Robust Bayesian analysis of loss reserving data using scale mixtures distributions

It is vital for insurance companies to have appropriate levels of loss reserving to pay outstanding claims and related settlement costs. With many uncertainties and time lags inherently involved in the claims settlement process, loss reserving therefore must be based on estimates. Existing models and methods cannot cope with irregular and extreme claims and hence do not offer an accurate prediction of loss reserving. This paper extends the conventional normal error distribution in loss reserving modeling to a range of heavy-tailed distributions which are expressed by certain scale mixtures forms. This extension enables robust analysis and, in addition, allows an efficient implementation of Bayesian analysis via Markov chain Monte Carlo simulations. Various models for the mean of the sampling distributions, including the log-Analysis of Variance (ANOVA), log-Analysis of Covariance (ANCOVA) and state space models, are considered and the straightforward implementation of scale mixtures distributions is demonstrated using OpenBUGS.     

Precise large deviations of aggregate claims in a compound size-dependent renewal risk model

In this paper, we introduce a compound size-dependent renewal risk model driven by two sequences of random sources. The individual claim sizes and their inter-arrival times form a sequence of independent and identically distributed random pairs with each pair obeying a specific dependence structure. The numbers of claims caused by individual events form another sequence of independent and identically distributed positive integer-valued random variables, independent of the random pairs above. Precise large deviations of aggregate claims for the compound size-dependent renewal risk model are investigated in the case of dominatedly varying claim sizes.     

Overcoming biases and misconceptions in ecological studies

The aggregate data study design provides an alternative group level analysis to ecological studies in the estimation of individual level health risks. An aggregate model is derived by aggregating a plausible individual level relative rate model within groups, such that population-based disease rates are modelled as functions of individual level covariate data. We apply an aggregate data method to a series of fictitious examples from a review paper by Greenland and Robins which illustrated the problems that can arise when using the results of ecological studies to make inference about individual health risks. We use simulated data based on their examples to demonstrate that the aggregate data approach can address many of the sources of bias that are inherent in typical ecological analyses, even though the limited between-region covariate variation in these examples reduces the efficiency of the aggregate study. The aggregate method has the potential to estimate exposure effects of interest in the presence of non-linearity, confounding at individual and group levels, effect modification, classical measurement error in the exposure and non-differential misclassification in the confounder.     

On MCMC sampling in hierarchical longitudinal models

Markov chain Monte Carlo (MCMC) algorithms have revolutionized Bayesian practice. In their simplest form (i.e., when parameters are updated one at a time) they are, however, often slow to converge when applied to high-dimensional statistical models. A remedy for this problem is to block the parameters into groups, which are then updated simultaneously using either a Gibbs or Metropolis-Hastings step. In this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.     

Using Convex Combination of Copulas and EM Algorithm for Credibility

In this article, using longitudinal data, we develop the theory of credibility by copula model. The convex combination of copulas is used to describe the dependencies among claims. Finally, for comparing with the results of a single copula, using EM algorithm, some simulations of Massachusetts automobile claims are presented.     

Ties in one block comparison experiments: a generalization of the Mallows–Bradley–Terry ranking model

This study is concerned with the extension of the Mallows–Bradley–Terry ranking model for one block comparison consisting of all the items of interest to situations which allow an expression of no preference. We consider a modification of the Mallows–Bradley–Terry ranking model by introducing an additional parameter, called an index of discrimination, in the model. This permits ties in the model. The maximum likelihood estimates of the parameters are found using a Maximization–Minimization algorithm: the evaluation of the mathematical expectations involved in the log-likelihood equation is obtained by generating samples of Monte Carlo Markov chain from the stationary distribution. In addition, a simulation study for asymptotic properties assessment has been made. The proposed method is applied to analyze data election.     

Large deviations for the stochastic present value of aggregate claims in the nonstandard compound renewal risk model with widely upper Orthant dependent claims

Abstract

Recently, Jiang et al. (Statist. Probab. Lett. 101, 83–91) obtained the asymptotic formulas for the large deviations for the stochastic present value of aggregate claims in the renewal risk model with Pareto-type claims and stochastic return on investments, where the price process of the investment portfolio is described as a geometric Lévy process. In the paper, we extend the above results to a nonstandard compound renewal risk model with widely upper orthant dependent and dominatedly-varying-tailed claims.     

Addressing misclassification for binary data: probit and t-link regressions

Generalized linear models are addressed to describe the dependence of data on explanatory variables when the binary outcome is subject to misclassification. Both probit and t-link regressions for misclassified binary data under Bayesian methodology are proposed. The computational difficulties have been avoided by using data augmentation. The idea of using a data augmentation framework (with two types of latent variables) is exploited to derive efficient Gibbs sampling and expectation–maximization algorithms. Besides, this formulation has allowed to obtain the probit model as a particular case of the t-link model. Simulation examples are presented to illustrate the model performance when comparing with standard methods that do not consider misclassification. In order to show the potential of the proposed approaches, a real data problem arising when studying hearing loss caused by exposure to occupational noise is analysed.     

