Delong, Ł., Wüthrich, 2023, The role of the variance function in mean estimation and validation
Working Paper,
01-07-2023
Regression modeling for insurance pricing mostly focuses on mean estimation. Using a
strictly consistent loss function implies that the mean estimates are asymptotically correct.
However, this is a limiting statement and insurance prices are calculated on finite samples. It is
known that under heteroskedasticity suitable variance estimates can significantly improve the
regression model estimation. In this paper we investigate isotonic regression which is a nonparametric
rank-preserving regression approach. This isotonic regression is used to (1) explore
the power variance parameter of the variance function within Tweedie’s family of distributions,
(2) derive a semi-parametric bootstrap under heteroskedasticity, (3) provide a test for autocalibration,
(4) explore a quasi-likelihood approach to benefit from best-asymptotic estimation,
(5) deal with several difficulties under lognormal assumptions. In all these problems we verify
that variance estimation using an isotonic regression is very beneficial.
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Delong, Ł., Kozak, A., 2023, The use of autoencoders for training neural networks with mixed categorical and numerical features
Published Paper,
01-02-2023
We focus on modelling categorical features and improving predictive power of neural networks with mixed categorical and numerical features in supervised learning tasks. The goal of this paper is to challenge the current dominant approach in actuarial data science with a new architecture of a neural network and a new training algorithm. The key proposal is to use a joint embedding for all categorical features, instead of separate entity embeddings, to determine the numerical representation of the categorical features which is fed, together with all other numerical features, into hidden layers of a neural network with a target response. In addition, we postulate that we should initialize the numerical representation of the categorical features and other parameters of the hidden layers of the neural network with parameters trained with (denoising) autoencoders in unsupervised learning tasks, instead of using random initialization of parameters. Since autoencoders for categorical data play an important role in this research, they are investigated in more depth in the paper. We illustrate our ideas with experiments on a real data set with claim numbers, and we demonstrate that we can achieve a higher predictive power of the network.
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Delong, Ł., Lindholm, M., Wüthrich, M.V., 2021, Gamma Mixture Density Networks and their application to modelling insurance claim amounts
Published Paper,
05-10-2021
We discuss how mixtures of Gamma distributions with mixing probabilities, shape and rate parameters depending on features can be fitted with neural networks. We develop two versions of the EM algorithm for fitting so-called Gamma Mixture Density Networks, which we call the EM network boosting algorithm and the EM forward network algorithm, and we test their implementation together with the choices of hyperparameters. A simulation study shows that our algorithms perform very well on synthetic data sets. We further illustrate the application of the Gamma Mixture Density Network on a real data set of motor insurance claim amounts and conclude that Gamma Mixture Density Networks can improve the fit of the regression model and the predictions of the claim severities used for rate-making compared to classical actuarial techniques.
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