Một vấn đề khác phát sinh khi có các chức năng khả năng là rất phức tạp, bởi vì sau đó ma trận thông tin là khó khăn để tìm thấy phân tích. Trong những trường hợp này, chúng ta có thể sử dụng các phương pháp số. | 70 THE LEFT-TAIL CALIBRATION METHOD TABLE Bootstrap estimates of accumulation factor quantiles. Accumulation Period Bootstrap Estimate Approx. Standard Error 5 10 1-year 5-year 10-year Rather than sample 120 individual months for each hypothetical 10-year accumulation factor we have used 20 six-month blocks of successive values with random starting points to generate bootstrap estimates of quantiles for the 10-year accumulation factors from the TSE 300 monthly data. We have also generated bootstrap estimates of quantiles for the one-year and five-year accumulation factors again using six-month blocks. The bootstrap estimates are given in Table . They are remarkably consistent with the factors used in the SFTF 2000 report which were derived using stochastic volatility models fitted to the data with only the percent factor for the 10-year accumulation factor appearing a little low in the CIA table. Having given the case for the quantiles of the left tail of the accumulation factors we now discuss how to adjust the model parameters to comply with the calibration requirements. THE LOGNORMAL MODEL For the lognormal model with Yj N CT2 the one-year accumulation factor is 12 S12 exp Y1 Y2 Y12 3 log S12 i 1 3 logS12 N 12 12ff2 So the one-year accumulation factor has a lognormal distribution with parameters 12 fl and a ỰĨ2 a. It is possible to use any two of the table values to solve for the two unknown parameters p and a but this tends to give values that lie outside the acceptable range for the mean. So the recommended method from Appendix A of SFTF 2000 is to keep the mean constant and equal to the empirical mean of the data set is TSE 300 from 1956 to 1999 . This gives the first equation to solve for and r that exp a 2 2 The Lognormal Model 71 Then we can use each of the nine entries in Table as the other equation. Since each entry represents a separate test of the
