## Archive for the ‘**Operational Risk**’ Category

## Modeling Dollar Amounts in Regression Setting

After switching the role from the credit risk to the operational risk in 2015, I spent countless weekend hours in the Starbucks researching on how to model operational losses in the regression setting in light of the heightened scrutiny. While I feel very comfortable with various frequency models, how to model severity and loss remain challenging both conceptually and empirically. The same challenge also holds true for modeling other financial measures in dollar amounts, such as balance, profit, or cost.

Most practitioners still prefer modeling severity and loss under the Gaussian distributional assumption explicitly or implicitly. In practice, there are 3 commonly used approaches, as elaborated below.

– First of all, the simple OLS regression to model severity and loss directly without any transformation remains the number one choice due to the simplicity. Given the inconsistency between the empirical data range and the conceptual domain for a Gaussian distribution, it is evidential that this approach is problematic.

– Secondly, the OLS regression to model LOG transformed severity and loss under the Lognormal distributional assumption is also a common approach. In this method, Log(Y) instead of Y is estimated. However, given E(Log(Y)|X) != Log(E(Y|X)), the estimation bias is introduced and therefore should be corrected by MSE / 2. In addition, the positive domain of a Lognormal might not work well in cases of losses with a lower bound that can be either zero or a known threshold value.

– At last, the Tobit regression under the censored Normal distribution seems a viable solution that supports the non-negative or any above-threshold values shown in severity or loss measures. Nonetheless, the censorship itself is questionable given that the unobservability of negative or below-threshold values is not due to the censorship but attributable to the data nature governed by the data collection process. Therefore, the argument for the data censorship is not well supported.

Considering the aforementioned challenge, I investigated and experimented various approaches given different data characteristics observed empirically.

– In cases of severity or loss observed in the range of (0, inf), GLM under Gamma or Inverse Gaussian distributional assumption can be considered (https://statcompute.wordpress.com/2015/08/16/some-considerations-of-modeling-severity-in-operational-losses). In addition, the mean-variance relationship can be employed to assess the appropriateness of the correct distribution by either the modified Park test (https://statcompute.wordpress.com/2016/11/20/modified-park-test-in-sas) or the value of power parameter in the Tweedie distribution (https://statcompute.wordpress.com/2017/06/24/using-tweedie-parameter-to-identify-distributions).

– In cases of severity or loss observed in the range of [alpha, inf) with alpha being positive, then a regression under the type-I Pareto distribution (https://statcompute.wordpress.com/2016/12/11/estimate-regression-with-type-i-pareto-response) can be considered. However, there is a caveat that the conditional mean only exists when the shape parameter is large than 1.

– In cases of severity or loss observed in the range of [0, inf) with a small number of zeros, then a regression under the Lomax distribution (https://statcompute.wordpress.com/2016/11/13/parameter-estimation-of-pareto-type-ii-distribution-with-nlmixed-in-sas) or the Tweedie distribution (https://statcompute.wordpress.com/2017/06/29/model-operational-loss-directly-with-tweedie-glm) can be considered. For the Lomax model, it is worth pointing out that the shape parameter alpha has to be large than 2 in order to to have both mean and variance defined.

– In cases of severity or loss observed in the range of [0, inf) with many zeros, then a ZAGA or ZAIG model (https://statcompute.wordpress.com/2017/09/17/model-non-negative-numeric-outcomes-with-zeros) can be considered by assuming the measure governed by a mixed distribution between the point-mass at zeros and the standard Gamma or Inverse Gaussian. As a result, a ZA model consists of 2 sub-models, a nu model separating zeros and positive values and a mu model estimating the conditional mean of positive values.

## Estimating Parameters of A Hyper-Poisson Distribution in SAS

Similar to COM-Poisson, Double-Poisson, and Generalized Poisson distributions discussed in my previous post (https://statcompute.wordpress.com/2016/11/27/more-about-flexible-frequency-models/), the Hyper-Poisson distribution is another extension of the standard Poisson and is able to accommodate both under-dispersion and over-dispersion that are common in real-world problems. Given the complexity of parameterization and computation, the Hyper-Poisson is somewhat under-investigated. To the best of my knowledge, there is no off-shelf computing routine in SAS for the Hyper-Poisson distribution and only a R function available in http://www4.ujaen.es/~ajsaez/hp.fit.r written by A.J. Sáez-Castillo and A. Conde-Sánchez (2013).

