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MAXIMUM LIKELIHOOD ESTIMATION 397 dimension of the jth outcome. Then the loglikelihood function is n = 1, + I,. (14.2) 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 1, of the form n li = Clnfi(xi,j), i = 1,2 , , d where (14.3) is the loglikelihood function of the ith marginal distribution. Thus, we can first obtain all the parameter estimates for the marginal dis- tributions 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 (14.3) 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 (14.2) is ignored [110]. There are several approaches to maximizing the second teLm of loglikeli- hood (14.2). One way is to use the pseudo-MLEs. Let ;iii,3 = Fi(xi,j) denote the pseudo-estimates of the cdf of the marginal distributions at each observed value. Then the pseudo-likelihood of the copula function is (14.3) j=1 n (14.4) j=1 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 typi- cally, this will be only one parameter. The second stage is to maximize the loglikelihood (14.2) overall. This can be done by using all the pseudo-MLEs as starting values for the maximization procedure. This will lead to the true 398 NJTlNG COPULA MODELS MLEs of all parameters as long as the necessary regularity conditions are satisfied. Song et al. [110] suggest another algorithm for obtaining the MLEs. We denote the vector of parameters by @.Denote the true value of the parameter by 00. They suggest first obtaining the pseudo-estimates 81 by maximizing 1, as we did above or, by solving the equations d 88 -lw(8) = 0. Because the true MLEs satisfy a a lw(6)= Zc(8), d6 d@ they recommend solving ? for 6k iteratively for k = 2,3, , leading to the MLE 0 = 8,. They show that if the derivatives of the loglikelihoods are well-behaved, this iterative scheme will converge. 14.3 SEMIPARAMETRIC ESTIMATION OF THE COPULA There are several semiparametric or nonparametric procedures that can be used for estimating the copula parameters directly from the data without reference to the form of the marginal distributions. The first way is to use a nonparametric estimate of the cdf terms Fi(zi,j) using an empirical cdf estimator where rank(zi,j) is the rank (from lowest to highest) of the observed values zi,~, xi,2, , to the ordered values (from smallest to largest)'. The copula pseudo-MLEs are obtained by max- imizing the pseudo-likelihood (14.4). This method for estimating the copula parameters does not depend on the values of the parameters of the marginal distributions (only the observed ranks) and the resulting uncertainty intro- duced by estimation process of the marginals. from the ith marginal distribution. The empirical cdf assigns the values A, &, , 'Using n + 1 in the denominator provides a continuity correction and keeps the probilities away from 0 and 1. THE ROLE OF THRESHOLDS 399 Another approach to obtaining the copula parameter in the single-parameter case, is to obtain an estimate of the measure of association, Kendall’s tau, di- rectly from the data. From formula (8.3) in the bivariate case, Kendall’s tau can be written as where (X1,Xz) and (X;,Xz) are iid random variables. Consider a sample (zlj,rc2j), j = 1,2, , n. for each dimension, there are n(n - 1)/2 distinct pairs of points. Thus a natural estimator of Kendall’s tau is which is easily calculated. Because there is a one-to-one correspondence be- tweenLK and the single copula parameter $, we then can obtain the an esti- mate 8. Other techniques, or variations of the above techniques along with their properties have been discussed in detail by numerous authors including Genest and Rivest [46] and Genest, Ghoudri, and Rivest [44]. 14.4 THE ROLE OF THRESHOLDS In earlier chapters, we discussed thresholds below which losses are not recorded. As discussed in Chapter 1, the Base1 I1 framework document suggests using a threshold of 10,000 Euros for operational losses. However, in practice it may be beneficial to use different thresholds for different risk types. For ex- ample, for high-frequency losses, recording lower amounts will give a better understanding of aggregate losses of this type. When thresholds are used, losses below this level are completely ignored. In any estimation exercise, if we want to build models incorporating different thresholds or to estimate ground-up losses, it will be necessary to recognize the distribution below the threshold(s). This complicates the likelihood function somewhat. We now consider the impact on the likelihood function of thresholds either when the data are individual observations or when the data are grouped. Consider two ground-up loss random variable XI and X2 with