BATHTUB-TYPE CURVES IN RELIABILITY AND BEYOND

Finding optimal, or at least good, maintenance and repair policies is crucial in reliability engineering. Likewise, describing life phases of human mortality is important when determining social policy or insurance premiums. In these tasks, one searches for distributions to fit data and then makes inferences about the population(s). In the present paper, we focus on bathtub‐type distributions and provide a view of certain problems, methods and solutions, and a few challenges, that can be encountered in reliability engineering, survival analysis, demography and actuarial science.     

Estimating conditional tail expectation with actuarial applications in view

We develop statistical inferential tools for estimating and comparing conditional tail expectation (CTE) functions, which are of considerable interest in actuarial science. In particular, we construct estimators for the CTE functions, develop the necessary asymptotic theory for the estimators, and then use the theory for constructing confidence intervals and bands for the functions. Both parametric and non-parametric approaches are explored. Simulation studies illustrate the performance of estimators in various situations. Results are obtained under minimal assumptions, and the general Vervaat process plays a crucial role in achieving these goals.     

Convex rearrangements,generalized Lorenz curves,and correlated Gaussian data

We propose a statistical index for measuring the fluctuations of a stochastic process ξ$\xi$. This index is based on the generalized Lorenz curves and (modified) Gini indices of econometric theory.     

Estimating the turning point of a bathtub-shaped failure distribution

The turning point of a hazard rate function is useful in assessing the hazard in the useful life phase and helps to determine and plan appropriate burn-in, maintenance, and repair policies and strategies. For many bathtub-shaped distributions, the turning point is unique, and the hazard varies little in the useful life phase. We investigate the performance of an empirical estimator for the turning point in the case of the modified Weibull distribution, a bathtub-shaped generalization of the Weibull distribution, that has been found to be useful in reliability engineering and other areas concerned with life-time data. We illustrate the theory by means of an example, and also conduct a simulation study to assess the performance of the estimator in practice.     

Robust fitting of claim severity distributions and the method of trimmed moments

Many quantities arising in non-life insurance depend on claim severity distributions, which are usually modeled assuming a parametric form. Obtaining good estimates of the quantities, therefore, reduces to having good estimates of the model parameters. However, the notion of ‘good estimate’ depends on the problem at hand. For example, the maximum likelihood estimators (MLEs) are efficient, but they generally lack robustness. Since outliers are common in insurance loss data, it is therefore important to have a method that allows one to balance between efficiency and robustness. Guided by this philosophy, in the present paper we suggest a general estimation method that we call the method of trimmed moments (MTM). This method is appropriate for various model-fitting situations including those for which a close fit in one or both tails of the distribution is not required. The MTM estimators can achieve various degrees of robustness, and they also allow the decision maker to easily see the actions of the estimators on the data, which makes them particularly appealing. We illustrate these features with detailed theoretical analyses and simulation studies of the MTM estimators in the case of location–scale families and several loss distributions such as lognormal and Pareto. As a further illustration, we analyze a real data set concerning hurricane damages in the United States from 1925 to 1995.     

Identifying health inequalities between Māori and non‐Māori using mortality tables

Abstract

While there is a need for more detailed information on health inequality to guide public health policy, the most complete and easily available data remain those in mortality tables. We investigate, via a comparative analysis of data from New Zealand on Māori and non‐Māori mortality, whether more detailed information than raw life expectancy may be extracted from the mortality tables. Given a parametric distribution for the mortality capable of fitting irregularities in mortality table data, the curvature of the survival and hazard rates can identify changes in mortality rates, such as infant and late‐life adult mortality, which allows for straightforward comparisons between the two sub‐populations. Our results identify an exogenous effect in earlier mortality among Maori, which correlates well with many published observations of health and health‐care inequalities between Māori and non‐Māori. This “proof of concept” for our method of analysis indicates that examination of bulk data such as those in mortality tables has a potential role in the design of more detailed studies involving causes of mortality.     

The Asymptotic Distribution of the S–Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S –Gini index. It shows that the theory of L –statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S –Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S –Gini index can be estimated with a prescribed margin of error.     

Modeling lactation curves: classical parametric models re-examined and modified

A large number of methods for modeling lactation curves have been proposed – parametric and nonparametric, mathematically or biologically oriented. The most popular of these are methods that express the milk yield in terms of time via a parametric nonlinear functional equation. This is intuitive and allows for relatively easy mathematical and biological interpretations of the parameters involved. Interestingly, as far as we are aware, all such models generate nonzero milk yields on the whole positive time half-line, even though real lactation curves always have finite range, with spans of approximately 300 days for dairy cows. For this reason, we re-examine a number of existing parametric models, and modify them to produce finite-range lactation curves that fit remarkably well to data of milk yields from New Zealand cows. The use of daily or weekly yields rather than the monthly yields normally considered reveals considerable variation that is usually suppressed. Both individual and herd lactation curves are examined in the present paper, and median-based procedures explored as alternatives to the usual average-based methods. These suggestions offer further insights into the existing literature on modeling lactation curves.