Martin Crowder

Analysis of default data using hidden Markov models

The occurrence of defaults within a bond portfolio is modelled as a simple hidden Markov process. The hidden variable represents the risk state, which is assumed to be common to all bonds within one particular sector and region. After describing the model and recalling the basic properties of hidden Markov chains, we show how to apply the model to a simulated sequence of default events. Then, we consider a real scenario, with default events taken from a large database provided by Standard & Poors. We are able to obtain estimates for the model parameters and also to reconstruct the most likely sequence of the risk state. Finally, we address the issue of global versus industry-specific risk factors. By extending our model to include independent hidden risk sequences, we can disentangle the risk associated with the business cycle from that specific to the individual sector.

Analysis of Reliability and Warranty Claims in Products With Age and Usage Scales

Failures or other adverse events in systems or products may depend on the age and usage history of the unit. Motivated by motor vehicle reliability and warranty data issues, we present models that may be used to assess the dependence on age or usage in heterogeneous populations of products, and show how to estimate model parameters based on different types of field data. The setting in which the events in question are warranty claims is complicated because of the sparseness and incompleteness of the data, and we examine it in some detail. We consider some North American automobile warranty data and use these data to illustrate the methodology.

Identifiability Crises in Competing Risks

Summary Difficulties and pitfalls of dependency modelling in Statistics are very well illustrated by problems of identifiability in Competing Risks. This paper gives a review of such problems with examples intended to animate the theoretical results. The problems covered arise through the traditional way of modelling Competing Risks via latent failure times.

Elements of modern asymptotic theory with statistical applications

Probability and measure random variables and distributions in statistics concepts of asymptotic convergence further asymptotic theory with applications in regression likelihood and associated concepts maximum likelihood and asymptotic theory metric spaces and stochastic processes Brownian motion and weak convergence applications of weak convergence dependent random variables and mixing dependent sequences and martingales.

Models and estimation for systems with recurrent events and usage processes.

This paper considers settings where populations of units may experience recurrent events, termed failures for convenience, and where the units are subject to varying levels of usage. We provide joint models for the recurrent events and usage processes, which facilitate analysis of their relationship as well as prediction of failures. Data on usage are often incomplete and we show how to implement maximum likelihood estimation in such cases. Random effects models with linear usage processes and gamma usage processes are considered in some detail. Data on automobile warranty claims are used to illustrate the proposed models and estimation methodology.

Predicting the amount individuals withdraw at cash machines using a random effects multinomial model

Retail finance organizations use data on past behaviour to make predictions for customer value management strategies. Random-effects models, where each individual has a behavioural pattern drawn from an overall population distribution, are a natural statistical form in this context. The random effects models in this paper are used to predict how much individuals withdraw at a single cash machine visit. A multinomial distribution is taken for the distribution of amounts and the random effects are modelled by a Dirichlet distribution or the empirical distribution of individual maximum likelihood fits. A third model extends the multinomial distribution by incorporating a form of serial dependence and uses an empirical distribution for the random effects. Several prediction tests on a sample of 5000 UK high-street bank accounts find that the greatest benefit from the models is for accounts with a small number of past transactions; that little information may be lost by binning and that the Dirichlet distribution might overestimate the probability of previously unobserved withdrawal amounts. The empirical distribution of random effects is found to perform well because there are a large number of individual accounts.

Monitoring Warranty Claims With Cusums

Manufacturers monitor products in field use with respect to the quality and reliability of their performance, and warranty claims data are one valuable source of information. Updated warranty claims data are examined periodically by manufacturers, and formal monitoring procedures are a useful adjunct that can help identify emerging problems. This article presents cusum procedures for monitoring claims, designed to allow changes in claim rates to be detected in as timely a manner as possible. The determination of plans with given signal probabilities under nominal and increased rates is based on Markov chain calculations which are easy to implement. The procedures are motivated by and illustrated on warranty claims for North American automobiles. This article has online supplementary material.

BAYESIAN PRIORS BASED ON A PARAMETER TRANSFORMATION USING THE DISTRIBUTION FUNCTION

One of the tasks of the Bayesian consulting statistician is to elicit prior information from his client who may be unfamiliar with parametric statistical models. In some cases it may be more illuminating to base a prior distribution for parameter θ on the transformed version F(⋎/θ), where F is the data distribution function and v is a designated reference value, rather than on θ directly. This approach is outlined and explored in various directions to assess its implications. Some applications are given, including general linear regression and transformed linear models.

Two zs are Better than One

Given only a random sample of observations, the usual estimator for the population mean is the sample mean. If additional information is provided it might be possible in some situations to obtain a better estimator. The situation considered here is when the variable whose mean is sought is composed of factors that are themselves observable. In the basic case, the variable can be expressed as the product of two, independent, more basic variables, but we also consider the case of more than two, the effect of correlation, and when there are observation costs.

A method for exploratory repeated-measures analysis applied to a breast-cancer screening study

Published in at http://dx.doi.org/10.1214/11-AOAS481 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)