Overdispersion and Missing Telematics in a Claim-Frequency GLM

The residual deviance is 61,200 on 47,940 degrees of freedom, giving a dispersion statistic of 61,200/47,940 ≈ 1.28. For a Poisson GLM this should be near 1; 1.28 indicates modest overdispersion, so the model’s standard errors are too small by roughly sqrt(1.28) ≈ 1.13. The telematics coefficient has a naive z-statistic of 0.18/0.03 = 6.0; after inflating the standard error to about 0.034 the z-statistic is about 5.3, still significant, so the variable is unlikely to be noise. However, the deviance drop of 410 on 1 df is very large relative to a chi-square(1) reference and should be checked for data leakage or a coding error; with overdispersion an F-test or quasi-likelihood test is more appropriate. I would refit with quasi-Poisson or negative binomial standard errors, check the mean–variance relationship, and look for zero-inflation or duplicate policies. Only then would I decide whether the telematics score belongs in the pricing model.

Mean imputation is not neutral. Replacing missing telematics scores with the overall mean shrinks the variation in the score, biases its coefficient toward zero, and understates the standard error because the imputed values are treated as known. It also assumes the missing values are missing completely at random, which is unlikely: telematics may be missing because of vehicle type, policy duration or opt-out behaviour, all of which may relate to risk. A single imputation also ignores imputation uncertainty. Better options are: (i) multiple imputation by chained equations including the outcome and covariates, with imputations performed inside each cross-validation fold to avoid leakage; (ii) include a missing-score indicator and interact it with the observed score, which lets the model estimate a different effect for the telematics and non-telematics groups; (iii) fit separate models or a score with a missing category. The insurer should also test whether the missing group has different claim experience. If the missingness is informative, a pattern-mixture or selection model may be needed rather than imputation.

I would not choose on holdout deviance alone. First, check that the comparison is genuinely out-of-time and out-of-sample, with all preprocessing such as imputation, encoding and scaling fitted inside training folds. An 8% lower deviance is useful, but the GLM may be better calibrated in the tail and more stable across cells. I would compare: (i) out-of-time Poisson deviance and lift by risk decile; (ii) calibration curves and predicted-to-actual ratios by region and vehicle age; (iii) coefficient stability under bootstrapping; (iv) transparency and regulatory acceptability, since motor pricing often requires explainable relativities; and (v) fairness and proxy discrimination. The GLM with the telematics score and a missing indicator may capture most of the signal. The gradient-boosted model could be used as a benchmark, blended, or replaced by a monotonic or otherwise interpretable machine-learning model. The decision should be documented and monitored after deployment.