The SAS code presented below is the starting point of my attempt on the Hyper-Poisson and its potential applications. The purpose is to replicate the calculation result shown in the Table 6 of “On the Hyper-Poisson Distribution and its Generalization with Applications” by Bayo H. Lawal (2017) (http://www.journalrepository.org/media/journals/BJMCS_6/2017/Mar/Lawal2132017BJMCS32184.pdf). As a result, the parameterization employed in my SAS code will closely follow Bayo H. Lawal (2017) instead of A.J. Sáez-Castillo and A. Conde-Sánchez (2013).

data d1; input y n @@; datalines; 0 121 1 85 2 19 3 1 4 0 5 0 6 1 ; run; data df; set d1; where n > 0; do i = 1 to n; output; end; run; proc nlmixed data = df; parms lambda = 1 beta = 1; theta = 1; do k = 1 to 100; theta = theta + gamma(beta) * (lambda ** k) / gamma(beta + k); end; prob = (gamma(beta) / gamma(beta + y)) * ((lambda ** y) / theta); ll = log(prob); model y ~ general(ll); run; /* Standard Parameter Estimate Error DF t Value Pr > |t| Alpha lambda 0.3752 0.1178 227 3.19 0.0016 0.05 beta 0.5552 0.2266 227 2.45 0.0150 0.05 */

As shown, the estimated Lambda = 0.3752 and the estimated Beta = 0.5552 are identical to what is presented in the paper. The next step is be to explore applications in the frequency modeling as well as its value in business cases.

## Model Non-Negative Numeric Outcomes with Zeros

As mentioned in the previous post (https://statcompute.wordpress.com/2017/06/29/model-operational-loss-directly-with-tweedie-glm/), we often need to model non-negative numeric outcomes with zeros in the operational loss model development. Tweedie GLM provides a convenient interface to model non-negative losses directly by assuming that aggregated losses are the Poisson sum of Gamma outcomes, which however might not be well supported empirically from the data generation standpoint.

In examples below, we demonstrated another flexible option, namely Zero-Adjusted (ZA) models, in both scenarios of modeling non-negative numeric outcomes, one with a small number of zeros and the other with a large number of zeros. The basic idea of ZA models is very intuitive and similar to the concept of Hurdle models for count outcomes. In a nutshell, non-negative numeric outcomes can be considered two data generation processes, one for point-mass at zeros and the other governed by a statistical distribution for positive outcomes. The latter could be either Gamma or Inverse Gaussian.

First of all, we sampled down an auto-claim data in a way that only 10 claims are zeros and the rest are all positive. While 10 is an arbitrary choice in the example, other small numbers should show similar results.

pkgs <- list("cplm", "gamlss", "MLmetrics") lapply(pkgs, require, character.only = T) data(AutoClaim, package = "cplm") df1 <- na.omit(AutoClaim) # SMALL NUMBER OF ZEROS set.seed(2017) smp <- sample(seq(nrow(df1[df1$CLM_AMT == 0, ])), size = 10, replace = FALSE) df2 <- rbind(df1[df1$CLM_AMT > 0, ], df1[df1$CLM_AMT == 0, ][smp, ])

Next, we applied both Tweedie and zero-adjusted Gamma (ZAGA) models to the data with only 10 zero outcomes. It is worth mentioning that ZAGA doesn’t have to be overly complex in this case. As shown below, while we estimated the Gamma Mu parameter with model attributes, the Nu parameter to separate zeros is just a constant with the intercept = -5.4. Both Tweedie and GAZA models gave very similar estimated parameters and predictive measures with MAPE = 0.61.

tw <- cpglm(CLM_AMT ~ BLUEBOOK + NPOLICY, data = df2) # Estimate Std. Error t value Pr(>|t|) # (Intercept) 8.194e+00 7.234e-02 113.277 < 2e-16 *** # BLUEBOOK 2.047e-05 3.068e-06 6.671 3.21e-11 *** # NPOLICY 7.274e-02 3.102e-02 2.345 0.0191 * MAPE(df2$CLM_AMT, fitted(tw)) # 0.6053669 zaga0 <- gamlss(CLM_AMT ~ BLUEBOOK + NPOLICY, data = df2, family = "ZAGA") # Mu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 8.203e+00 4.671e-02 175.629 < 2e-16 *** # BLUEBOOK 2.053e-05 2.090e-06 9.821 < 2e-16 *** # NPOLICY 6.948e-02 2.057e-02 3.377 0.000746 *** # Nu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) -5.3886 0.3169 -17 <2e-16 *** MAPE(df2$CLM_AMT, (1 - fitted(zaga0, what = "nu")) * fitted(zaga0, what = "mu")) # 0.6053314

In the next case, we used the full data with a large number of zeros in the response and then applied both Tweedie and ZAGA models again. However, in ZAGA model, we estimated two sub-models this time, one for the Nu parameter to separate zeros from non-zeros and the other for the Mu parameter to model non-zero outcomes. As shown below, ZAGA outperformed Tweedie in terms of MAPE due to the advantage that ZAGA is able to explain two data generation schemes separately with different model attributes, which is the capability beyond what Tweedie can provide.

