Penalized likelihood regression for generalized linear models with non-quadratic penalties

One of the popular method for fitting a regression function is regularization: minimizing an objective function which enforces a roughness penalty in addition to coherence with the data. This is the case when formulating penalized likelihood regression for exponential families. Most of the smoothing methods employ quadratic penalties, leading to linear estimates, and are in general incapable of recovering discontinuities or other important attributes in the regression function. In contrast, non-linear estimates are generally more accurate. In this paper, we focus on non-parametric penalized likelihood regression methods using splines and a variety of non-quadratic penalties, pointing out common basic principles. We present an asymptotic analysis of convergence rates that justifies the approach. We report on a simulation study including comparisons between our method and some existing ones. We illustrate our approach with an application to Poisson non-parametric regression modeling of frequency counts of reported acquired immune deficiency syndrome (AIDS) cases in the UK.     

Bayesian quantile regression model for claim count data

Quantile regression model estimates the relationship between the quantile of a response distribution and the regression parameters, and has been developed for linear models with continuous responses. In this paper, we apply Bayesian quantile regression model for the Malaysian motor insurance claim count data to study the effects of change in the estimates of regression parameters (or the rating factors) on the magnitude of the response variable (or the claim count). We also compare the results of quantile regression models from the Bayesian and frequentist approaches and the results of mean regression models from the Poisson and negative binomial. Comparison from Poisson and Bayesian quantile regression models shows that the effects of vehicle year decrease as the quantile increases, suggesting that the rating factor has lower risk for higher claim counts. On the other hand, the effects of vehicle type increase as the quantile increases, indicating that the rating factor has higher risk for higher claim counts.     

Precision Matrix Estimation With ROPE

It is known that the accuracy of the maximum likelihood-based covariance and precision matrix estimates can be improved by penalized log-likelihood estimation. In this article, we propose a ridge-type operator for the precision matrix estimation, ROPE for short, to maximize a penalized likelihood function where the Frobenius norm is used as the penalty function. We show that there is an explicit closed form representation of a shrinkage estimator for the precision matrix when using a penalized log-likelihood, which is analogous to ridge regression in a regression context. The performance of the proposed method is illustrated by a simulation study and real data applications. Computer code used in the example analyses as well as other supplementary materials for this article are available online.     

Nonlife ratemaking and risk management with Bayesian generalized additive models for location,scale, and shape

Generalized additive models for location, scale and, shape define a flexible, semi-parametric class of regression models for analyzing insurance data in which the exponential family assumption for the response is relaxed. This approach allows the actuary to include risk factors not only in the mean but also in other key parameters governing the claiming behavior, like the degree of residual heterogeneity or the no-claim probability. In this broader setting, the Negative Binomial regression with cell-specific heterogeneity and the zero-inflated Poisson regression with cell-specific additional probability mass at zero are applied to model claim frequencies. New models for claim severities that can be applied either per claim or aggregated per year are also presented. Bayesian inference is based on efficient Markov chain Monte Carlo simulation techniques and allows for the simultaneous estimation of linear effects as well as of possible nonlinear effects, spatial variations and interactions between risk factors within the data set. To illustrate the relevance of this approach, a detailed case study is proposed based on the Belgian motor insurance portfolio studied in Denuit and Lang (2004).     

A priori ratemaking using bivariate Poisson regression models

In automobile insurance, it is useful to achieve a priori ratemaking by resorting to generalized linear models, and here the Poisson regression model constitutes the most widely accepted basis. However, insurance companies distinguish between claims with or without bodily injuries, or claims with full or partial liability of the insured driver. This paper examines an a priori ratemaking procedure when including two different types of claim. When assuming independence between claim types, the premium can be obtained by summing the premiums for each type of guarantee and is dependent on the rating factors chosen. If the independence assumption is relaxed, then it is unclear as to how the tariff system might be affected. In order to answer this question, bivariate Poisson regression models, suitable for paired count data exhibiting correlation, are introduced. It is shown that the usual independence assumption is unrealistic here. These models are applied to an automobile insurance claims database containing 80,994 contracts belonging to a Spanish insurance company. Finally, the consequences for pure and loaded premiums when the independence assumption is relaxed by using a bivariate Poisson regression model are analysed.     

Computationally Efficient Changepoint Detection for a Range of Penalties

In the multiple changepoint setting, various search methods have been proposed, which involve optimizing either a constrained or penalized cost function over possible numbers and locations of changepoints using dynamic programming. Recent work in the penalized optimization setting has focused on developing an exact pruning-based approach that, under certain conditions, is linear in the number of data points. Such an approach naturally requires the specification of a penalty to avoid under/over-fitting. Work has been undertaken to identify the appropriate penalty choice for data-generating processes with known distributional form, but in many applications the model assumed for the data is not correct and these penalty choices are not always appropriate. To this end, we present a method that enables us to find the solution path for all choices of penalty values across a continuous range. This permits an evaluation of the various segmentations to identify a suitable penalty choice. The computational complexity of this approach can be linear in the number of data points and linear in the difference between the number of changepoints in the optimal segmentations for the smallest and largest penalty values. Supplementary materials for this article are available online.     

