The Cox Regression Model for Claims Data m Non-Life Insurance

Abstract The Cox regression model is a standard tool in survival analysis for studying the dependence of a hazard rate on covariates (parametrically) and time (nonparametrically). This paper is a case study intended to indicate possible applications to non-life insurance, particularly occurrence of claims and rating. We studied individuals from one Danish county holding policies in auto, property and household insurance simultaneously at some point during the four year period 1988-1991 in one company. The hazard of occurrence of claims of each type was studied as function of calendar time, time since the last claim of each type, age of policy holder, urbanization and detailed type of insurance. Particular emphasis was given to the technical advantages and disadvantages (particularly the complicated censoring patterns) of considering the nonparametrically underlying time as either calendar time or time since last claim. In the former case the theory is settled, but the results are somewhat complicated. The latter choice leads to several issues still under active methodological development. We develop a goodness-of-fit criterion which shows the lack of fit of some models, for which the practical conclusions might otherwise have been useful.

[1]  P. Hougaard,et al.  Frailty models for survival data , 1995, Lifetime data analysis.

[2]  Torben Martinussen,et al.  Retrospective Ascertainment of Recurrent Events: An Application to Time to Pregnancy , 1999 .

[3]  James Lindsey,et al.  Fitting Parametric Counting Processes by using Log-linear Models , 1995 .

[4]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[5]  David Oakes,et al.  Frailty Models For Multiple Event Times , 1992 .

[6]  S. McClean,et al.  A nonparametrie maximum likelihood estimator for incomplete renewal data , 1995 .

[7]  Richard D. Gill,et al.  Nonparametric estimation based on censored observations of a Markov renewal process , 1978 .

[8]  Jerald F. Lawless,et al.  A point-process model incorporating renewals and time trends, with application to repairable systems , 1996 .

[9]  Y. Vardi.  Nonparametric Estimation in Renewal Processes , 1982 .

[10]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[11]  Dorota M. Dabrowska,et al.  Cox regression in a Markov renewal model : an application to the analysis of bone marrow transplant data , 1994 .

[12]  O. Aalen,et al.  Statistical analysis of repeated events forming renewal processes. , 1991, Statistics in medicine.

[13]  David Oakes and Lu Cui.  On semiparametric inference for modulated renewal processes , 1994 .

[14]  Michael Woodroofe,et al.  Nonparametric estimation and consistency for renewal processes , 1996 .

[15]  David R. Cox,et al.  Regression models and life tables (with discussion , 1972 .

[16]  Dorota M. Dabrowska,et al.  Estimation of transition probabilities and bootstrap in a semiparametric markov renewal model , 1995 .

[17]  A. V. Peterson,et al.  On the regression analysis of multivariate failure time data , 1981 .

[18]  N. Keiding.  Historical controls and modern survival analysis , 1995, Lifetime data analysis.

[19]  H. Blossfeld.  Techniques of event history modeling , 1989 .

[20]  R. Gill,et al.  Cox's regression model for counting processes: a large sample study : (preprint) , 1982 .

[21]  H. Ramlau-Hansen.  Smoothing Counting Process Intensities by Means of Kernel Functions , 1983 .