Operational Risk Modeling Analytics phần 4

(Ví dụ 7,8 tiếp tục) Giả sử thay vì được sự mất mát cá nhân Pareto phân phối với dfSau đó, sự phân bố của sự mất mát tối đa cho một khoảng thời gian năm k có dfGumbel, Frkchet, và phân phối Weibull có một tài sản, được gọi là "sự ổn định của tối đa" hay | THE a 6 1 CLASS 121 If the original values were all available then the zero-truncated probabilities could have all been obtained by multiplying the original values by 1 1 . For the zero-modified random variable pM arbitrarily. From pM 1 - 0 6 l - . Then p I II p I II . In this case each original negative binomial probability has been multiplied by 1 - l - . Also note that for j 1 p QApT. Although we have only discussed the zero-modified distributions of the a 6 0 class the a 6 1 class admits additional distributions. The a 6 parameter space can be expanded to admit an extension of the negative binomial distribution to include cases where 1 r 0. For the a 6 0 class r 0 is required. By adding the additional region to the sample space the extended truncated negative binomial ETNB distribution has parameter restrictions 3 0 r 1 r 0. To show that the recursive equation Pk Pk l k 2 3 . with Po 0 defines a proper distribution it is sufficient to show that for any value of Pl the successive values of Pfc obtained recursively are each positive and that Pk 00. For the ETNB this must be done for the parameter space . 3 a 1 3 3 0 6 r 1 3 r 1 r Ạ 0 see Exercise . When r 0 the limiting case of the ETNB is the logarithmic distribution with J W l fe . Pi - 41a l J - 5 9 see Exercise . The pgf of the logarithmic distribution is pT . 1 _ M1 - 3 z - 1 z 1 ln l 3 see Exercise . The zero-modified logarithmic distribution is created by assigning an arbitrary probability at zero and reducing the remaining probabilities. 122 MODELS FOR THE NUMBER OF LOSSES COUNTING DISTRIBUTIONS It is also interesting that the special extreme case with 1 r 0 and 3 - oo is a proper distribution sometimes called the Sibuya distribution. It has pgf F z 1 1 z r and no moments exist see Exercise . Distributions with no moments are not particularly interesting for .