tailieunhanh - Operational Risk Modeling Analytics phần 10

và một cuộc thảo luận về các vấn đề xung quanh việc lựa chọn các phương pháp Bayesian so với frequentist, xem Efron [26]. Một nguồn tin cho một điều trị toàn diện toán học của phương pháp Bayesian là văn bản bằng cách Berger [15]. Trong những năm gần đây tiến bộ nhiều trong các tính toán Bayesian đã xảy ra. Một nguồn lực tốt là [21]. | MAXIMUM LIKELIHOOD ESTIMATION 397 dimension of the jth outcome. Then the loglikelihood function is n z In IJ j i nd n fi ij 52 nc F-i xTj Fdtxdj j l j l j l lw zc. The maximum likelihood estimates are the values of the parameters that maximize the loglikelihood function. This form of the loglikelihood suggests obtaining approximate estimates of the parameters by first maximizing the first term the. marginals term and then maximizing the second term the. copula term . Maximizing the marginals term involves maximizing the d different terms in lw of the form li i ỉ 2 . d j i where is the loglikelihood function of the ith marginal distribution. Thus we can first obtain all the parameter estimates for the marginal distributions using the univariate methods described earlier. It should be noted that these are not the ultimate maximum likelihood estimates because the ultimate estimates depend also on the estimates of the copula parameter s which have not yet been estimated. We shall refer to the estimates arising from the maximization of as. pseudo-MLEs. The efficiency of these estimates may be low because the information about the parameters contained in the second term of the loglikelihood is ignored 110 . There are several approaches to maximizing the second term of loglikelihood . One way is to use the pseudo-MLEs. Let Fị xịj denote the pseudo-estimates of the cdf of the marginal distributions at each observed value. Then the pseudo-likelihood of the copula function is n ĩc 521nc Ũ j . ũdj . j i This is then maximized with respect to the copula parameters to obtain the pseudo-MLEs of the copula parameters. This maximization can be done by any method although we prefer the simplex method because it is very stable especially with few parameters. We expect that in most cases in applications where there are not large amounts of data the principle of parsimony will dictate that very few parameters should be used for the copula. Most