Borek Puza

The health expectancies of older Australians

Health expectancies of the states ‘Disability-free’ and ‘Disabled’ are estimated for Australian females and males aged 60 and over, both by cohort from 1980 and current for survey years 1981, 1988, 1993 and 1998. Modifications of recently developed logistic regression techniques are used rather than the standard 1971 method due to Sullivan. Results from the three later surveys are broadly similar and differ in important respects from those of the 1981 survey. Based on the last three surveys our estimates support the view that, depending on age, two-thirds or more of the increase in female life expectancy over the decade 1988–1998 is spent in the Disabled state. The situation is worse for elderly men, for whom all of the increased years of expected life are estimated to be spent in the Disabled state. The findings do not support rectangularization of the survival curve or Disability-free survival curve.

Generalised Clopper–Pearson confidence intervals for the binomial proportion

In this paper we develop some new confidence intervals for the binomial proportion. The Clopper–Pearson interval is interpreted as an outcome of randomised confidence interval theory. The problem of randomised intervals possibly being empty is solved using a new technique involving ‘tail functions’, with the offshoot being a new class of randomised and Clopper–Pearson intervals. Some of the new intervals are investigated and shown to have attractive frequentist properties. Coverage probabilities and expected widths are compared and guidelines are established for constructing the optimal generalised Clopper–Pearson interval in any given situation.

Bayesian Truncated Poisson Regression with Application to Dutch Illegal Immigrant Data

This article presents a Bayesian approach to the regression analysis of truncated data, with a focus on zero-truncated counts from the Poisson distribution. The approach provides inference not only on the regression coefficients but also on the total sample size and the parameters of the covariate distribution. The theory is applied to some illegal immigrant data from The Netherlands. Several models are fitted with the aid of Markov chain Monte Carlo methods and assessed via posterior predictive p-values. Inferences are compared with those obtained elsewhere using other approaches.

A Bayesian approach to modeling the interaction between air pollution and temperature.

PURPOSE Investigating the interaction between particulate matter air pollution (PM) and temperature is important for quantifying the effects of PM on mortality. One approach is stratification-estimating the effect of PM within different temperature strata-but this treats the cutpoints that define the strata as fixed, when in fact they are unknown. The purpose of this paper is to propose a new approach that appropriately accounts for uncertainty regarding the cutpoints, and to apply this approach to data from two Australian cities. METHODS We propose a Bayesian model which allows the effects of PM to differ within different temperature strata. The cutpoints that define the strata are parameters that are jointly estimated along with the other model parameters. This is in contrast with the standard stratification approach, where cutpoints are specified a priori and treated as fixed constants. Also, the Bayesian model is formulated in a way that ensures continuity in the effects of PM at the stratum boundaries. Markov chain Monte Carlo methods are used to perform the inferences. RESULTS Analysis of daily data over several years provides evidence for an interactive effect between PM and temperature in Sydney and no support for such an effect in Melbourne. CONCLUSIONS The proposed Bayesian model provides a means for investigating interactions between PM and temperature which appropriately incorporates uncertainty.

Constrained confidence intervals in time series studies of mortality and air pollution.

This paper focuses on constrained confidence intervals in the context of environmental time series studies where one seeks to ascertain the effects of ambient air pollution on human mortality. If the regression parameter representing such effects is non-negative, corresponding to a belief that more pollution cannot be beneficial, a desirable goal is to produce a constrained confidence interval for the parameter which is entirely non-negative. We show how this goal can be achieved using the method of tail functions. The proposed methodology is illustrated by the application to an environmental study of 100 cities in the United States involving regressions of mortality counts on levels of particulate matter air pollution. The large number of constrained CIs that contain zero is an indication that for the majority of the 100 cities there is not enough evidence to conclude a positive association between air pollution and mortality.

Approximations of Value-at-Risk As an Extreme Quantile of a Random Sum of Heavy-Tailed Random Variables

This paper studies the approximation of extreme quantiles of random sums of heavy-tailed random variables, or more specifically, subexponential random variables. A key application of this approximation is the calculation of operational VaR (value at risk) for financial institutions, to determine operational risk capital requirements. The paper follows work by Bocker & Kluppelberg (2005) & Bocker and Sprittulla (2006) and makes several advances. These include two new approximations of VaR and an extension to multiple loss types where the VaR relates to a sum of random sums, each of which is defined by different distributions. The proposed approximations are assessed via a simulation study.

Two Birthday Problems

ABSTRACT This article focuses on two birthday problems which naturally follow from the most common birthday problems taught in statistics courses. The solutions to these new problems provide a good illustration of several combinatorial techniques.

A series of two-urn biased sampling problems

ABSTRACT Much of the literature on matching problems in statistics has focused on single items chosen from independent, but fully overlapping sets. This paper considers a more general problem of sampling without replacement from partially overlapping sets and presents new theory on probability problems involving two urns and the selection of balls from these urns according to a biased without-replacement sampling scheme. The strength of the sampling bias is first considered as known, and then as unknown, with a discussion of how that strength may be assessed using observed data. Each scenario is illustrated with a numerical example, and the theory is applied to a gender bias problem in committee selection, and to a problem where competing retailers select products to place on promotion.

Improved confidence intervals for the exponential mean via tail functions

Abstract The method of tail functions is applied to confidence estimation of the exponential mean in the presence of prior information. It is shown how the “ordinary” confidence interval can be generalized using a class of tail functions and then engineered for optimality, in the sense of minimizing prior expected length over that class, whilst preserving frequentist coverage. It is also shown how to derive the globally optimal interval, and how to improve on this using tail functions when criteria other than length are taken into consideration. Probabilities of false coverage are reported for some of the intervals under study, and the theory is illustrated by application to confidence estimation of a reliability coefficient based on some survival data.

Length-biased, with-replacement sampling from an exponential finite population

In this article, we study a statistical model which features a finite population of exponentially distributed values and a length-biased, with-replacement sampling mechanism. This mechanism is such that units compete with one another for selection at each draw. It is shown how inference on a number of quantities can be performed using both frequentist and Bayesian strategies. A Monte Carlo study is used to assess the performance of the proposed point and interval estimators.