带注资的二维复合泊松模型的最优分红

研究建立两类理赔关系的二维复合泊松模型的最优分红与注资问题,目标为最大化分红减注资的折现. 该问题由随机控制问题刻画, 通过解相应的哈密尔顿-雅克比-贝尔曼(HJB)方程,得到了最优分红策略,并在指数理赔时明确地解决该问题.     

Bayesian multivariate Poisson models for insurance ratemaking

When actuaries face the problem of pricing an insurance contract that contains different types of coverage, such as a motor insurance or a homeowner’s insurance policy, they usually assume that types of claim are independent. However, this assumption may not be realistic: several studies have shown that there is a positive correlation between types of claim. Here we introduce different multivariate Poisson regression models in order to relax the independence assumption, including zero-inflated models to account for excess of zeros and overdispersion. These models have been largely ignored to date, mainly because of their computational difficulties. Bayesian inference based on MCMC helps to resolve this problem (and also allows us to derive, for several quantities of interest, posterior summaries to account for uncertainty). Finally, these models are applied to an automobile insurance claims database with three different types of claim. We analyse the consequences for pure and loaded premiums when the independence assumption is relaxed by using different multivariate Poisson regression models together with their zero-inflated versions.     

Support vector regression for warranty claim forecasting

Forecasting the number of warranty claims is vitally important for manufacturers/warranty providers in preparing fiscal plans. In existing literature, a number of techniques such as log-linear Poisson models, Kalman filter, time series models, and artificial neural network models have been developed. Nevertheless, one might find two weaknesses existing in these approaches: (1) they do not consider the fact that warranty claims reported in the recent months might be more important in forecasting future warranty claims than those reported in the earlier months, and (2) they are developed based on repair rates (i.e., the total number of claims divided by the total number of products in service), which can cause information loss through such an arithmetic-mean operation.To overcome the above two weaknesses, this paper introduces two different approaches to forecasting warranty claims: the first is a weighted support vector regression (SVR) model and the second is a weighted SVR-based time series model. These two approaches can be applied to two scenarios: when only claim rate data are available and when original claim data are available. Two case studies are conducted to validate the two modelling approaches. On the basis of model evaluation over six months ahead forecasting, the results show that the proposed models exhibit superior performance compared to that of multilayer perceptrons, radial basis function networks and ordinary support vector regression models.     

Capture-Recapture模型与车险事故总体预测

车险事故总体预测问题一直是车辆保险公司研究的重点内容之一,目前最为常用的方法是与泊松分布相关的模型.基于车辆保险中索赔数据的结构特征,构建了Capture-Recapture模型,并使用一组车辆保险数据,利用Capture-Recapture及常用的零膨胀泊松等模型分别建模分析,得出了一些新的结论,即Capture-Recapture模型拟合效果整体较优,从而为车辆保险公司更好预测事故总体提供一定的理论依据.     

Analysis of the Gerber-Shiu function and dividend barrier problems for a risk process with two classes of claims

In this paper we consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, the Poisson and the generalized Erlang(2) process. We prove that the Gerber-Shiu function satisfies some defective renewal equations. Exact representations for the solutions of these equations are derived through an associated compound geometric distribution and an analytic expression for this quantity is given when the claim severities have rationally distributed Laplace transforms. Further, the same risk model is considered in the presence of a constant dividend barrier. A system of integro-differential equations with certain boundary conditions for the Gerber-Shiu function is derived and solved. Using systems of integro-differential equations for the moment-generating function as well as for the arbitrary moments of the discounted sum of the dividend payments until ruin, a matrix version of the dividends-penalty is derived. An extension to a risk model when the two independent claim counting processes are Poisson and generalized Erlang(ν), respectively, is considered, generalizing the aforementioned results.     

广义线性模型在汽车延保风险保费测算中的应用

基于保险公司2010年1月—2019年3月的实际保单数据样本,分别运用广义线性模型中的泊松模型和伽玛模型测算出险频率和案均赔款,构建风险保费测算模型,对影响风险保费的因素进行定量研究及分析.结果表明:该方法能够构建多个变量与风险保费的数值关系,减少了信息的损失,得到的费率表可作为实际应用的参考.最后,通过该方法测算结果与市场定价的实例比较对方法的合理性与优越性进行了说明.     

