Skewed bivariate models and nonparametric estimation for the CTE risk measure

In this paper, we illustrate the use of the Conditional Tail Expectation (CTE) risk measure on a set of bivariate real data consisting of two types of auto insurance claim costs. Several continuous bivariate distributions (normal, lognormal, skew-normal with the alternative log-skew-normal) are fitted to the data. Besides, a bivariate nonparametric transformed kernel estimation is presented. CTE formulas are given for all these, and numerical results on the real data are discussed and compared.     

Smoothing noisy data for irregular regions using penalized bivariate splines on triangulations

The penalized spline method has been widely used for estimating univariate smooth functions based on noisy data. This paper studies its extension to the two-dimensional case. To accommodate the need of handling data distributed on irregular regions, we consider bivariate splines defined on triangulations. Penalty functions based on the second-order derivatives are employed to regularize the spline fit and generalized cross-validation is used to select the penalty parameters. A simulation study shows that the penalized bivariate spline method is competitive to some well-established two-dimensional smoothers. The method is also illustrated using a real dataset on Texas temperature.     

A class of bivariate negative binomial distributions with different index parameters in the marginals

In this paper, we consider a new class of bivariate negative binomial distributions having marginal distributions with different index parameters. This feature is useful in statistical modelling and simulation studies, where different marginal distributions and a specified correlation are required. This feature also makes it more flexible than the existing bivariate generalizations of the negative binomial distribution, which have a common index parameter in the marginal distributions. Various interesting properties, such as canonical expansions and quadrant dependence, are obtained. Potential application of the proposed class of bivariate negative binomial distributions, as a bivariate mixed Poisson distribution, and computer generation of samples are examined. Numerical examples as well as goodness-of-fit to simulated and real data are also given here in order to illustrate the application of this family of bivariate negative binomial distributions.     

独立非同分布二元高斯三角阵最大值的渐近性及相关统计推断

考虑二元独立非同分布高斯随机向量三角阵列最大值分布的渐近性及相关统计推断.此高斯三角阵的第n列的第i个向量服从二元高斯分布,其相关系数为i/n的函数并单调连续.首先建立了此高斯三角阵最大值分布的一阶和二阶渐近展开式.其次,分析相关系数参数估计及估计量的渐近性质.最后,通过随机模拟说明了相关系数之参数估计的有效性,并将该二元非同分布三角阵列模型应用于实际数据,得到了满意的结果.     

Statistical post-processing of forecasts for extremes using bivariate brown-resnick processes with an application to wind gusts

To improve the forecasts of weather extremes, we propose a joint spatial model for the observations and the forecasts, based on a bivariate Brown-Resnick process. As the class of stationary bivariate Brown-Resnick processes is fully characterized by the class of pseudo cross-variograms, we contribute to the theorical understanding of pseudo cross-variograms refining the knowledge of the asymptotic behaviour of all their components and introducing a parsimonious, but flexible parametric model. Both findings are of interest in classical geostatistics on their own. The proposed model is applied to real observation and forecast data for extreme wind gusts at 119 stations in Northern Germany.     

A measure of dependence for stable distributions

A distance based measure of dependence is proposed for stable distributions that completely characterizes independence for a bivariate stable distribution. Properties of this measure are analyzed, and contrasted with the covariation and co-difference. A sample analog of the measure is defined and demonstrated on simulated and real data, including time series and distributions in the domain of attraction of a stable law.     

Efficient Spatial Modeling Using the SPDE Approach With Bivariate Splines

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to nonstationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency. Supplementary materials are available online.     

Univariate and bivariate GPD methods for predicting extreme wind storm losses

Wind storm and hurricane risks are attracting increased attention as a result of recent catastrophic events. The aim of this paper is to select, tailor, and develop extreme value methods for use in wind storm insurance. The methods are applied to the 1982-2005 losses for the largest Swedish insurance company, the Länsförsäkringar group. Both a univariate and a new bivariate Generalized Pareto Distribution (GPD) gave models which fitted the data well. The bivariate model led to lower estimates of risk, except for extreme cases, but taking statistical uncertainty into account the two models lead to qualitatively similar results. We believe that the bivariate model provided the most realistic picture of the real uncertainties. It additionally made it possible to explore the effects of changes in the insurance portfolio, and showed that loss distributions are rather insensitive to portfolio changes. We found a small trend in the sizes of small individual claims, but no other trends. Finally, we believe that companies should develop systematic ways of thinking about “not yet seen” disasters.     

Compound negative binomial shared frailty models for bivariate survival data

In this paper, we introduce a new shared frailty model called the compound negative binomial shared frailty model with three different baseline distributions namely, Weibull, generalized exponential and exponential power distribution. To estimate the parameters involved in these models we adopt Markov Chain Monte Carlo (MCMC) approach. Also we apply these three models to a real life bivariate survival data set of McGrilchrist and Aisbett (1991) related to kidney infection and suggest a better model for the data.     

