14.33. It might be (erroneously) supposed that an unbiased estimate may always be found for an unknown parameter. That this is not so is illustrated by the following example. Suppose that n repetitions of an experiment are made and a particular event A occurs precisely k times. If there is hypothesized a constant probability p = P(A) that A occurs whenever the experiment is performed, we might be interested in estimating the ratio r = p/(1 - p). To see that no unbiased estimate of p/(1 - p) exists [based on the observation of kA's and (n - k)A's], suppose that in fact such an estimate does exist. That is, suppose that = h(k) is a statistic for which E() = p/(1 - p). Specifi- cally, suppose that n = 2 and hence k = 0, 1, or 2. Denote the corresponding three values of by

a, b, and

c. Show that E() = p/(1 - p) yields a contradiction by noting what happens to the left- and the right-hand side of this equation as p 1.