Random self-decomposability and autoregressive processes

We introduce the notion of random self-decomposability and discuss its relation to the concepts of self-decomposability and geometric infinite divisibility. We present its connection with time series autoregressive schemes with a regression coefficient that randomly turns on and off. In particular, we provide a characterization of random self-decomposability as well as that of marginal distributions of stationary time series that follow this scheme. Our results settle an open question related to the existence of such processes.     

Loss reserving using loss aversion functions

This article discusses the determination of risk capital based on “aversion” functions. Aversion functions weigh different outcomes according to perceived severity. Many practical and popular risk measures are usefully viewed in terms of aversion functions including those arising from distortion operators and risk margin loadings. The approach of this paper builds on, unifies, and extends existing disparate approaches discussed in the literature. Analytical and computer generated illustrations are given as well as suggestions for the practical determination of aversion functions.     

Robust loss reserving in a log-linear model

It is well known that the presence of outlier events can overestimate or underestimate the overall reserve when using the chain-ladder method. The lack of robustness of loss reserving estimators leads to the development of this paper. The appearance of outlier events (including large claims—catastrophic events) can offset the result of the ordinary chain ladder technique and perturb the reserving estimation. Our proposal is to apply robust statistical procedures to the loss reserving estimation, which are insensitive to the occurrence of outlier events in the data. This paper considers robust log-linear and ANOVA models to the analysis of loss reserving by using different type of robust estimators, such as LAD-estimators, M-estimators, LMS-estimators, LTS-estimators, MM-estimators (with initial S-estimate) and Adaptive-estimators. Comparisons of these estimators are also presented, with application of a well known data set.     

The discrete-time queue with autoregressive inputs revisited

In this paper, we present an exact queuing analysis of a discrete-time queue whose arrival process is correlated and consists of a discrete autoregressive model of order 1 (DAR(1)). The functional equation describing this DAR(1)/D/1 queuing model, originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41), is manipulated and transformed into a mathematical tractable form. By using simple analytical transform techniques, we show how our proposed approach allows us to derive an equivalent (yet simpler) expression for the steady-state probability generating function (pgf) of the queue length, as originally derived in Hwang and Sohraby (Queuing Systems 43 (2003)29–41). From this pgf, we characterize the distribution of the packet delay. New numerical results related to packet loss ratio and mean delay of the DAR(1)/D/1 queue are also presented. The proposed approach outlines an alternate solution technique and a general framework under which more complex time-series based queuing models can be analyzed. AMS Subject Classifications 60K25     

The estimation of multivariate random coefficient autoregressive models

Using a two stage regression procedure estimates of the unknown parameters of a class of multivariate random coefficient autoregressive models are obtained. The estimates are shown, under fairly general conditions, to be strongly consistent and to have a distribution which converges to that of a normally distributed random vector.     

Semiparametric model for prediction of individual claim loss reserving

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Conventional methods, such as ladder or separation methods based on aggregated or grouped claims of the so-called “run-off triangle”, have been illustrated to have some drawbacks. Recently, individual claim loss models have attracted a great deal of interest in actuarial literature, which can overcome the shortcomings of aggregated claim loss models. In this paper, we propose an alternative individual claim loss model, which has a semiparametric structure and can be used to fit flexibly the claim loss reserving. Local likelihood is employed to estimate the parametric and nonparametric components of the model, and their asymptotic properties are discussed. Then the prediction of the IBNR claim loss reserving is investigated. A simulation study is carried out to evaluate the performance of the proposed methods.     

Applying copula models to individual claim loss reserving methods

The estimation of loss reserves for incurred but not reported (IBNR) claims presents an important task for insurance companies to predict their liabilities. Recently, individual claim loss models have attracted a great deal of interest in the actuarial literature, which overcome some shortcomings of aggregated claim loss models. The dependence of the event times with the delays is a crucial issue for estimating the claim loss reserving. In this article, we propose to use semi-competing risks copula and semi-survival copula models to fit the dependence structure of the event times with delays in the individual claim loss model. A nonstandard two-step procedure is applied to our setting in which the associate parameter and one margin are estimated based on an ad hoc estimator of the other margin. The asymptotic properties of the estimators are established as well. A simulation study is carried out to evaluate the performance of the proposed methods.     