# LARGE NUMBER OF ZEROS tw <- cpglm(CLM_AMT ~ BLUEBOOK + NPOLICY + CLM_FREQ5 + MVR_PTS + INCOME, data = df1) # Estimate Std. Error t value Pr(>|t|) # (Intercept) 6.854e+00 1.067e-01 64.241 < 2e-16 *** # BLUEBOOK 1.332e-05 4.495e-06 2.963 0.00305 ** # NPOLICY 4.380e-02 3.664e-02 1.195 0.23196 # CLM_FREQ5 2.064e-01 2.937e-02 7.026 2.29e-12 *** # MVR_PTS 1.066e-01 1.510e-02 7.063 1.76e-12 *** # INCOME -4.606e-06 8.612e-07 -5.348 9.12e-08 *** MAPE(df1$CLM_AMT, fitted(tw)) # 1.484484 zaga1 <- gamlss(CLM_AMT ~ BLUEBOOK + NPOLICY, nu.formula = ~(CLM_FREQ5 + MVR_PTS + INCOME), data = df1, family = "ZAGA") # Mu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 8.203e+00 4.682e-02 175.218 < 2e-16 *** # BLUEBOOK 2.053e-05 2.091e-06 9.816 < 2e-16 *** # NPOLICY 6.948e-02 2.067e-02 3.362 0.000778 *** # Nu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 1.153e+00 5.077e-02 22.72 <2e-16 *** # CLM_FREQ5 -3.028e-01 2.283e-02 -13.26 <2e-16 *** # MVR_PTS -1.509e-01 1.217e-02 -12.41 <2e-16 *** # INCOME 7.285e-06 6.269e-07 11.62 <2e-16 *** MAPE(df1$CLM_AMT, (1 - fitted(zaga1, what = "nu")) * fitted(zaga1, what = "mu")) # 1.470228

Given the great flexibility of ZA models, we also have the luxury to explore other candidates than ZAGA. For instance, if the positive part of non-negative outcomes demonstrates a high variance, we can also try a zero-inflated Inverse Gaussian (ZAIG) model, as shown below.

zaig1 <- gamlss(CLM_AMT ~ BLUEBOOK + NPOLICY, nu.formula = ~(CLM_FREQ5 + MVR_PTS + INCOME), data = df1, family = "ZAIG") # Mu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 8.205e+00 5.836e-02 140.591 < 2e-16 *** # BLUEBOOK 2.163e-05 2.976e-06 7.268 3.97e-13 *** # NPOLICY 5.898e-02 2.681e-02 2.200 0.0278 * # Nu Coefficients: # Estimate Std. Error t value Pr(>|t|) # (Intercept) 1.153e+00 5.077e-02 22.72 <2e-16 *** # CLM_FREQ5 -3.028e-01 2.283e-02 -13.26 <2e-16 *** # MVR_PTS -1.509e-01 1.217e-02 -12.41 <2e-16 *** # INCOME 7.285e-06 6.269e-07 11.62 <2e-16 *** MAPE(df1$CLM_AMT, (1 - fitted(zaig1, what = "nu")) * fitted(zaig1, what = "mu")) # 1.469236

## Model Operational Losses with Copula Regression

In the previous post (https://statcompute.wordpress.com/2017/06/29/model-operational-loss-directly-with-tweedie-glm), it has been explained why we should consider modeling operational losses for non-material UoMs directly with Tweedie models. However, for material UoMs with significant losses, it is still beneficial to model the frequency and the severity separately.

In the prevailing modeling practice for operational losses, it is often convenient to assume a functional independence between frequency and severity models, which might not be the case empirically. For instance, in the economic downturn, both the frequency and the severity of consumer frauds might tend to increase simultaneously. With the independence assumption, while we can argue that same variables could be included in both frequency and severity models and therefore induce a certain correlation, the frequency-severity dependence and the its contribution to the loss distribution might be overlooked.

