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**Extra info for Nonparametric Function Estimation, Modeling, and Simulation**

**Sample text**

Xn) e Rn. Then the sequence (0n(xi> x 2 , . . , xn)} converges almost surely to $0. More generally, Wald [17] has considered the case where there may be a multitude of relative maxima of the log likelihood. He has shown that, assuming a number of side conditions, if we take for our estimator a value of 9n e 0, which gives an absolute maximum of the likelihood, then 9n will converge almost surely to 90. We have presented the less general result of Wilks because of its relative brevity. , [9, pp.

Let us suppose we have some knowledge of the true value of 6, which can be characterized by a prior probability density p(6). The joint density of jc = (x l5 x 2 , . . , xn} and 0 is given by where fn(x\6) is the likelihood function. , the posterior density of 9) is then This is simply a version of Bayes's Theorem. To obtain a good estimate of 9 we might use some measure of centrality of the posterior distribution of 9. For example, if we attempt to minimize we may do so by selecting for each x that value which minimizes Now, this may be obtained by differentiating with respect to the real valued 9(x) and setting the derivative equal to zero to give or, simply the mean of the posterior distribution g(9\x).

If the development of statistics proceeded by a sequence of steady Teutonic increments, one might suppose that following Pearson's breakthrough, the main channel of statistics would have proceeded from his work. Accordingly, we might expect to have seen a succession of important generalizations of Pearson's family. , [7], [10], [19]. However, these studies did not actually push toward the natural goal, namely, the creation of practical algorithms which enable the stable and consistent estimation of probability densities in very general classes.