# More about Flexible Frequency Models

Modeling the frequency is one of the most important aspects in operational risk models. In the previous post (https://statcompute.wordpress.com/2016/05/13/more-flexible-approaches-to-model-frequency), the importance of flexible modeling approaches for both under-dispersion and over-dispersion has been discussed.

In addition to the quasi-poisson regression, three flexible frequency modeling techniques, including generalized poisson, double poisson, and Conway-Maxwell poisson, with their implementations in R should also be demonstrated below. While the example is specifically related to the over-dispersed data simulated with the negative binomial distributional assumption, these approaches can be generalized to the under-dispersed data as well given their flexibility. However, as demonstrated below, the calculation of parameters for these modeling approaches is not straight-forward.

Over-Dispersed Data Simulation

```> set.seed(1)
> ### SIMULATE NEG. BINOMIAL WITH MEAN(X) = MU AND VAR(X) = MU + MU ^ 2 / THETA
> df <- data.frame(y = MASS::rnegbin(1000, mu = 10, theta = 5))
> ### DATA MEAN
> mean(df\$y)
[1] 9.77
> ### DATA VARIANCE
> var(df\$y)
[1] 30.93003003
```

Generalized Poisson Regression

```> library(VGAM)
> gpois <- vglm(y ~ 1, data = df, family = genpoisson)
> gpois.theta <- exp(coef(gpois)[2])
> gpois.lambda <- (exp(coef(gpois)[1]) - 1) / (exp(coef(gpois)[1]) + 1)
> ### ESTIMATE MEAN = THETA / (1 - LAMBDA)
> gpois.theta / (1 - gpois.lambda)
(Intercept):2
9.77
> ### ESTIMATE VARIANCE = THETA / ((1 - LAMBDA) ^ 3)
> gpois.theta / ((1 - gpois.lambda) ^ 3)
(Intercept):2
31.45359991
```

Double Poisson Regression

```> ### DOUBLE POISSON
> library(gamlss)
> dpois <- gamlss(y ~ 1, data = df, family = DPO, control = gamlss.control(n.cyc = 100))
> ### ESTIMATE MEAN
> dpois.mu <- exp(dpois\$mu.coefficients)
> dpois.mu
(Intercept)
9.848457877
> ### ESTIMATE VARIANCE = MU * SIGMA
> dpois.sigma <- exp(dpois\$sigma.coefficients)
> dpois.mu * dpois.sigma
(Intercept)
28.29229702
```

Conway-Maxwell Poisson Regression

```> ### CONWAY-MAXWELL POISSON
> library(CompGLM)
> cpois <- glm.comp(y ~ 1, data = df)
> cpois.lambda <- exp(cpois\$beta)
> cpois.nu <- exp(cpois\$zeta)
> ### ESTIMATE MEAN = LAMBDA ^ (1 / NU) - (NU - 1) / (2 * NU)
> cpois.lambda ^ (1 / cpois.nu) - (cpois.nu - 1) / (2 * cpois.nu)
(Intercept)
9.66575376
> ### ESTIMATE VARIANCE = LAMBDA ** (1 / NU) / NU
> cpois.lambda ^ (1 / cpois.nu) / cpois.nu
(Intercept)
29.69861239
```