By Dr. J. G. Kalbfleisch (auth.)
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Example text
In general. there is no simple relationship between relative likelihoods and probabilities. The following example shows that in some cases the 50% LI will certainly contain the true parameter value. and in other cases it will almost certainly not. 1. • n. and m-l extra cards from some unknown demonination S. One card is selected at random. and its denomination is found to be x. The problem is to estimate S. There are m cards of denomination S. and one of each of the other n denominations. Hence the probability of drawing a card of denomination x is m P(x;S) = 1 n+m if x = S; if x ...!...
Since maximum likeliheed estimaters are invariant under enete-ene parameter changes, they cannet in general be unbiassed. Hewever, it can be shewn that under suitable regularity condi thms, the bias ef maximum likeliheed estimaters tends to' zerO' as n + 00. 4 is to' estimate the success prebability to' be first trial, and to' be 1 0 if a failure occurs en the if a success eccurs en the first trial. This is net a very sensible estimatien precedure. 5, but to achieve the correct long-run average, we must e to be 1.
8 2 Show that p = P(X(2k) $ 13~/5). 8/~. 2 to determine 6 and a 10% likelihood interval for 13. Sufficient Statistics Consider an experiment for which the probability model involves an unknown parameter 8. Let x denote a typical outcome. In 48 most of the examples we shall consider, x will be a vector of n counts or measurements. The probability of outcome x will be a function of s, P(x;S), and the likelihood function L(S) is propor- tional to P(x;S). The likelihood function is defined only up to a multiplicative constant, and two likelihood functions which are proportional are considered to be the same.