thresholds dl and dz, respectively. The joint cdf is and the pdf is 400 FlTTlNG COPULA MODELS where c(u1,uz) is the copula density function. We denote the derivatives of the copula function as For grouped (interval) data in setting up the likelihood function, we need to consider only the interval into which an observation falls. We denote the lower and upper limits of the interval for Xlby u1 and w1 and for Xz by v2 and w2. We now consider the four possible cases and express the contribution to the likelihood function by a single bivariate observation expressed in terms of the distributions of X and Y and also expressed in terms of the copula distribution functions and derivatives. Writing down the likelihood contribution is a non- trivial. One needs to be careful about conditioning. If the outcome Xi falls below its threshold dl, then the outcome (Xl,Xz) is not observed. Hence observations need to be conditioned on X1 > dl and also on X2 > d2. Case 1. Individual observation for both XI and X2 If the outcome X falls below its threshold dl, then the outcome (Xl,X2) is not observed. Hence observations need to be conditioned on X1 > dl; also on X2 > d2 (14.5) Case 2. Individual observation for X1 and grouped observation for Xz GOODNESS-OF-F/T TESTING 401 Case 4. Individual observation for XI and grouped observation for Xz The likelihood function is the product of the contributions of all observa- tions, in this case bivariate observations. The separation into two terms that allow a two-stage process (as in the previous section) to get approximate es- timates of the parameters is not possible. In this case, it may be advisable to choose a representative point within each interval for each grouped obser- vation, simplifying the problem considerably. This will lead to approximate estimates using the two-stage process. Then these estimates can be used as initial values for maximizing the likelihood function using the simplex method described in Appendix C. 14.5 GOODNESS-OF-FIT TESTING Klugman and Parsa [70] address the issue of testing the fit of a bivariate copula. They point out that it is possible to use a standard chi-square test of fit. However, to do so requires that we group data into intervals, in this case rectangles over the unit square. Because the data may be concentrated in certain parts of the square, there are likely to be large areas where there are fewer than five expected observations falling into a rectangle. Following methods used in Chapter 11, it would seem logical to group adjacent intervals into larger areas until a minimum of five observations are expected. In two dimensions there is no obviously logical way of combining intervals. Thus we try a different strategy. Consider two random variables X1 and Xz with cdfs F,(z) and Fz(z) re- spectively. The random variables U1 = Fl(X1) and Uz = Fz(X2) are both uniform (0,l) random variables. (This is key in simulation!) Now introduce the conditional random variables V1 = Flz(X1 I XZ) and VZ = FZI(XZ 1 XI). Then the random variables V1 and Uz are mutually independent uniform (0, 402 NTTlNG COPULA MODELS 1) random variables. This can be argued as follows. Consider the random variable Vl = F12(X1 I X2 = z). Because it is a cdf, it must have a uniform (0,l) distribution. This is true for any value of z. Therefore, the distribution of V1 does not depend on the value of X2 or the value of U2 = F2(X2). An identical argument shows that the random variables Vz and U1 are mutually independent uniform (0, 1) random variables. The observed value of distribution function of the conditional random vari- able X2 given XI = z1 is F21(z2 I x1 = 51) = c1 (FXl(Zl), Fx,(z2)). (14.9) The observed value v2 of the random variable V2 can be obtained from the observed values of the bivariate random variables (XI, X2) from Thus, we can generate a univariate set of data that should look like a sample from a uniform (0,l) distribution if the combination of marginal distributions and the copula fits the data well. Klugman and Parsa [70] suggest the following procedure for testing the fit based entirely on univariate methods: Step 1. Fit and select the marginal distributions using univariate meth- ods Step 2. Test the conditional distribution of V1 for uniformity Step 3. Test the conditional distribution of V2 for uniformity The tests for uniformity can be done using a formal goodness-of-fit test such as a Kolmogorov-Smirnov test. Alternatively, one can plot the cdf of the empirical distributions, which should be linear (or close to it). This is equivalent to doing a p-p plot for the uniform distribution. In higher dimensions, the problems become more complicated. However, by following the above procedures for all pairs of random variables, one can be reasonably satisfied about the overall fit of the model (both marginals and copula). This requires a significant effort, but can be automated relatively easily. 