Eva Boj

Distance-based local linear regression for functional predictors

The problem of nonparametrically predicting a scalar response variable from a functional predictor is considered. A sample of pairs (functional predictor and response) is observed. When predicting the response for a new functional predictor value, a semi-metric is used to compute the distances between the new and the previously observed functional predictors. Then each pair in the original sample is weighted according to a decreasing function of these distances. A Weighted (Linear) Distance-Based Regression is fitted, where the weights are as above and the distances are given by a possibly different semi-metric. This approach can be extended to nonparametric predictions from other kinds of explanatory variables (e.g., data of mixed type) in a natural way.

Interaction Terms in Distance-Based Regression

We propose a method of including polynomial and interaction terms in Distance-Based Regression (Cuadras and Arenas, 1990), relying on properties of a semi-Hadamard or Khatri-Rao product of matrices. We demonstrate its application to real data examples.

Local Linear Functional Regression Based on Weighted Distance-based Regression

We consider the problem of nonparametrically predicting a scalar response variable y from a functional predictor ´. We have n observations (´i;yi) and we assign a weight wi / K (d(´;´i)=h) to eachi, where d(¢; ¢) is a semi-metric, K is a kernel function and h is the bandwidth. Then we flt a Weighted (Linear) Distance-Based Regression, where the weights are as above and the distances are given by a possibly difierent semi-metric. This approach can be extended to nonparametric predictions from other kind of explanatory variables (e.g., data of mixed type) in a natural way.

Prediction Error in Distance-Based Generalized Linear Models

In generalized linear models, the mean squared prediction error can be approximated by the sum of two components: variability in the data (process variance) and variability due to estimation (estimation variance). The estimation variance can be calculated by using the corresponding formula or, alternatively, it can be approximated by using bootstrap methodology. When we use bootstrap methodology we are able to obtain, in addition, the predictive distribution of estimations. We apply these concepts to the actuarial problem of claim reserving, where data are collected in a run-off triangle, and it is of interest the use of generalized linear models and the calculus of prediction error. We illustrate computations with a well-known data set. Distance-based generalized linear model is fitted using the dbglm function of the dbstats package for R.

Claim Reserving Using Distance-Based Generalized Linear Models

Generalized linear models (GLM) can be considered a stochastic version of the classical Chain-Ladder (CL) method of claim reserving in nonlife insurance. In particular, the deterministic CL model is reproduced when a GLM is fitted assuming over-dispersed Poisson error distribution and logarithmic link. Our aim is to propose the use of distance-based generalized linear models (DB-GLM) in the claim reserving problem. DB-GLM can be considered a generalization of the classical GLM to the distance-based analysis, because DB-GLM contains as a particular instance ordinary GLM when the Euclidean, \(l^2\), metric is applied. Then, DB-GLM can be considered too a stochastic version of the CL claim reserving method. In DB-GLM, the only information required is a predictor distance matrix. DB-GLM can be fitted using the dbstats package for R. To estimate reserve distributions and standard errors, we propose a nonparametric bootstrap technique adequate to the distance-based regression models. We illustrate the method with a well-known actuarial dataset.

Logistic Classification for New Policyholders Taking into Account Prediction Error

An expression of the mean squared error, MSE, of prediction for new observations when using logistic regression is showed. First, MSE is approximated by the sum of the process variance and of the estimation variance. The estimation variance can be estimated by applying the delta method and/or by using bootstrap methodology. When using bootstrap, e.g. bootstrap residuals, it is possible to obtain an estimation of the distribution for each predicted value. Confidence intervals can be calculated taking into account the bootstrapped distributions of the predicted new values to help us in the knowledge of their randomness. The general formulas of prediction error (the square root of MSE of prediction), PE, in the cases of the power family of error distributions and of the power family of link functions for generalized linear models were obtained in previous works. Now, the expression of the MSE of prediction for the generalized linear model with Binomial error distribution and logit link function, the logistic regression, is obtained. Its calculus and usefulness are illustrated with real data to solve the problem of Credit Scoring, where policyholders are classified into defaulters and non-defaulters.

Provisions for Outstanding Claims with Distance-Based Generalized Linear Models

In previous works we developed the formulas of the prediction error in generalized linear model (GLM) for the future payments by calendar years assuming the logarithmic link and the parametric family of error distributions named power family. In the particular case of assuming (overdispersed) Poisson and logarithmic link the GLM gives the same provision estimations as those of the Chain-Ladder deterministic method. Now, we are studying the possibility to use distance-based generalized linear models (DB-GLM) to solve the problem of claim reserving in the same way as GLM is used in this context. DB-GLM can be fitted by using the function dbglm of the dbstats package for R. In this study we calculate the prediction error associated to the accident years future payments and total payment, and also to the calendar years future payments using DB-GLM in the general case of the power families of error distributions and link functions. We make an application with the well known run-off triangle of Taylor and Ashe.