Generalized additive models in insurance pricing

V magistrskem delu je predstavljen linearni model ter njegovi razširitvi: posplošeni linearni model - GLM (iz ang. generalized linear model) ter posplošeni aditivni model - GAM (iz ang. generalized additive model). Linearni modeli so široko uporabljani statistični modeli, kjer je pričakovana vrednost slučajne spremenljivke modelirana kot seštevek linearnih prediktorjev. Ta je odvisen od napovednih spremenljivk in nekaterih parametrov, ki jih je potrebno oceniti. Ključna predpostavka je, da je prediktor linearno odvisen od parametrov, predpostavljeno pa je tudi, da je slučajna spremenljivka porazdeljena normalno. V GLM je predpostavka o linearnosti prediktorja posplošena in pričakovana vrednost slučajne spremenljivke je tako odvisna od poljubne gladke monotone funkcije linearnega prediktorja. Pri tem lahko slučajna spremenljivka pripada družini eksponentno porazdeljenih spremenljivk (npr. normalne, Poissonove, binomske, Gamma, itd.). GAM je še dodatna posplošitev, kjer pa je linearni prediktor predstavljen kot gladka funkcija napovednih spremenljivk. Te gladke funkcije praviloma nimajo eksaktnega zapisa, zato jih je treba predstaviti in obravnavati njihovo gladkost. V delu je predstavljena osnova teorija linearnih modelov, GLM in GAM ter primer njihove uporabe v določanju cen zavarovanj avtomobilske odgovornosti.In master\u27s thesis we will present the linear model and its extensions: the generalized linear model - GLM and the generalized additive model - GAM. Linear models are widely used statistical models in which a univariate response is modelled as the sum of a linear predictors. The linear predictor depends on predictor variables and unknown parameters, which must be estimated. A key feature of the models is that the linear predictor depends linearly on the parameters. Statistical inference is usually based on the assumption that the response variable is normally distributed. GLM somewhat relaxes the strict linearity assumption of linear models by allowing the expected value of the response to depend on a smooth monotonic function of the linear predictor. Similarly the assumption that the response is normally distributed is relaxed by allowing it to follow any distribution from the exponential family (for example, normal, Poisson, binomial, gamma, etc.). The GAM is a GLM where the linear predictor depends linearly on smooth functions of predictor variables. The exact parametric form of these functions is unknown, as is the degree of smoothness appropriate for them. A short theoretical introduction to linear models, GLM and GAM will be presented as well as their use for pricing in MTPL insurance