14.6 AN EXAMPLE We illustrate some of the concepts in this chapter using simulated data. The data consist of 100 pairs {(zj, yj), j = 1,2, , 100) that are simulated from the bivariate distribution with a Gumbel (0 = 3) copula and marginal dis- tributions loglogistic (0 = 1, 7 = 3) and Weibull (0 = 1, 7 = 3). This is a five-parameter model. We first use maximum likelihood to fit the same. “correct” five- parameter distribution but with all parameters treated as un- known. We then attempt to fit an “incorrect” distribution with marginals of the same form but a misspecified copula. AN EXAMPLE 403 Given the 100 points, the 5-parameter joint distribution is easy to fit di- rectly using maximum likelihood. The loglikelihood function is 100 where fl(z) and fi(x) are the marginal distributions and c(z1,uz) is the copula density function. The first term was maximized with the following results Distribution 8 7 Loglogistic 1.00035 3.27608 Weibull 0.74106 3.22952 Gumbel copula - These are the maximum likelihood estimates of the marginal distributions. The entire likelihood was then maximized. This resulted in the following estimates of the five parameters. Distribution 8 T Loglogistic 1.00031 3.25611 Weibull 0.75254 3.08480 Gumbel copula 2.84116 - Note that the parameter estimates for the marginal distribution changed slightly as a result of simultaneously estimating the copula parameter. The overall negative loglikelihood was 10.06897. To illustrate the impact of esti- mation errors, we now simulate, using the same random numbers, 100 points from the fitted distribution. The results are illustrated in Figure 14.1, where both sets of simulated data are plotted. The key observation from Figure 14.1 this plot is that the points from the fitted distribution are quite close to the original points. We repeat this exercise but using the Joe copula as an alternative. The results of the simulta- neous maximum likelihood estimation of all five parameters gave the following estimates: Distribution 8 r Loglogistic 0.98330 3.12334 Weibull 0.74306 2.89547 Joe copula 3.85403 - The overall negative loglikelihood increased to 15.68361. This is a quite large increase over that using the Gumbel copula. Note also that the estimates of the parameters of the marginal distributions are also changed. To illustrate 404 FITTING COPULA MODELS 01 0 02 04 06 08 1 Logloglstlc Fig. 14.1 MLEfitted marginals and Gumbel copula the impact of misspecification of the copula together with estimation errors, we simulated, using the same random numbers, 100 points from the fitted distribution. The results are illustrated in Figure 14.2, where both sets of simulated data are plotted. Note that the second set of points are further from the original set of simulated points. Rather than use the observed values of the marginal distribution to estimate the cop- ula parameter, we used the ranks of those values. The ranks are independent of the choice of marginal distribution. Using these values, together with the “correct” specification of the copula, we also calculated the value of the neg- ative loglikelihood with these estimates. Of course, the negative loglikelihood will be higher because the MLE method gave the lowest possible value. It is 13.67761 which is somewhat greater than the minimum of 10.06897. The new estimate of the Gumbel copula parameter is 2.69586. The corresponding simulated values are shown in Figure 14.3. Finally, we also used the nonparametric approach with the misspecified copula function, the Joe copula. The estimate of the Joe copula parameter is 3.31770 with a corresponding likelihood of 21.58245, which is quite a lot greater than the other likelihood values. The corresponding simulated values are plotted in Figure ??. It is quite interesting to note that a visual assessment of the scatterplots is not very helpful. It is impossible to distinguish the different plot in terms of the fit to the original data. All four plots look good. However, the values For the same data, we also used the semiparametric approach. AN EXAMPLE 405 Fig. 14.2 MLE-fitted marinals and Joe copula 406 F/TT/NG COPULA MODELS Logloglstlc fig. 14.4 Semiparametric-fitted Joe copula of the likelihood function for the four cases are quite different. This suggest that it is important to carry out serious technical analysis of the data rather than relying on pure judgement based on observation.0 [...]... 