In the context of Copula, the distribution of operational losses can be considered a joint distribution determined by both marginal distributions and a parameter measuring the dependence between marginals, of which marginal distributions can be Poisson for the frequency and Gamma for the severity. Depending on the dependence structure in the data, various copula functions might be considered. For instance, a product copula can be used to describe the independence. In the example shown below, a Gumbel copula is considered given that it is often used to describe the positive dependence on the right tail, e.g. high severity and high frequency. For details, the book “Copula Modeling” by Trivedi and Zimmer is a good reference to start with.

In the demonstration, we simulated both frequency and severity measures driven by the same set of co-variates. Both are positively correlated with the Kendall’s tau = 0.5 under the assumption of Gumbel copula.

library(CopulaRegression) # number of observations to simulate n <- 100 # seed value for the simulation set.seed(2017) # design matrices with a constant column X <- cbind(rep(1, n), runif(n), runif(n)) # define coefficients for both Poisson and Gamma regressions p_beta <- g_beta <- c(3, -2, 1) # define the Gamma dispersion delta <- 1 # define the Kendall's tau tau <- 0.5 # copula parameter based on tau theta <- 1 / (1 - tau) # define the Gumbel Copula family <- 4 # simulate outcomes out <- simulate_regression_data(n, g_beta, p_beta, X, X, delta, tau, family, zt = FALSE) G <- out[, 1] P <- out[, 2]

After the simulation, a Copula regression is estimated with Poisson and Gamma marginals for the frequency and the severity respectively. As shown in the model estimation, estimated parameters with related inferences are different between independent and dependent assumptions.

m <- copreg(G, P, X, family = 4, sd.error = TRUE, joint = TRUE, zt = FALSE) coef <- c("_CONST", "X1", "X2") cols <- c("ESTIMATE", "STD. ERR", "Z-VALUE") g_est <- cbind(m$alpha, m$sd.alpha, m$alpha / m$sd.alpha) p_est <- cbind(m$beta, m$sd.beta, m$beta / m$sd.beta) g_est0 <- cbind(m$alpha0, m$sd.alpha0, m$alpha0 / m$sd.alpha0) p_est0 <- cbind(m$beta0, m$sd.beta0, m$beta0 / m$sd.beta0) rownames(g_est) <- rownames(g_est0) <- rownames(p_est) <- rownames(p_est0) <- coef colnames(g_est) <- colnames(g_est0) <- colnames(p_est) <- colnames(p_est0) <- cols # estimated coefficients for the Gamma regression assumed dependence print(g_est) # ESTIMATE STD. ERR Z-VALUE # _CONST 2.9710512 0.2303651 12.897141 # X1 -1.8047627 0.2944627 -6.129003 # X2 0.9071093 0.2995218 3.028526 # estimated coefficients for the Gamma regression assumed dependence print(p_est) # ESTIMATE STD. ERR Z-VALUE # _CONST 2.954519 0.06023353 49.05107 # X1 -1.967023 0.09233056 -21.30414 # X2 1.025863 0.08254870 12.42736 # estimated coefficients for the Gamma regression assumed independence # should be identical to GLM() outcome print(g_est0) # ESTIMATE STD. ERR Z-VALUE # _CONST 3.020771 0.2499246 12.086727 # X1 -1.777570 0.3480328 -5.107478 # X2 0.905527 0.3619011 2.502140 # estimated coefficients for the Gamma regression assumed independence # should be identical to GLM() outcome print(p_est0) # ESTIMATE STD. ERR Z-VALUE # _CONST 2.939787 0.06507502 45.17536 # X1 -2.010535 0.10297887 -19.52376 # X2 1.088269 0.09334663 11.65837

If we compare conditional loss distributions under different dependence assumptions, it shows that the predicted loss with Copula regression tends to have a fatter right tail and therefore should be considered more conservative.

df <- data.frame(g = G, p = P, x1 = X[, 2], x2 = X[, 3]) glm_p <- glm(p ~ x1 + x2, data = df, family = poisson(log)) glm_g <- glm(g ~ x1 + x2, data = df, family = Gamma(log)) loss_dep <- predict(m, X, X, independence = FALSE)[3][[1]][[1]] loss_ind <- fitted(glm_p) * fitted(glm_g) den <- data.frame(loss = c(loss_dep, loss_ind), lines = rep(c("DEPENDENCE", "INDEPENDENCE"), each = n)) ggplot(den, aes(x = loss, fill = lines)) + geom_density(alpha = 0.5)