用Poisson-Tweedie模型拟合索赔次数

本文主要考虑用Poisson Tweedie分布族来近似实际索赔次数的分布。在这里,我们把使X2拟合优度统计量的值的大小作为评估的标准,最后用来自B櫣hlmann的一组数据对此方法进行了验证。     

Adaptive basis expansion via \ell _1 trend filtering

We propose a new approach for nonlinear regression modeling by employing basis expansion for the case where the underlying regression function has inhomogeneous smoothness. In this case, conventional nonlinear regression models tend to be over- or underfitting, where the function is more or less smoother, respectively. First, the underlying regression function is roughly approximated with a locally linear function using an \(\ell _1\) penalized method, where this procedure is executed by extending an algorithm for the fused lasso signal approximator. We then extend the fused lasso signal approximator and develop an algorithm. Next, the residuals between the locally linear function and the data are used to adaptively prepare the basis functions. Finally, we construct a nonlinear regression model with these basis functions along with the technique of a regularization method. To select the optimal values of the tuning parameters for the regularization method, we provide an explicit form of the generalized information criterion. The validity of our proposed method is then demonstrated through several numerical examples.     

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??Inspired by intuitive meanings of truncated power basis's coefficients, the local penalization based on range's linear decreasing function is given in penalized spline regression model. This method gives less penalization to fitting curve where data is with more volatility, which makes fitted curve controls tradeoff between goodness-of-fit and smoothness better. Simulations show that regression models with local penalized spline obtain lower information rules' scores than global penalized spline when the data is with heteroskedasticity.     

Penalized maximum likelihood estimation of a stochastic multivariate regression model

We study the large-sample properties of the penalized maximum likelihood estimator of a multivariate stochastic regression model with contemporaneously correlated data. The penalty is in terms of the square norm of some (vector) linear function of the regression coefficients. The model subsumes the so-called common transfer function model useful for extracting common signals in a panel of short time series. We show that, under mild regularity conditions, the penalized maximum likelihood estimator is consistent and asymptotically normal. The asymptotic bias of the regression coefficient estimator is also derived.     

Autocovariance Function Estimation via Penalized Regression

The work revisits the autocovariance function estimation, a fundamental problem in statistical inference for time series. We convert the function estimation problem into constrained penalized regression with a generalized penalty that provides us with flexible and accurate estimation, and study the asymptotic properties of the proposed estimator. In case of a nonzero mean time series, we apply a penalized regression technique to a differenced time series, which does not require a separate detrending procedure. In penalized regression, selection of tuning parameters is critical and we propose four different data-driven criteria to determine them. A simulation study shows effectiveness of the tuning parameter selection and that the proposed approach is superior to three existing methods. We also briefly discuss the extension of the proposed approach to interval-valued time series. Supplementary materials for this article are available online.     

Generalized linear models for dependent frequency and severity of insurance claims

Traditionally, claim counts and amounts are assumed to be independent in non-life insurance. This paper explores how this often unwarranted assumption can be relaxed in a simple way while incorporating rating factors into the model. The approach consists of fitting generalized linear models to the marginal frequency and the conditional severity components of the total claim cost; dependence between them is induced by treating the number of claims as a covariate in the model for the average claim size. In addition to being easy to implement, this modeling strategy has the advantage that when Poisson counts are assumed together with a log-link for the conditional severity model, the resulting pure premium is the product of a marginal mean frequency, a modified marginal mean severity, and an easily interpreted correction term that reflects the dependence. The approach is illustrated through simulations and applied to a Canadian automobile insurance dataset.     

Variable selection for the partial linear single-index model

In this paper, we consider the issue of variable selection in partial linear single-index models under the assumption that the vector of regression coefficients is sparse. We apply penalized spline to estimate the nonparametric function and SCAD penalty to achieve sparse estimates of regression parameters in both the linear and single-index parts of the model. Under some mild conditions, it is shown that the penalized estimators have oracle property, in the sense that it is asymptotically normal with the same mean and covariance that they would have if zero coefficients are known in advance. Our model owns a least square representation, therefore standard least square programming algorithms can be implemented without extra programming efforts. In the meantime, parametric estimation, variable selection and nonparametric estimation can be realized in one step, which incredibly increases computational stability. The finite sample performance of the penalized estimators is evaluated through Monte Carlo studies and illustrated with a real data set.     

The Markovian regime-switching risk model with a threshold dividend strategy

In this paper, we study a regime-switching risk model with a threshold dividend strategy, in which the rate for the Poisson claim arrivals and the distribution of the claim amounts are driven by an underlying (external) Markov jump process. The purpose of this paper is to study the unified Gerber-Shiu discounted penalty function and the moments of the total dividend payments until ruin. We adopt an approach which is akin to the one used in [Lin, X.S., Pavlova, K.P., 2006. The compound Poisson risk model with a threshold dividend strategy. Insu.: Math. and Econ. 38, 57-80] to extend the results for the classical risk model with a threshold dividend strategy to our model. The matrix form of systems of integro-differential equations is presented and the analytical solutions to these systems are derived. Finally, numerical illustrations with exponential claim amounts are also given.