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.     

TEACHING THE INDEPENDENCE OF X AND S2 IN APPLIED STATISTICS*

In this note, the independence of X and S² is proved without the prerequisite knowledge of differential calculus. The purpose is to compensate for the lack of a text in applied statistics that proves the independence of such statistics.     

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).     

Using age and spatial flow structures in the indirect estimation of migration streams

This article outlines a formal model-based approach for inferring interregional age-specific migration streams in settings where such data are incomplete, inadequate, or unavailable. The estimation approach relies heavily on log-linear models, using them to impose some of the regularities exhibited by past age and spatial structures or to combine and borrow information drawn from other sources. The approach is illustrated using data from the 1990 and 2000 U.S. and Mexico censuses.     

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.     

Decision settings analysis – a tool for analysis and design of human activity systems

The paper describes a methodology to be used for analysis and design of human activity systems. The methodology is based on an analysis of the decision settings whereas most other decision analysis methodologies are analysing the process. The decision concept is analysed and discussed. A distinction between programmed and programmable as well as non-programmed and non-programmable decisions is proposed. A classification of different information types for decision making is presented. A methodology based on a systemic and systematic analysis of the information requirements of an organization is proposed. This methodology also indicates organizational discrepancies and information imbalances. The methodology focuses the settings of the decisions on all levels of organizations. The methodology can be regarded as a dynamic, learning system. The author proposes further research on the individuals decision making abilities.     

Correlation in a Bayesian framework

The authors consider the correlation between two arbitrary functions of the data and a parameter when the parameter is regarded as a random variable with given prior distribution. They show how to compute such a correlation and use closed form expressions to assess the dependence between parameters and various classical or robust estimators thereof, as well as between p‐values and posterior probabilities of the null hypothesis in the one‐sided testing problem. Other applications involve the Dirichlet process and stationary Gaussian processes. Using this approach, the authors also derive a general nonparametric upper bound on Bayes risks.     

Differential equation models for statistical functions

Differential equations have been used in statistics to define functions such as probability densities. But the idea of using differential equation formulations of stochastic models has a much wider scope. The author gives several examples, including simultaneous estimation of a regression model and residual density, monotone smoothing, specification of a link function, differential equation models of data, and smoothing over complicated multidimensional domains. This paper aims to stimulate interest in this approach to functional estimation problems, rather than provide carefully worked out methods.