## Model Operational Loss Directly with Tweedie GLM

In the development of operational loss forecasting models, the Frequency-Severity modeling approach, which the frequency and the severity of a Unit of Measure (UoM) are modeled separately, has been widely employed in the banking industry. However, sometimes it also makes sense to model the operational loss directly, especially for UoMs with non-material losses. First of all, given the low loss amount, the effort of developing two models, e.g. frequency and severity, might not be justified. Secondly, for UoMs with low losses due to low frequencies, modeling the frequency and the severity separately might overlook the internal connection between the low frequency and the subsequent low loss amount. For instance, when the frequency N = 0, then the loss L = $0 inevitably.

The Tweedie distribution is defined as a Poisson sum of Gamma random variables. In particular, if the frequency of loss events N is assumed a Poisson distribution and the loss amount L_i of an event i, where i = 0, 1 … N, is assumed a Gamma distribution, then the total loss amount L = SUM[L_i] would have a Tweedie distribution. When there is no loss event, e.g. N = 0, then Prob(L = $0) = Prob(N = 0) = Exp(-Lambda). However, when N > 0, then L = L_1 + … + L_N > $0 is governed by a Gamma distribution, e.g. sum of I.I.D. Gamma also being Gamma.

For the Tweedie loss, E(L) = Mu and VAR(L) = Phi * (Mu ** P), where P is called the index parameter and Phi is the dispersion parameter. When P approaches 1 and therefore VAR(L) approaches Phi * E(L), the Tweedie would be similar to a Poisson-like distribution. When P approaches 2 and therefore VAR(L) approaches Phi * (E(L) ** 2), the Tweedie would be similar to a Gamma distribution. When P is between 1 and 2, then the Tweedie would be a compound mixture of Poisson and Gamma, where P and Phi can be estimated.

To estimate a regression with the Tweedie distributional assumption, there are two implementation approaches in R with cplm and statmod packages respectively. With the cplm package, the Tweedie regression can be estimated directly as long as P is in the range of (1, 2), as shown below. In the example, the estimated index parameter P is 1.42.

> library(cplm) > data(FineRoot) > m1 <- cpglm(RLD ~ Zone + Stock, data = FineRoot) > summary(m1) Deviance Residuals: Min 1Q Median 3Q Max -1.0611 -0.6475 -0.3928 0.1380 1.9627 Estimate Std. Error t value Pr(>|t|) (Intercept) -1.95141 0.14643 -13.327 < 2e-16 *** ZoneOuter -0.85693 0.13292 -6.447 2.66e-10 *** StockMM106 0.01177 0.17535 0.067 0.947 StockMark -0.83933 0.17476 -4.803 2.06e-06 *** --- Estimated dispersion parameter: 0.35092 Estimated index parameter: 1.4216 Residual deviance: 203.91 on 507 degrees of freedom AIC: -157.33

The statmod package provides a more general and flexible solution with the two-stage estimation, which will estimate the P parameter first and then estimate regression parameters. In the real-world practice, we could do a coarse search to narrow down a reasonable range of P and then do a fine search to identify the optimal P value. As shown below, all estimated parameters are fairly consistent with ones in the previous example.

> library(tweedie) > library(statmod) > prof <- tweedie.profile(RLD ~ Zone + Stock, data = FineRoot, p.vec = seq(1.1, 1.9, 0.01), method = "series") 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.2 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.3 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.5 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.6 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.7 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.8 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.9 .................................................................................Done. > prof$p.max [1] 1.426531 > m2 <- glm(RLD ~ Zone + Stock, data = FineRoot, family = tweedie(var.power = prof$p.max, link.power = 0)) > summary(m2) Deviance Residuals: Min 1Q Median 3Q Max -1.0712 -0.6559 -0.3954 0.1380 1.9728 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.95056 0.14667 -13.299 < 2e-16 *** ZoneOuter -0.85823 0.13297 -6.454 2.55e-10 *** StockMM106 0.01204 0.17561 0.069 0.945 StockMark -0.84044 0.17492 -4.805 2.04e-06 *** --- (Dispersion parameter for Tweedie family taken to be 0.4496605) Null deviance: 241.48 on 510 degrees of freedom Residual deviance: 207.68 on 507 degrees of freedom AIC: NA

## Using Tweedie Parameter to Identify Distributions

In the development of operational loss models, it is important to identify which distribution should be used to model operational risk measures, e.g. frequency and severity. For instance, why should we use the Gamma distribution instead of the Inverse Gaussian distribution to model the severity?