An algorithm for the Buckley–James estimator with interval-censored data

The Buckley–James estimator (BJE) [J. Buckley and I. James, Linear regression with censored data, Biometrika 66 (1979), pp. 429–436] has been extended from right-censored (RC) data to interval-censored (IC) data by Rabinowitz et al. [D. Rabinowitz, A. Tsiatis, and J. Aragon, Regression with interval-censored data, Biometrika 82 (1995), pp. 501–513]. The BJE is defined to be a zero-crossing of a modified score function H(b), a point at which H(·) changes its sign. We discuss several approaches (for finding a BJE with IC data) which are extensions of the existing algorithms for RC data. However, these extensions may not be appropriate for some data, in particular, they are not appropriate for a cancer data set that we are analysing. In this note, we present a feasible iterative algorithm for obtaining a BJE. We apply the method to our data.     

Estimation of parameters in incomplete data models defined by dynamical systems

Parametric incomplete data models defined by ordinary differential equations (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Accurate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochastic approximation EM algorithm (SAEM) in the maximum likelihood estimation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings–Metropolis (H–M) algorithms. A modified H–M algorithm, including an original local linearization scheme to solve the ODEs, is proposed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacokinetic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates.     

The random partition masking model for interval-censored and masked competing risks data

We study the nonparametric maximum likelihood estimate (NPMLE) of the cdf or sub-distribution functions of the failure time for the failure causes in a series system. The study is motivated by a cancer research data (from the Memorial Sloan-Kettering Cancer Center) with interval-censored time and masked failure cause. The NPMLE based on this data set suggests that the existing masking models are not appropriate. We propose a new model called the random partition masking model, which does not rely on the commonly used symmetry assumption (namely, given the failure cause, the probability of observing the masked failure causes is independent of the failure time; see Flehinger et al. Inference about defects in the presence of masking, Technometrics 38 (1996), pp. 247–255). The RPM model is easier to implement in simulation studies than the existing models. We discuss the algorithms for computing the NPMLE and study its asymptotic properties. Our simulation and data analysis indicate that the NPMLE is feasible for a moderate sample size.     

The Generalized Birnbaum–Saunders Distribution and Its Theory,Methodology, and Application

In this paper, we discuss the class of generalized Birnbaum–Saunders distributions, which is a very flexible family suitable for modeling lifetime data as it allows for different degrees of kurtosis and asymmetry and unimodality as well as bimodality. We describe the theoretical developments on this model including properties, transformations and related distributions, lifetime analysis, and shape analysis. We also discuss methods of inference based on uncensored and censored data, diagnostics methods, goodness-of-fit tests, and random number generation algorithms for the generalized Birnbaum–Saunders model. Finally, we present some illustrative examples and show that this distribution fits the data better than the classical Birnbaum–Saunders model.     

Model‐based clustering of longitudinal data

A new family of mixture models for the model‐based clustering of longitudinal data is introduced. The covariance structures of eight members of this new family of models are given and the associated maximum likelihood estimates for the parameters are derived via expectation–maximization (EM) algorithms. The Bayesian information criterion is used for model selection and a convergence criterion based on the Aitken acceleration is used to determine the convergence of these EM algorithms. This new family of models is applied to yeast sporulation time course data, where the models give good clustering performance. Further constraints are then imposed on the decomposition to allow a deeper investigation of the correlation structure of the yeast data. These constraints greatly extend this new family of models, with the addition of many parsimonious models. The Canadian Journal of Statistics 38:153–168; 2010 © 2010 Statistical Society of Canada     

Analyses of infectious disease data from household outbreaks by Markov chain Monte Carlo methods

The analysis of infectious disease data presents challenges arising from the dependence in the data and the fact that only part of the transmission process is observable. These difficulties are usually overcome by making simplifying assumptions. The paper explores the use of Markov chain Monte Carlo (MCMC) methods for the analysis of infectious disease data, with the hope that they will permit analyses to be made under more realistic assumptions. Two important kinds of data sets are considered, containing temporal and non-temporal information, from outbreaks of measles and influenza. Stochastic epidemic models are used to describe the processes that generate the data. MCMC methods are then employed to perform inference in a Bayesian context for the model parameters. The MCMC methods used include standard algorithms, such as the Metropolis–Hastings algorithm and the Gibbs sampler, as well as a new method that involves likelihood approximation. It is found that standard algorithms perform well in some situations but can exhibit serious convergence difficulties in others. The inferences that we obtain are in broad agreement with estimates obtained by other methods where they are available. However, we can also provide inferences for parameters which have not been reported in previous analyses.     

Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models

Bimodal truncated count distributions are frequently observed in aggregate survey data and in user ratings when respondents are mixed in their opinion. They also arise in censored count data, where the highest category might create an additional mode. Modeling bimodal behavior in discrete data is useful for various purposes, from comparing shapes of different samples (or survey questions) to predicting future ratings by new raters. The Poisson distribution is the most common distribution for fitting count data and can be modified to achieve mixtures of truncated Poisson distributions. However, it is suitable only for modeling equidispersed distributions and is limited in its ability to capture bimodality. The Conway–Maxwell–Poisson (CMP) distribution is a two-parameter generalization of the Poisson distribution that allows for over- and underdispersion. In this work, we propose a mixture of CMPs for capturing a wide range of truncated discrete data, which can exhibit unimodal and bimodal behavior. We present methods for estimating the parameters of a mixture of two CMP distributions using an EM approach. Our approach introduces a special two-step optimization within the M step to estimate multiple parameters. We examine computational and theoretical issues. The methods are illustrated for modeling ordered rating data as well as truncated count data, using simulated and real examples.     

Bayesian Analysis of Skew Gaussian Spatial Models Based on Censored Data

The customary approach to spatial data modeling in the presence of censored data, is to assume the underlying random field is Gaussian. However, in practice, we often faced data that the exploratory data analysis shows the skewness and consequently, it violates the normality assumption. In such setting, the skew Gaussian (SG) spatial model is used to overcome this issue. In this article, the SG model is fitted based on censored observations. For this purpose, we adopt the Bayesian approach and utilize the Markov chain Monte Carlo algorithms and data augmentations to carry out calculations. A numerical example illustrates the methodology.     

An MM Algorithm for Constrained Estimation in a Road Safety Measure Modeling

We model the effect of a road safety measure on a set of target sites with a control area for each site, and we suppose that the accident data recorded at each site are classified in different mutually exclusive types. We adopt the before–after technique and we assume that at any one target site the total number of accidents recorded is multinomially distibuted between the periods and types of accidents. In this article, we propose a minorization–majorization (MM) algorithm for obtaining the constrained maximum likelihood estimates of the parameter vector. We compare it with a gradient projection–expectation maximization (GP-EM) algorithm, based on gradient projections. The performance of the algorithms is examined through a simulation study of road safety data.     

Parameter estimation of nonlinear mixed-effects models using first-order conditional linearization and the EM algorithm

Nonlinear mixed-effects (NLME) models are flexible enough to handle repeated-measures data from various disciplines. In this article, we propose both maximum-likelihood and restricted maximum-likelihood estimations of NLME models using first-order conditional expansion (FOCE) and the expectation–maximization (EM) algorithm. The FOCE-EM algorithm implemented in the ForStat procedure SNLME is compared with the Lindstrom and Bates (LB) algorithm implemented in both the SAS macro NLINMIX and the S-Plus/R function nlme in terms of computational efficiency and statistical properties. Two realworld data sets an orange tree data set and a Chinese fir (Cunninghamia lanceolata) data set, and a simulated data set were used for evaluation. FOCE-EM converged for all mixed models derived from the base model in the two realworld cases, while LB did not, especially for the models in which random effects are simultaneously considered in several parameters to account for between-subject variation. However, both algorithms had identical estimated parameters and fit statistics for the converged models. We therefore recommend using FOCE-EM in NLME models, particularly when convergence is a concern in model selection.     

The value of the aggregate data approach in meta-analysis with time-to-event outcomes

Collecting individual patient data has been described as the 'gold standard' for undertaking meta-analysis. If studies involve time-to-event outcomes, conducting a meta-analysis based on aggregate data can be problematical. Two meta-analyses of randomized controlled trials with time-to-event outcomes are used to illustrate the practicality and value of several proposed methods to obtain summary statistic estimates. In the first example the results suggest that further effort should be made to find unpublished trials. In the second example the use of aggregate data for trials where no individual patient data have been supplied allows the totality of evidence to be assessed and indicates previously unrecognized heterogeneity.     

Fitting finite mixture models using iterative Monte Carlo classification

Parameters of a finite mixture model are often estimated by the expectation–maximization (EM) algorithm where the observed data log-likelihood function is maximized. This paper proposes an alternative approach for fitting finite mixture models. Our method, called the iterative Monte Carlo classification (IMCC), is also an iterative fitting procedure. Within each iteration, it first estimates the membership probabilities for each data point, namely the conditional probability of a data point belonging to a particular mixing component given that the data point value is obtained, it then classifies each data point into a component distribution using the estimated conditional probabilities and the Monte Carlo method. It finally updates the parameters of each component distribution based on the classified data. Simulation studies were conducted to compare IMCC with some other algorithms for fitting mixture normal, and mixture t, densities.