By Piet de Jong, Gillian Z. Heller
This is often the one booklet actuaries have to comprehend generalized linear types (GLMs) for coverage functions. GLMs are utilized in the coverage to help serious judgements. earlier, no textual content has brought GLMs during this context or addressed the issues particular to coverage facts.
Using coverage facts units, this functional, rigorous e-book treats GLMs, covers all average exponential kin distributions, extends the method to correlated info buildings, and discusses contemporary advancements which transcend the GLM. the problems within the publication are particular to assurance information, equivalent to version choice within the presence of huge information units and the dealing with of various publicity occasions.
Exercises and data-based practicals support readers to consolidate their abilities, with recommendations and knowledge units given at the better half web site. even if the e-book is package-independent, SAS code and output examples characteristic in an appendix and at the site. furthermore, R code and output for all of the examples are supplied at the site.
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Additional info for Generalized Linear Models for Insurance Data (International Series on Actuarial Science)
Y = 0, 1, 2, . . 1. 4 μ=1 0 5 10 y 15 20 0 5 10 y 15 20 0 5 10 y 15 20 Fig. 2. 3. There is good agreement of the data with the Poisson model. In the sample there are more women with no children, and fewer with one child, compared to the Poisson model predictions. 5 Negative binomial The classic derivation of the negative binomial distribution is as the number of failures in Bernoulli trials until r successes. Having y failures implies that trial r + y is a success, and in the previous r + y − 1 trials there were exactly r − 1 successes and y failures.
Mathematical derivation. To avoid notational confusion here the mean of the conditional distribution of y is denoted as λ (rather than μ). Given λ, assume the distribution of y is Poisson with mean λ: y | λ ∼ P(λ) ⇒ f (y | λ) = e−λ λy . y! Suppose λ is regarded as a continuous random variable with probability function g(λ) where g(λ) = 0 for λ < 0. Then the unconditional probability function of y is ∞ f (y) = f (y | λ) g(λ) dλ . e. 6): ∞ e−λ λy λ−1 λν e−λν/μ dλ y! Γ (ν) μ 0 ν ∞ ν 1 λy+ν−1 e−λ(1+ν/μ) dλ = y!
This displays linearity and approximate homoskedasticity, and therefore a linear model based on log claims as the response and log accidents as the explanatory variable, is more amenable to analysis using the normal linear model than one based on the raw data. 2. Linear modeling 8 6 0 2 4 Log claims 4000 2000 0 Claims 6000 50 0 2000 4000 6000 8000 3 4 Accidents 5 6 7 8 9 Log accidents 0 0 Frequency 20 40 60 80 Frequency 20 40 60 Fig. 1. Scatterplots of number of accidents and number of claims, raw and log scales 2000 0 2 6000 Accidents 10000 0 2000 4000 Claims 6000 0 0 Frequency 5 10 15 Frequency 5 10 15 20 20 0 4 6 8 Log accidents 10 0 2 4 6 Log claims 8 10 Fig.
Generalized Linear Models for Insurance Data (International Series on Actuarial Science) by Piet de Jong, Gillian Z. Heller