William S. Jewell

Further Results on Recursive Evaluation of Compound Distributions

A recent result by Panjer provides a recursive algorithm for the compound distribution of aggregate claims when the counting law belongs to a special recursive family. In the present paper we first give a characterization of this recursive family, then describe some generalizations of Panjers result.

Simulation of urban vehicle-monitoring systems

The results of experimentally based computer simulations of phase-ranging and pulse-ranging urban vehicle-monitoring systems are given. These show that such systems are quite feasible even in the worst environments.

Credible Means are exact Bayesian for Exponential Families

The credibility formula used in casualty insurance experience rating is known to be exact for certain prior-likelihood distributions, and is the minimum least-squares unbiased estimator for all others. We show that credibility is, in fact, exact for all simple exponential families where the mean is the sufficient statistic, and is also exact in an extended sense for all regular distributions with their natural conjugate priors where there is a fixed-dimensional sufficient statistic.

A Simple Proof of: L = λW

A simple proof of the fundamental queuing formula L = λW is given that is based on renewal theory. The basic assumptions that are needed are: 1 the event system is empty is recurrent, and 2 the arrival and waiting-time mechanisms are reset by the next arrival after this event occurs.

Bayesian Extensions to a Basic Model of Software Reliability

A Bayesian analysis of the software reliability model of Jelinski and Moranda is given, based upon Meinhold and Singpurwalla. Important extensions are provided to the stopping rule and prior distribution of the number of defects, as well as permitting uncertainty in the failure rate. It is easy to calculate the predictive distribution of unfound errors at the end of software testing, and to see the relative effects of uncertainty in the number of errors and in the detection efficiency. The behavior of the predictive mode and mean over time are examined as possible point estimators, but are clearly inferior to calculating the full predictive distribution.

Predicting Ibnyr Events and Delays: I. Continuous Time

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. Both the rate at which such events occur and the parameters of the delay distribution are unknown random quantities. Given the number of events that have been reported during some observation interval, plus various secondary data on the dates of the events, the problem is to estimate the true values of the unknown parameters and to predict the number of events that are still unreported. A full-distributional Bayesian model is used, and it is shown that the amount of secondary data is critical. A recursive procedure calculates the predictive density; however, an explicit formula for the predictive mode can be obtained. The main computational work is the evaluation of an integral involving the prior density of the delay parameters, but this can be simplified in the exponential case using Gammoid approximations.

Regularity Conditions For Exact Credibility

Conditions under which the natural conjugate prior is not zero on its boundary are given, correcting an argument about conditions for exact credibility given in another paper.

A General Framework for Learning Curve Reliability Growth Models

In reliability growth models, system performance improves during prototype testing, as design changes are made and operating procedures and the environment are modified. There is great interest in predicting the ultimate performance of the system, using only the epochs of the failures that occur early in the testing program. This paper constructs a general framework for learning-curve models of reliability growth, including many different model variations that have previously been analyzed. Numerical trials indicate the difficulty of estimating ultimate performance; the maximum likelihood estimator is unstable for small testing intervals with a small number of systems on test. Bayesian procedures are recommended for implementation.

Predicting IBNYR Events and Delays II. Discrete Time

An IBNYR event is one that occurs randomly during some fixed exposure interval and incurs a random delay before it is reported. A previous paper developed a continuous-time model of the IBNYR process in which both the Poisson rate at which events occur and the parameters of the delay distribution are unknown random quantities; a full-distributional Bayesian method was then developed to predict the number of unreported events. Using a numerical example, the success of this approach was shown to depend upon whether or not the occurrence dates were available in addition to the reporting dates. This paper considers the more usual practical situation in which only discretized epoch information is available; this leads to a loss of predictive accuracy, which is investigated by considering various levels of quantization for the same numerical example.

Two Classes of Covariance Matrices Giving Simple Linear Forecasts

Two special classes of covariance matrices are considered which give simplified computations for linear forecasts without continued reinversion of the matrix. In the first class, the optimal coefficients in the forecast can be computed in advance for every time period by simple closed formulas. In the second class, which is a generalization of the first, the optimal coefficients are obtained through a simple first-order linear recursive relation between forecasts of successive time periods. Collective risk forecasting models which give rise to these classes of covariances are presented.