In my previous post https://statcompute.wordpress.com/2016/11/20/modified-park-test-in-sas, it is shown how to use the Modified Park test to identify the mean-variance relationship and then decide the corresponding distribution of operational risk measures. Following the similar logic, we can also leverage the flexibility of the Tweedie distribution to accomplish the same goal. Based upon the parameterization of a Tweedie distribution, the variance = Phi * (Mu ** P), where Mu is the mean and P is the power parameter. Depending on the specific value of P, the Tweedie distribution can accommodate several important distributions commonly used in the operational risk modeling, including Poisson, Gamma, Inverse Gaussian. For instance,

- With P = 0, the variance would be independent of the mean, indicating a Normal distribution.
- With P = 1, the variance would be in a linear form of the mean, indicating a Poisson-like distribution
- With P = 2, the variance would be in a quadratic form of the mean, indicating a Gamma distribution.
- With P = 3, the variance would be in a cubic form of the mean, indicating an Inverse Gaussian distribution.

In the example below, it is shown that the value of P is in the neighborhood of 1 for the frequency measure and is near 3 for the severity measure and that, given P closer to 3, the Inverse Gaussian regression would fit the severity better than the Gamma regression.

library(statmod) library(tweedie) profile1 <- tweedie.profile(Claim_Count ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE) print(profile1$p.max) # [1] 1.216327 # The P parameter close to 1 indicates that the claim_count might follow a Poisson-like distribution profile2 <- tweedie.profile(Severity ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE) print(profile2$p.max) # [1] 2.844898 # The P parameter close to 3 indicates that the severity might follow an Inverse Gaussian distribution BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = Gamma(link = log))) # [1] 360.8064 BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = inverse.gaussian(link = log))) # [1] 350.2504

Together with the Modified Park test, the estimation of P in a Tweedie distribution is able to help us identify the correct distribution employed in operational loss models in the context of GLM.

## Double Poisson Regression in SAS

In the previous post (https://statcompute.wordpress.com/2016/11/27/more-about-flexible-frequency-models), I’ve shown how to estimate the double Poisson (DP) regression in R with the gamlss package. The hurdle of estimating DP regression is the calculation of a normalizing constant in the DP density function, which can be calculated either by the sum of an infinite series or by a closed form approximation. In the example below, I will show how to estimate DP regression in SAS with the GLIMMIX procedure.

First of all, I will show how to estimate DP regression by using the exact DP density function. In this case, we will approximate the normalizing constant by computing a partial sum of the infinite series, as highlighted below.

data poi; do n = 1 to 5000; x1 = ranuni(1); x2 = ranuni(2); x3 = ranuni(3); y = ranpoi(4, exp(1 * x1 - 2 * x2 + 3 * x3)); output; end; run; proc glimmix data = poi; nloptions tech = quanew update = bfgs maxiter = 1000; model y = x1 x2 x3 / link = log solution; theta = exp(_phi_); _variance_ = _mu_ / theta; p_u = (exp(-_mu_) * (_mu_ ** y) / fact(y)) ** theta; p_y = (exp(-y) * (y ** y) / fact(y)) ** (1 - theta); f = (theta ** 0.5) * ((exp(-_mu_)) ** theta); do i = 1 to 100; f = f + (theta ** 0.5) * ((exp(-i) * (i ** i) / fact(i)) ** (1 - theta)) * ((exp(-_mu_) * (_mu_ ** i) / fact(i)) ** theta); end; k = 1 / f; prob = k * (theta ** 0.5) * p_y * p_u; if log(prob) ~= . then _logl_ = log(prob); run;

Next, I will show the same estimation routine by using the closed form approximation.

proc glimmix data = poi; nloptions tech = quanew update = bfgs maxiter = 1000; model y = x1 x2 x3 / link = log solution; theta = exp(_phi_); _variance_ = _mu_ / theta; p_u = (exp(-_mu_) * (_mu_ ** y) / fact(y)) ** theta; p_y = (exp(-y) * (y ** y) / fact(y)) ** (1 - theta); k = 1 / (1 + (1 - theta) / (12 * theta * _mu_) * (1 + 1 / (theta * _mu_))); prob = k * (theta ** 0.5) * p_y * p_u; if log(prob) ~= . then _logl_ = log(prob); run;

While the first approach is more accurate by closely following the DP density function, the second approach is more efficient with a significantly lower computing cost. However, both are much faster than the corresponding R function gamlss().