Distorted Copulas: Constructions and Tail Dependence

Given a copula C, we examine under which conditions on an order isomorphism ψ of [0, 1] the distortion C ψ: [0, 1]2 → [0, 1], C ψ(x, y) = ψ{C[ψ−1(x), ψ−1(y)]} is again a copula. In particular, when the copula C is totally positive of order 2, we give a sufficient condition on ψ that ensures that any distortion of C by means of ψ is again a copula. The presented results allow us to introduce in a more flexible way families of copulas exhibiting different behavior in the tails

The Use of Official Statistics in Self-Selection Bias Modeling

Official statistics are a fundamental source of publicly available information that periodically provides a great amount of data on all major areas of citizens’ lives, such as economics, social development, education, and the environment. However, these extraordinary sources of information are often neglected, especially by business and industrial statisticians. In particular, data collected from small businesses, like small and medium-sized enterprizes (SMEs), are rarely integrated with official statistics data

A note on deficit analysis in dependency models involving Coxian claim amounts

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Sample selection models for count data in R

We provide a detailed hands-on tutorial for the R package SemiParSampleSel (version 1.5). The package implements selection models for count responses fitted by penalized maximum likelihood estimation. The approach can deal with non-random sample selection, flexible covariate effects, heterogeneous selection mechanisms and varying distributional parameters. We provide an overview of the theoretical background and then demonstrate how SemiParSampleSel can be used to fit interpretable models of different complexity. We use data from the German Socio-Economic Panel survey (SOEP v28, 2012. doi: 10.5684/soep.v28) throughout the tutorial

A Bivariate Timing Model of Customer Acquisition and Retention

Two widely recognized components, central to the calculation of customer value, are acquisition and retention propensities. However, while extant research has incorporated such components into different types of models, limited work has investigated the kinds of associations that may exist between them. In this research, we focus on the relationship between a prospective customer\u27s time until acquisition of a particular service and the subsequent duration for which he retains it, and examine the implications of this relationship on the value of prospects and customers. To accomplish these tasks, we use a bivariate timing model to capture the relationship between acquisition and retention. Using a split-hazard model, we link the acquisition and retention processes in two distinct yet complementary ways. First, we use the Sarmonov family of bivariate distributions to allow for correlations in the observed acquisition and retention times within a customer; next, we allow for differences across customers using latent classes for the parameters that govern the two processes. We then demonstrate how the proposed methodology can be used to calculate the discounted expected value of a subscription based on the time of acquisition, and discuss possible applications of the modeling framework to problems such as customer targeting and resource allocation

Copula models for epidemiological research and practice

Investigating associations between random variables (rvs) is one of many topics in the heart of statistical science. Graphical displays show emerging patterns between rvs, and the strength of their association is conventionally quantified via correlation coefficients. When two or more of these rvs are thought of as outcomes, their association is governed by a joint probability distribution function (pdf). When the joint pdf is bivariate normal, scalar correlation coefficients will produce a satisfactory summary of the association, otherwise alternative measures are needed. Local dependence functions, together with their corresponding graphical displays, quantify and show how the strength of the association varies across the span of the data. Additionally, the multivariate distribution function can be explicitly formulated and explored. Copulas model joint distributions of varying shapes by combining the separate (univariate) marginal cumulative distribution functions of each rv under a specified correlation structure. Copula models can be used to analyse complex relationships and incorporate covariates into their parameters. Therefore, they offer increased flexibility in modelling dependence between rvs. Copula models may also be used to construct bivariate analogues of centiles, an application for which few references are available in the literature though it is of particular interest for many paediatric applications. Population centiles are widely used to highlight children or adults who have unusual univariate outcomes. Whilst the methodology for the construction of univariate centiles is well established there has been very little work in the area of bivariate analogues of centiles where two outcomes are jointly considered. Conditional models can increase the efficiency of centile analogues in detection of individuals who require some form of intervention. Such adjustments can be readily incorporated into the modelling of the marginal distributions and of the dependence parameter within the copula model

BAYESIAN ANALYSIS OF THE COMPOUND COLLECTIVE MODEL; THE VARIANCE PREMIUM PRINCIPLE WITH EXPONENTIAL POISSON AND GAMMA-GAMMA DISTRIBUTIONS

The distribution of the aggregate claim size is the considerable importance in insurance theory since, for example, it is needed as an input in premium calculation principles and reserve calculation which plays an important paper in ruin theory. In this paper a Bayesian study for the collective risk model by incorporating a prior distribution for both, the parameter of the claim number distribution and the parameter of the claim size distribution is made and applied to the variance premium principle. Later a sensitivity study is to carry out on both parameters using Bayesian global robustness. Despite the complicated form of the collective risk model it is shown how the robustness study can be treated in an easy way. We illustrate the results obtained with numerical examples.Bayesian Robustness, Contamination Class, Variance Principle.

Statistical modeling and reliability analysis for multi-component systems with dependent failures

Reliability analysis of systems based on component reliability models has drawn the great interest of many researchers so far, as one of the fundamental aspects of reliability assessment issues. In particular, reliability analysis considering dependent failure occurrences of system components is important because the components may fail mutually due to sharing workloads such as heat, tasks and so on. In such a situation, we are liable to incorrectly estimate the reliability of the system unless we consider the possibility of the dependent failure occurrence phenomena. Thus, there are many publications about this topic in the literature. Most of the existing studies deal with the dependent failure between any two components in a multi-component system since its mathematical formulation is comparatively easy. However, the dependent failure may occur among two or more components in actual cases.In this thesis, we aim at developing reliability analysis techniques when several components of a system break down dependently. First, we newly formulate a reliability model of systems with the dependent failure by using a multivariate Farlie-Gumbel-Morgenstern (FGM) copula. Based on the model, we investigate the effect of the dependent failure occurrence on the system\u27s reliability. Secondly, we deal with the parameter estimation for the model in order to evaluate the dependence among the components by using their failure times. To do so, we propose a useful estimation algorithm for the multivariate FGM copula. In addition, we theoretically reveal the asymptotic normality of the proposed estimators and numerically investigate the estimation accuracy. Finally, we present a new method for the detection of the dependent failure occurrence in an n-component parallel system. These results are helpful to both quantitative and qualitative reliability assessment of the system under the possibility of the dependent failure occurrences. Also, our estimation method is especially applicable not only the reliability analysis but also other research fields.博士(工学)法政大学 (Hosei University