A New,Fast Algorithm to Find the Regions of Possible Support for Bivariate Interval-Censored Data

The estimation of the nonparametric maximum likelihood estimate (NPMLE) of the bivariate distribution function on interval-censored data is a recent topic of research. Among other things, it provides a basic tool for checking a parametric model for the bivariate failure times. As a first step in the estimation of the NPMLE for bivariate interval-censored data, the regions of possible support—that is, the rectangles with nonzero mass—are calculated. For this step a new, fast algorithm is introduced here and compared with two existing algorithms. The advantages of our algorithm will be illustrated on the emergence times of permanent teeth on data from the longitudinal Signal® Tandmobiel study.     

Estimation Procedures for a Semiparametric Family of Bivariate Copulas

This article proposes simple estimation methods dedicated to a semiparametric family of bivariate copulas. These copulas can be simply estimated through the estimation of their univariate generating function. We use this result to estimate the associated measures of association as well as the high probability regions of the copula. These procedures are illustrated using both simulations and real data.     

Nonparametric Estimation of the Dependence Function in Bivariate Extreme Value Distributions

The paper considers the problem of estimating the dependence function of a bivariate extreme survival function with standard exponential marginals. Nonparametric estimators for the dependence function are proposed and their strong uniform convergence under suitable conditions is demonstrated. Comparisons of the proposed estimators with other estimators are made in terms of bias and mean squared error. Several real data sets from various applications are used to illustrate the procedures.     

实分片代数曲线的某些研究

分片代数曲线作为二元样条函数的零点集合是经典代数曲线的推广. 利用代数的基本知识, 本文对实分片代数曲线的基本性质进行了初步讨论, 并且将实分片代数曲线与相应的二元样条分类进行讨论. 最后, 对实分片代数曲线上的孤立点进行了研究.     

Indirect assessment of the bivariate survival function

Estimating the bivariate survival function has been a major goal of many researchers. For that purpose many methods and techniques have been published. However, most of these techniques and methods rely heavily on bivariate failure data. There are situations in which failure time data are difficult to obtain and thus there is a growing need to assess the bivariate survival function for such cases. In this paper we propose two techniques for generating families of bivariate processes for describing several variables that can be used to indirectly assess the bivariate survival function. An estimation procedure is provided and a simulation study is conducted to evaluate the performance of our proposed estimator.     

Time–frequency analysis of bivariate signals

Many phenomena are described by bivariate signals or bidimensional vectors in applications ranging from radar to EEG, optics and oceanography. We show that an adequate quaternion Fourier transform permits to build relevant time–frequency representations of bivariate signals that naturally identify geometrical or polarization properties. First, a bivariate counterpart of the usual analytic signal of real signals is introduced, called the quaternion embedding of bivariate signals. Then two fundamental theorems ensure that a quaternion short term Fourier transform and a quaternion continuous wavelet transform are well defined and obey desirable properties such as conservation laws and reconstruction formulas. The resulting spectrograms and scalograms provide meaningful representations of both the time–frequency and geometrical/polarization content of the signal. Moreover the numerical implementation remains simply based on the use of FFT. A toolbox is available for reproducibility. Synthetic and real-world examples illustrate the relevance and efficiency of the proposed approach.     

Controlling jumps in correlated processes of Poisson counts

Processes of autocorrelated Poisson counts can often be modelled by a Poisson INAR(1) model, which proved to apply well to typical tasks of SPC. Statistical properties of this model are briefly reviewed. Based on these properties, we propose a new control chart: the combined jumps chart. It monitors the counts and jumps of a Poisson INAR(1) process simultaneously. As the bivariate process of counts and jumps is a homogeneous Markov chain, average run lengths (ARLs) can be computed exactly with the well‐known Markov chain approach. Based on an investigation of such ARLs, we derive design recommendations and show that a properly designed chart can be applied nearly universally. This is also demonstrated by a real‐data example from the insurance field. Copyright © 2008 John Wiley & Sons, Ltd.     

A bivariate interval censorship model for partnership formation

We consider a statistical problem of estimating a bivariate age distribution of newly formed partnership. The study is motivated by a type of data that consist of uncensored, right-censored, left-censored, interval-censored and missing observations in the coordinates of a bivariate random vector. A model is proposed for formulating such type of data. A feasible algorithm to estimate the generalized MLE (GMLE) of the bivariate distribution function is also proposed. We establish asymptotic properties for the GMLE under a discrete assumption on the underlying distributions and apply the method to the data set.     

Clustering bivariate mixed-type data via the cluster-weighted model

The cluster-weighted model (CWM) is a mixture model with random covariates that allows for flexible clustering/classification and distribution estimation of a random vector composed of a response variable and a set of covariates. Within this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart in the family of latent class models is also defined. Maximum likelihood parameter estimates are derived using the expectation-maximization algorithm and some computational issues are detailed. Through Monte Carlo experiments, the classification performance of the proposed model is compared with other mixture-based approaches, consistency of the estimators of the regression coefficients is evaluated, and several likelihood-based information criteria are compared for selecting the number of mixture components. An application to real data is also finally considered.