers work in Statistical Inference, Chapter 8.]

1. What multiple of the sample mean $\overline{X}$ estimates the population mean with minimum mean square error? In particular, if there is a known relationship between $\mu$ and $\sigma^2$, say , what is this multiple?
2. $X_1, \ldots, X_n$ is a random sample from the distribution with
   
   
   
   Find the Cramer-Rao lower bound for the variance of an unbiased estimator of $\theta$. Identify the estimator that has this variance.
3. For the Cauchy distribution with location parameter $\theta$,
   
   
   
   show that the MVB cannot be attained.
4. Show that for $n$ independent observations $x_1, \ldots, x_n$ from a distribution with pdf of the regular case of the exponential class,
   
   
   
   the statistic attains the Cramer-Rao lower bound for an unbiased estimator of its expectation. Give the variance of the estimator.
5. Find the smallest value of $n$ for which
   
   
   
   where $Y_1, \ldots, Y_n$ are the order statistics of a random sample of size $n$ from a continuous distribution.
6. Compute where are the order statistics of a random sample of size from a continuous distribution.
7. Let be the order statistics of a random sample of size from a continuous distribution. Compute approximately