Dispersion Parameter

Study guide: Exam MAS-I

The Dispersion Parameter $\phi$ is the scale factor in an Exponential Family variance: $\text{Var}(Y) = \phi\,V(\mu)$ . It separates how the variance depends on the mean (the variance function $V$ , fixed by the chosen distribution) from how large the variance is overall (the scale, estimated from the data).

$\text{Var}(Y_i) = \frac{\phi\,V(\mu_i)}{w_i}$

$\hat\phi = \frac{1}{n-p}\sum_{i=1}^{n}\frac{w_i\,(y_i - \hat\mu_i)^2}{V(\hat\mu_i)} \qquad \text{(Pearson estimate)}$

For the Normal response $\phi = \sigma^2$ ; for Poisson and binomial the theory fixes $\phi = 1$ ; for Gamma $\phi$ is the squared coefficient of variation
$\phi$ does not affect the fitted coefficients $\hat\beta$ — it scales their standard errors by $\sqrt{\hat\phi}$ , and so changes every p-value and Confidence Interval in Parameter Estimate Tables
$\hat\phi > 1$ in a Poisson or binomial model is overdispersion: more spread than the distribution permits, usually from variation in risk level among policyholders that the model's predictors do not explain
Two estimators are in common use — the Pearson $\chi^2$ statistic over degrees of freedom (above) and the unscaled residual Deviance over degrees of freedom, $D/(n-p)$ . The Pearson version is the standard approach
With $D$ the unscaled deviance, the scaled deviance $D^* = D/\phi$ is what is compared against $\chi^2$ ; when $\phi$ has to be estimated, comparing nested models that differ by $r$ parameters uses the $F$ statistic $\frac{\Delta D/r}{\hat\phi}$ instead, with $\hat\phi$ from the larger model
Prior weights $w_i$ (e.g. exposure or claim count) divide the variance: a group average built from many observations is more precisely known, which is how grouped data is fitted correctly

Estimating and Applying the Dispersion A Poisson GLM on $500$ observations with $8$ parameters has a Pearson $\chi^2$ of $1{,}230$ . Estimate $\phi$ and adjust a coefficient with $\hat\beta = 0.40$ and unscaled $\text{SE} = 0.15$ .

Answer $\hat\phi = \frac{1{,}230}{500 - 8} = \frac{1{,}230}{492} = 2.50$ $\text{SE}_{\text{scaled}} = 0.15\sqrt{2.50} = 0.237$ The Wald statistic falls from $0.40/0.15 = 2.67$ (significant) to $0.40/0.237 = 1.69$ (not significant at $5\%$ ). Ignoring overdispersion would have kept a predictor the data does not support.

Dispersion in a Gamma Severity Model A Gamma GLM for severity returns $\hat\phi = 0.64$ . What does that say about the spread of individual claims?

Answer For a Gamma, $\text{Var}(Y) = \phi\mu^2$ , so the coefficient of variation is $\sqrt{\phi} = 0.80$ — the standard deviation of a claim is about $80\%$ of its expected size, whatever the size. It also corresponds to a Gamma shape of $\alpha = 1/\phi = 1.56$ , a right-skewed severity distribution, and here $\phi$ is a genuine parameter of the model rather than a symptom of anything wrong.