METHODS FOR QUANTIFYING SUBJECTIVE PROBABILITIES AND MULTI-ATTRIBUTE UTILITIES*†

In many situations where normative decision-aiding techniques could be usefully applied, historical data are inadequate for estimating the required outcome probabilities, and economic methodologies are inadequate for estimating the aggregate utility derived from the several outcome attributes. In such cases it is often useful to obtain the required estimates in the form of expert judgments, i.e. to obtain subjective probabilities and multi-attribute utilities. Similarly, in many situations where behavioral decision processes are to be studied, it is necessary to scale the expectations and perceived values of the decision makers. This article describes the methods for eliciting subjective probabilities and multi-attribute utilities whose usefulness has been empirically studied and reported in the research literature. It also contains summary guidelines concerning the elicitation and use of such judgments.     

Information theory and risk in capital markets

The present empirical study applies the methodology of information theory to the problem of assessing and separating capital market risk, which is separated into its systematic and unsystematic components. Monthly return relatives for all securities traded on the New York Stock Exchange are examined for the period 1926 to 1971, which is segmented into six 7-year subperiods. The securities are combined into portfolios of various sizes and ranked. It is concluded that although both systematic and unsystematic risks have increased over the 45-year interval—particularly between the pre-1940 and post-1940 periods—they have maintained their relative share of the total risk over the same period.     

The Assessment of Probability Distributions from Expert Opinions with an Application to Seismic Fragility Curves

A method is developed for estimating a probability distribution using estimates of its percentiles provided by experts. The analyst's judgment concerning the credibility of these expert opinions is quantified in the likelihood function of Bayes'Theorem. The model considers explicitly the random variability of each expert estimate, the dependencies among the estimates of each expert, the dependencies among experts, and potential systematic biases. The relation between the results of the formal methods of this paper and methods used in practice is explored. A series of sensitivity studies provides insights into the significance of the parameters of the model. The methodology is applied to the problem of estimation of seismic fragility curves (i.e., the conditional probability of equipment failure given a seismically induced stress).     

Widowhood,family size,and post-reproductive Mortality: a comparative Analysis of three populations in Nineteenth-Century Europe

Researchers from a number of disciplines have offered competing theories about the effects oJ childbearing on parents 'postreproductive longevity. The "disposable soma theory" argues that investments in somatic maintenance increase longevity but reduce childbearing. "Maternal depletion" models suggest that the rigors of childrearing increase mortality in later years. Other researchers consider continued childbearing a sign of healthy aging and a predictor of future longevity. Empirical studies have produced inconsistent and contradictory results. Our focus is on the experience of widowhood, which has been ignored in previous studies. We hypothesize that the death of a spouse is a stressful event with long-term consequences for health, especially for women with small children. Data are drawn from historical sources in Sweden, Belgium, and the Netherlands from 1766 to 1980. Postreproductive mortality was highest among young widows with larger families in all three samples. Age at last birth had little or no effect. We conclude that raising children under adverse circumstances can have long-lasting, harmful effects on a mother's health.     

The limitations of using school league tables to inform school choice

Summary. In England, so-called 'league tables' based on examination results and test scores are published annually, ostensibly to inform parental choice of secondary schools. A crucial limitation of these tables is that the most recent published information is based on the current performance of a cohort of pupils who entered secondary schools several years earlier, whereas for choosing a school it is the future performance of the current cohort that is of interest. We show that there is substantial uncertainty in predicting such future performance and that incorporating this uncertainty leads to a situation where only a handful of schools' future performances can be separated from both the overall mean and from one another with an acceptable degree of precision. This suggests that school league tables, including value-added tables, have very little to offer as guides to school choice.     

Central Limit Theorem approximations for the number of runs in Markov-dependent binary sequences

We consider Markov-dependent binary sequences and study various types of success runs (overlapping, non-overlapping, exact, etc.) by examining additive functionals based on state visits and transitions in an appropriate Markov chain. We establish a multivariate Central Limit Theorem for the number of these types of runs and obtain its covariance matrix by means of the recurrent potential matrix of the Markov chain. Explicit expressions for the covariance matrix are given in the Bernoulli and a simple Markov-dependent case by expressing the recurrent potential matrix in terms of the stationary distribution and the mean transition times in the chain. We also obtain a multivariate Central Limit Theorem for the joint number of non-overlapping runs of various sizes and give its covariance matrix in explicit form for Markov dependent trials.     

Characterizations of Student's t-distribution via regressions of order statistics

Utilizing regression properties of order statistics, we characterize a family of distributions introduced by Akhundov et al. [New characterizations by properties of midrange and related statistics, Commun. Stat. Theory Methods 33(12) (2004), pp. 3133–3143], which includes the t-distribution with two degrees of freedom as one of its members. Then we extend this characterization result to t-distribution with more than two degrees of freedom.     

Estimable functions in the nonlinear model

When the method of least squares is used to estimate the parameters in a general model and the generated system of normal equations is linearly dependent, the estimate of the vector of parameters which satisfies the criterion is not unique. However, there exist certain functions of the estimated vector of parameters which are invariant to the least squares solution obtained from the normal equations. We define those invariant functions to be estimable, and present a technique to determine the functions of the parameters which are estimable for the general model. The method results in solving either a linear first order partial differential equation or a system of linear first order partial differential equations corresponding, respectively, to a single or multiple dependency between columns of the Jacobian matrix of the mean of the model. The usual results concerning estimability for linear models are a special case of the general results developed.