Nonparametric Estimation of Copula Regression Models with Discrete Outcomes

Multivariate discrete outcomes are common in a wide range of areas including insurance, finance, and biology. When the interplay between outcomes is significant, quantifying dependencies among interrelated variables is of great importance. Due to their ability to accommodate dependence flexibly, copulas are being applied increasingly. Yet, the application of copulas on discrete data is still in its infancy; one of the biggest barriers is the nonuniqueness of copulas, calling into question model interpretations and predictions. In this article, we study copula estimation with discrete outcomes in a regression context. As the marginal distributions vary with covariates, inclusion of continuous regressors expands the region of support for consistent estimation of copulas. Because some properties of continuous outcomes do not carry over to discrete outcomes, specification of a copula model has been a problem. We propose a nonparametric estimator of copulas to identify the “hidden” dependence structure for discrete outcomes and develop its asymptotic properties. The proposed nonparametric estimator can also serve as a diagnostic tool for selecting a parametric form for copulas. In the simulation study, we explore the performance of the proposed estimator under different scenarios and provide guidance on when the choice of copulas is important. The performance of the estimator improves as discreteness diminishes. A practical bandwidth selector is also proposed. An empirical analysis examines a dataset from the Local Government Property Insurance Fund (LGPIF) in the state of Wisconsin. We apply the nonparametric estimator to model the dependence among claim frequencies from different types of insurance coverage

A multilevel analysis of intercompany claim counts

For the construction of a fair tariff structure in automobile insurance, insurers classify the risks that they underwrite. The idea behind this risk classification is to subdivide the portfolio into classes of risks with similar profiles. While some insurers may have sufficient historical data, several others may not have significant volume of experience data in order to produce reliable claims predictions to help enhance their risk classification systems. A database containing a pooled experience of several insurers thereby helps to produce a more fair, reliable, and equitable premium structure for all risks concerned. Research and analysis of such "intercompany" insurance experience data is lacking in both the actuarial and statistical literature. Its benefits goes beyond the insurer; reinsurers (i.e. insurers of insurers) together with regulators also benefit from statistical models of this type of data because they typically deal with analyzing the experience of a collection of insurers. In this paper, we use multilevel models to analyze the data on claim counts provided by the General Insurance Association of Singapore, an organization consisting of most of the general insurers in Singapore. Our data comes from the financial records of automobile insurance policies followed over a period of nine years. The multilevel nature of the data is due to the following: a certain vehicle is observed over a period of years and is insured by a particular insurance company under a certain ‘fleet’ policy. Fleet policies are umbrella-type policies issued to customers whose insurance covers more than a single vehicle with a taxicab company being a typical example. We show how intercompany data lead to a priori premiums and a posteriori corrections to these initial premiums. Specific focus is made in understanding the intercompany effects using various count distribution models (Poisson, negative binomial, zero-inflated and hurdle Poisson). The performance of these various models is compared; we also investigated how to use the historical claims of a company, fleet and/or vehicle in order to correct for the premium initially set

Conditional expectation formulae for copulas

© 2008 Australian Statistical Publishing Association Inc.Not only are copula functions joint distribution functions in their own right, they also provide a link between multivariate distributions and their lower-dimensional marginal distributions. Copulas have a structure that allows us to characterize all possible multivariate distributions, and therefore they have the potential to be a very useful statistical tool. Although copulas can be traced back to 1959, there is still much scope for new results, as most of the early work was theoretical rather than practical. We focus on simple practical tools based on conditional expectation, because such tools are not widely available. When dealing with data sets in which the dependence throughout the sample is variable, we suggest that copula-based regression curves may be more accurate predictors of specific outcomes than linear models. We derive simple conditional expectation formulae in terms of copulas and apply them to a combination of simulated and real data.Glenis J. Crane and John van der Hoe

Multilevel Model Prediction

best linear unbiased predictors, linear mixed-effects models, hierarchical linear models, random coefficients, school effectiveness,

Bivariate frequency analysis of floods using copulas

[[abstract]]Bivariate flood frequency analysis offers improved understanding of the complex flood process and useful information in preparing flood mitigation measures. However, difficulties arise from limited bivariate distribution functions available to jointly model the correlated flood peak and volume that have different univariate marginal distributions. Copulas are functions that link univariate distribution functions to form bivariate distribution functions, which can overcome such difficulties. The objective of this study was to analyze bivariate frequency of flood peak and volume using copulas. Separate univariate distributions of flood peak and volume are first fitted from observed data. Copulas are then employed to model the dependence between flood peak and volume and join the predetermined univariate marginal distributions to construct the bivariate distribution. The bivariate probabilities and associated return periods are calculated in terms of univariate marginal distributions and copulas. The advantage of using copulas is that they can separate the effect of dependence from the effects of the marginal distributions. In addition, explicit relationships between joint and univariate return periods are made possible when copulas are employed to construct bivariate distribution of floods. The annual floods of Tongtou flow gauge station in the Jhuoshuei River, Taiwan, are used to illustrate bivariate flood frequency analysis.[[notice]]補正完