Question: A random sample is drawn from a population with density function f(x|0) = -e-e-x (0 x 1). Show that the mean of the sample is

A random sample is drawn from a population with density function f(x|0) = -e-e-x (0 x 1). Show that the mean of the sample is a sufficient statistic for the parameter and verify that the maximum likelihood estimate is a function of the sample mean. Let this estimate be denoted by 01. Suppose that the only information available about each sample member concerns whether or not it is greater than a half. Derive the maximum likelihood estimate in this case and compare its posterior variance with that of 01. (Wales Dip.) 3. A continuous random variable x, defined in the range 0 < x 0. Find the density function of x. Given a random sample of n observations from this distribution, derive the maximum likelihood equation for , the estimate of

a. Indicate very briefly how this equation can be solved numerically. Also prove that the posterior variance of a is 44*sinh*l& n(4sinh-)* (Leic. Gen.)

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