78 joint, 282 k-point mixture, 68 log-t, 61 logarithmic, 110, 120 logistic, 87 loglogzstic, 56 lognormal, 54, 56, 65, 79, 152, 179 loss count, 148 marganal, 282 medium tailed, 180 mixed frequency, 129 mzxture/mixzng, 68, 79 modified negative binomial, 204 negative binomial 100 -101 , 104 , 109 , 113, 119, 133, 198, 209, 307 as Poisson mixture, 101 extended truncated, 11 0 negative hypergeometric, 190 Neyman... Statistics, New York: Wiley 103 Ross, S (1996) Stochastic Processes, 2nd ed., New York: Wiley 104 Ross, S (2003), Introduction to Probability Models, 8th ed., San Diego: Academic Press 105 Scollnik, D (2002) Modeling size-of-loss distributions for exact data in WinBUGS,” Journal of Actuarial Practice, 10, 193-218 106 Self, S and Liang, K (1987) “Asymptotic properties of maximum likelihood estimators and... (1997) “Thinking coherently,” RISK, 10, 11, 68-71 7 Balkema, A and de Haan, L (1974) “Residual life at great ages,” Annals of Probability, 2, 792-804 8 Baker, C (1977) The Numerical Treatment of Integral Equations, Oxford: Clarendon Press 417 418 REFERENCES 9 Basel Committee on Banking Supervision (1998) Operational Risk Management, Basel: Bank for International Settlements 10 Basel Committee on Banking... censored, 35 skewness, 29 standard deviation, 29 support, 21 variance, 29 Recursion, 104 , 107 , 109 , 111, 115-116, 120, 122, 159, 161-1 64 Recursive formula aggregate loss dzstribution, 159 continuous severity distribution, 379 Jor compound Jreqency, 11 6 Relutzve efficiency, 259 Reputational risk, 44 Right censored tiaraable, 33 Risk measure coherent, 4 3 P-value, 258 Panjer recursion, 11 5-1 17, 122, 159... binomaal ( E T N B ) , 110 extended truncated negative binomial, 110 extreme value, 188, 209, 354 frailty, 8 2 Frcchet, 189 gamma, 30, 96, 53, 72, 7'5, 78, 81, 83, 89, 91 generalized beta, 62 generalized extreme value, 189 generahted Pareto, 59, 202, 209 generalized Poisson-Pascal, 123 yeneruhzed Waring, 130, 300 geometric-Poisson, 122 geometric- E T N B , 122 geometric, 101 , 104 , 119, 171 Gumbel, 189... Springer 100 Rioux, J and Klugman S (2006) “Toward a unified approach to fitting loss models,” to appear in North American Actuarial Journal 101 Robertson, J (1992) “The computation of aggregate loss distributions,” Proceedings of the Casualty Actuarial Society, 79, 57-133 424 REFERENCES 102 Rohatgi, V (1976) A n Introduction to Probability Theory and Mathematical Statistics, New York: Wiley 103 Ross,... nonstandard conditions,” Journal of the American Statistical Association 82, 605- 610 107 Schwarz, G (1978) “Estimating the dimension of a model,” Annals of Statistics, 6 , 461-464 108 Simon, L (1961), “Fitting negative binomial distributions by the method of maximum likelihood,” Proceedings of the Casualty Actuarial Society, 48, 45-53 109 Sklar, A (1959) “Functions de rhparation h n dimensions et leur marges,”... Mikosch, T (1997) Modelling Extremal Events for Insurance and Finance, Berlin: Springer 31 Embrechts, P., MacNeil, A and Straumann, D (2002) “Correlation and dependency in risk management: Properties and pitfalls,” in Risk Management: Value at Risk and Beyond, M Dempster (ed), Cambridge; Cambridge University Press 32 Embrechts, P., Maejima, M and Teugels, J (1985) “Asymptotic behaviour of compound distributions,”... (1990) “Asymptotic behaviour of Poisson mixtures with applications,” Advances in Applied Probability, 22, 147-159 128 Wirch J (1999) “Raising Value at Risk, ” North American Actuarial Journal 3, 106 -115 129 Wirch, J., and Hardy, M (1999) “A synthesis of risk measures for capital adequacy,” Insurance: Mathematics and Economics, 2 5 , 337-348 Index A Aggregate loss distribution, 148, 154 approximating... variable, 21 Distribution function, 18 Distribution function plot, 323 Distribution of the maximum, 190 Distribution (a b 0 ) class, 104 , 119 (a b 1 ) class, lor, 120 t , 47 aggregate loss, 148 Bernoulli, 202 beta, 62, 130, 201 binomial-beta, 130 binomial, 103 , 119, 138, 310 zero-truncated, 121 BUTT, 83 59, compound, 150 compound frequency, 111, 122 recursive formula, 116 compound geometric-exponential, . document suggests using a threshold of 10, 000 Euros for operational losses. However, in practice it may be beneficial to use different thresholds for different risk types. For ex- ample, for high-frequency. of the concepts in this chapter using simulated data. The data consist of 100 pairs {(zj, yj), j = 1,2, , 100 ) that are simulated from the bivariate distribution with a Gumbel (0. AN EXAMPLE 403 Given the 100 points, the 5-parameter joint distribution is easy to fit di- rectly using maximum likelihood. The loglikelihood function is 100 where fl(z) and fi(x)