Statistik für bivariate gemischte Poisson–Prozesse am Beispiel der Kraftfahrthaftpflichtversicherung

Zusammenfassung: In diesem Artikel wird der Weg von einem univariaten gemischten Poisson–Prozess, der in vielen Bereichen zum Z?hlen von Ereignissen benutzt wird, zu einem bivariaten gemischten Poisson–Prozess aufgezeigt. Dazu werden einige Eigenschaften des bivariaten Prozesses angegeben. Im zweiten Teil der Arbeit wird gezeigt, wie mit Hilfe dieses Prozesses der übergang von einem herk?mmlichen Bonus–Malus–System in der Kraftfahrthaftpflichtversicherung zu einem Bonus–Malus–System mit Berücksichtigung der Schadenart beschritten werden kann. Dazu wird zuerst eine Modellprüfung der gegebenen Daten vorgenommen und sodann werden für verschiedene mischende Verteilungen die Verteilungsparameter gesch?tzt und Nettopr?mien angegeben sowie die Prognosegenauigkeit getestet.

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Summary: In this paper we show that the model of the bivariate mixed Poisson process arises in a natural way from the univariate mixed Poisson process, which is used in several areas for counting certain events. Furthermore we state some properties of the bivariate process. In the second part of the paper we illustrate how by means of the bivariate mixed Poisson process a bonus–malus system handling different types of accidents can be derived from the classical bonus–malus system in third–party liability insurance. To this end we first check the model on the given data and then estimate distribution parameters and compute net premiums for different mixing distributions as well as test the prediction probabilities.

* Vortrag am Dresdner Forum zur Versicherungsmathematik: Tarifierung in Erst- und Rückversicherung am 25. Juni 2004. Für die Unterstützung zu dieser Arbeit m?chte der Autor Lothar Partzsch, Klaus D. Schmidt (beide Dresden) und Friedemann Spies (München) recht herzlich danken.     

Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

A class of weighted dependence measures for bivariate failure time data

This paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow-up region. The class consists of measures that are weighted averages of local dependence measures, and includes the cross-ratio-measure and finite region version of Kendall's τ; recently proposed by the authors. Two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators. Nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated. Properties of selected estimators are evaluated in a simulation study, and the method is illustrated through an analysis of Australian Twin Study data.     

Fitting competing risks data to bivariate Pareto models

This paper revisits two bivariate Pareto models for fitting competing risks data. The first model is the Frank copula model, and the second one is a bivariate Pareto model introduced by Sankaran and Nair (1993 Sankaran, P. G., and N. U. Nair. 1993. A bivariate Pareto model and its applications to reliability. Naval Research Logistics 40 (7):1013–1020. doi:10.1002/1520-6750(199312)40:7%3c1013::AID-NAV3220400711%3e3.0.CO;2-7.[Crossref], [Web of Science ®] , [Google Scholar]). We discuss the identifiability issues of these models and develop the maximum likelihood estimation procedures including their computational algorithms and model-diagnostic procedures. Simulations are conducted to examine the performance of the maximum likelihood estimation. Real data are analyzed for illustration.     

Linear scalar-on-surface random effects regression models

Many research fields increasingly involve analyzing data of a complex structure. Models investigating the dependence of a response on a predictor have moved beyond the ordinary scalar-on-vector regression. We propose a regression model for a scalar response and a surface (or a bivariate function) predictor. The predictor has a random component and the regression model falls in the framework of linear random effects models. We estimate the model parameters via maximizing the log-likelihood with the ECME (Expectation/Conditional Maximization Either) algorithm. We use the approach to analyze a data set where the response is the neuroticism score and the predictor is the resting-state brain function image. In the simulations we tried, the approach has better performance than two other approaches, a functional principal component regression approach and a smooth scalar-on-image regression approach.     

Communication in Statistics-Theory and methods improved GQL estimation method for the generalised BINMA(1) model

In a recent research, the quasi-likelihood estimation methodology was developed to estimate the regression effects in the Generalized BINMA(1) (GBINMA(1)) process. The method provides consistent parameter estimates but, in the intermediate computations, moment estimating equations were used to estimate the serial- and cross-correlation parameters. This procedure may not result optimal parameter estimates, in particular, for the regression effects. This paper provides an alternative simpler GBINMA(1) process based on multivariate thinning properties where the main effects are estimated via a robust generalized quasi-likelihood (GQL) estimation approach. The two techniques are compared through some simulation experiments. A real-life data application is studied.     

A new nonparametric bivariate test for two sample location problem

A strictly nonparametric bivariate test for two sample location problem is proposed. The proposed test is easy to apply and does not require the stringent condition of affine-symmetry or elliptical symmetry which is required by some of the major tests available for the same problem. The power function of the proposed test is calculated. The asymptotic distribution of the proposed test statistic is found to be normal. The power of proposed test is compared with some of the well-known tests under various distributions using Monte Carlo simulation technique. The power study shows that the proposed test statistic performs better than most of the test statistics for almost all the distributions considered here. As soon as the underlying population structure deviates from normality, the ability of the proposed test statistic to detect the smallest shift in location increases as compared to its competitors. The application of the test is shown by using a data set.     

Order statistics from trivariate normal and -distributions in terms of generalized skew-normal and skew- distributions

We consider here a generalization of the skew-normal distribution, GSN(λ₁,λ₂,ρ), defined through a standard bivariate normal distribution with correlation ρ, which is a special case of the unified multivariate skew-normal distribution studied recently by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574]. We then present some simple and useful properties of this distribution and also derive its moment generating function in an explicit form. Next, we show that distributions of order statistics from the trivariate normal distribution are mixtures of these generalized skew-normal distributions; thence, using the established properties of the generalized skew-normal distribution, we derive the moment generating functions of order statistics, and also present expressions for means and variances of these order statistics.Next, we introduce a generalized skew-t_ν distribution, which is a special case of the unified multivariate skew-elliptical distribution presented by Arellano-Valle and Azzalini [2006. On the unification of families of skew-normal distributions. Scand. J. Statist. 33, 561–574] and is in fact a three-parameter generalization of Azzalini and Capitanio's [2003. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J. Roy. Statist. Soc. Ser. B 65, 367–389] univariate skew-t_ν form. We then use the relationship between the generalized skew-normal and skew-t_ν distributions to discuss some properties of generalized skew-t_ν as well as distributions of order statistics from bivariate and trivariate t_ν distributions. We show that these distributions of order statistics are indeed mixtures of generalized skew-t_ν distributions, and then use this property to derive explicit expressions for means and variances of these order statistics.     

A computationally simple bivariate survival estimator for efficacy and safety

Both treatment efficacy and safety are typically the primary endpoints in Phase II, and even in some Phase III, clinical trials. Efficacy is frequently measured by time to response, death, or some other milestone event and thus is a continuous, possibly censored, outcome. Safety, however, is frequently measured on a discrete scale; in Eastern Cooperative Oncology Group clinical trial E2290, it was measured as the number of weekly rounds of chemotherapy that were tolerable to colorectal cancer patients. For the joint analysis of efficacy and safety, we propose a non-parametric, computationally simple estimator for the bivariate survival function when one time-to-event is continuous, one is discrete, and both are subject to right-censoring. The bivariate censoring times may depend on each other, but they are assumed to be independent of both event times. We derive a closed-form covariance estimator for the survivor function which allows for inference to be based on any of several possible statistics of interest. In addition, we derive its covariance with respect to calendar time of analysis, allowing for its use in sequential studies.