THE USE OF AGGREGATE DATA TO ESTIMATE GOMPERTZ‐TYPE OLD‐AGE MORTALITY IN HETEROGENEOUS POPULATIONS

We consider two related aspects of the study of old‐age mortality. One is the estimation of a parameterized hazard function from grouped data, and the other is its possible deceleration at extreme old age owing to heterogeneity described by a mixture of distinct sub‐populations. The first is treated by half of a logistic transform, which is known to be free of discretization bias at older ages, and also preserves the increasing slope of the log hazard in the Gompertz case. It is assumed that data are available in the form published by official statistical agencies, that is, as aggregated frequencies in discrete time. Local polynomial modelling and weighted least squares are applied to cause‐of‐death mortality counts. The second, related, problem is to discover what conditions are necessary for population mortality to exhibit deceleration for a mixture of Gompertz sub‐populations. The general problem remains open but, in the case of three groups, we demonstrate that heterogeneity may be such that it is possible for a population to show decelerating mortality and then return to a Gompertz‐like increase at a later age. This implies that there are situations, depending on the extent of heterogeneity, in which there is at least one age interval in which the hazard function decreases before increasing again.     

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.     

ESTIMATING PROCESSES OF COUNTS FROM CROSS-SECTIONAL AGGREGATE DATA, WITH AN APPLICATION TO MULTISTATE LIFE TABLES AND HEALTH EXPECTANCIES

This paper discusses the estimation of time‐dependent probabilities of a finite state‐space discrete‐time process using aggregate cross‐sectional data. A large‐sample version of multistate logistic regression is described and shown to be useful for analysing multistate life tables. The technique is applied to the estimation of disability‐free, severely disabled and other disabled survival curves and health expectancies in Australia, based on data from national health surveys in 1988, 1993 and 1998. A conclusion is that there has been a general upward trend in the future time expected to be spent in a state of disability, the picture being more pessimistic for males than females. For example, during 1988‐1998 the estimated increase in male life expectancy at birth of 2.7 years is decomposed as a decrease of 1.2 years in disability‐free health (life) expectancy and increases of 1.3 and 2.6 years, respectively, in states of severe disability and other disability.     

The Evaluation of Long Term Trend I

A procedure is presented for the determination of the long term trend in series of data. Robust data smoothing procedures are used to decompose the data into trend, seasonal and irregular components. Confirmatory data analysis is then used to model the deseasonalized data and hence determine the long term trend. A series of riverflow data is analysed using these methods.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE II1

Numerical results are presented for estimates of the parameters in the linear model Y =βX +ε in which X is normally distributed and ε is symmetric stable. The study complements an earlier paper of the same title and the main concern is with numerical comparisons between four estimates of β; the least squares estimate, the minimum absolute deviations estimate, and two moment estimates of the form derived in Chambers and Heathcote (1975). The generation of fifty independent sets of observations (X_j, Y_j), j = 1,2, …, n for each of n = 100, 500 and selected combinations of parameter values provided the basis of the results. It is indicated that the moment estimators and the minimum absolute deviation estimator performed comparably, and are a significant improvement on the least squares estimator. The main conclusion is that one of the moment estimates, based on a two stage adaptive procedure and denoted by β¯_n(t_a) below, is generally the most useful of the four.     

ESTIMATING A NON-GAUSSIAN REGRESSION MODEL WITH MULTICOLLINEARITY

The paper discusses practical issues that arose in estimating a Linear model subject to heteroscedasticity, skewness and multicollinearity. The context of the problem required that linearity of the model as originally formulated be preserved. A seemingly satisfactory fit was obtained using a Gamma model and principal components.     

Working life expectancies: the case of Finland 1980–2006

Summary.  Working life expectancy is the future time that a person is expected to spend in employment. The paper is concerned with its estimation jointly with the expected times spent in the related states of 'on disability pension' and 'other alive'. The method, which is novel in this field, first estimates year- and age-dependent probabilities of being in the states of interest by large sample multivariate logistic regression. Estimates of probabilities, and subsequently expectancies, are given for the case of Finnish women and men aged 16–64 years for selected years in the period 1980–2001, together with projections for 2006. Since 1996 the decline in the employment of males has largely been due to the increasing popularity of early retirement. It was not due to an increase in disability. There has been no such decline for women, and the working life expectancy for males has been predicted to decline to or to fall below the initially lower figure for females by 2006. Considering that the Finnish population is aging rapidly, these trends could entail serious social and economic consequences for society in the coming years because of a looming shortage in the labour force that could undermine the sustainability of a welfare state.     

A LINEAR MODEL WITH ERRORS LACKING A VARIANCE1

The problem discussed is that of estimating β= (β¹, …, β_k) in the model Y=βX +ε when X has a specified multivariate distribution and the error ε does not necessarily have a finite second moment, for example, ε symmetric stable. We construct a moment estimator based on the empirical characteristic function and establish asymptotic unbiassedness and normality. Most of the paper is concerned with the case when X is normal. Forms of the suggested estimator are given in (2.5), (4.6) and (5.5).