An individual loss reserving model with independent reporting and settlement

The main purpose of this paper is to assess and demonstrate the advantage of claims reserving models based on individual data in forecasting future liabilities over traditional models on aggregate data both theoretically and numerically. The available information consists of the reporting delays, settlement delays and claim payments. The model settings include Poisson distributed frequency of claims produced by each policy, claims payable at the settlement time, and the amount of payment depending only on its settlement delay. While such settings are applicable to certain but not all practical cases, the principal purpose of the paper is to examine the efficiency of individual data against aggregate data. We refer to loss reserving as to estimate the projections of the outstanding liabilities on observed information. The efficiency of the individual loss reserving against classical aggregate loss reservings, namely Chain-Ladder (C-L) and Bornhuetter–Ferguson (B–F), is assessed by comparing the asymptotic variances of the errors in estimating the conditional expectation (projection) of the outstanding liability between individual, C-L and B–F reservings. The research shows a significant increase in the accuracy of loss reserving by using individual data compared with aggregate data.     

Social welfare and the gini coefficient revisited

The question of conflict between the rankings of income distributions with the same mean by the Gini coefficient and by individualistic social welfare functions is re-examined. The negative result of Newbery (1970) is extended. However, positive results are obtainable which reverse Newbery's conclusion by admitting into the individual's utility index a measure of his position in the income distribution, or of his deprivation with respect to others' incomes.     

The mixing property of bilinear and generalised random coefficient autoregressive models

The paper gives sufficient conditions for the absolute regularity of bilinear models. Our approach is based on their Markovian representation. The above property is a direct consequence of the geometric ergodicity of the Markovian process in this representation. The latter process belongs to what we call the generalised random coefficients autoregressive models. Conditions for the geometric ergodicity and also for the existence of moments for this model are given. Our results generalise that of Feigin and Tweedie.     

Bayesian analysis of loss reserving using dynamic models with generalized beta distribution

A Bayesian approach is presented in order to model long tail loss reserving data using the generalized beta distribution of the second kind (GB2) with dynamic mean functions and mixture model representation. The proposed GB2 distribution provides a flexible probability density function, which nests various distributions with light and heavy tails, to facilitate accurate loss reserving in insurance applications. Extending the mean functions to include the state space and threshold models provides a dynamic approach to allow for irregular claims behaviors and legislative change which may occur during the claims settlement period. The mixture of GB2 distributions is proposed as a mean of modeling the unobserved heterogeneity which arises from the incidence of very large claims in the loss reserving data. It is shown through both simulation study and forecasting that model parameters are estimated with high accuracy.     

Random coefficient differential equation models for bacterial growth

In the mathematical modeling of population growth, and in particular of bacterial growth, parameters are either measured directly or determined by curve fitting. These parameters have large variability that depends on the experimental method and its inherent error, on differences in the actual population sample size used, as well as other factors that are difficult to account for. In this work the parameters that appear in the Monod kinetics growth model are considered random variables with specified distributions. A stochastic spectral representation of the parameters is used, together with the polynomial chaos method, to obtain a system of differential equations, which is integrated numerically to obtain the evolution of the mean and higher-order moments with respect to time.     

Limit theory for random coefficient first-order autoregressive process under martingale difference error sequence

Phillips and Magdalinos (2007) [1] gave the asymptotic theory for autoregressive time series with a root of the form ρ_n=1+c/k_n, where k_n is a deterministic sequence. In this paper, an extension to the more general case where the coefficients of an AR(1) model is a random variable and the error sequence is a sequence of martingale differences is discussed. A conditional least squares estimator of the autoregressive coefficient is derived and shown to be asymptotically normal. This extends the result of Phillips and Magdalinos (2007) [1] for stationary and near-stationary cases.     

Weak limits of random coefficient autoregressive processes and their application in ruin theory

We prove that a large class of discrete-time insurance surplus processes converge weakly to a generalized Ornstein–Uhlenbeck process, under a suitable re-normalization and when the time-step goes to 0. Motivated by ruin theory, we use this result to obtain approximations for the moments, the ultimate ruin probability and the discounted penalty function of the discrete-time process.     

Shrinkage estimation and variable selection in multiple regression models with random coefficient autoregressive errors

In this paper, we consider improved estimation strategies for the parameter vector in multiple regression models with first-order random coefficient autoregressive errors (RCAR(1)). We propose a shrinkage estimation strategy and implement variable selection methods such as lasso and adaptive lasso strategies. The simulation results reveal that the shrinkage estimators perform better than both lasso and adaptive lasso when and only when there are many nuisance variables in the model.