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Univariate and multivariate total positivity, generalized convexity and related inequalities. (English) Zbl 0534.90052
Generalized concavity in optimization and economics, Proc. NATO Adv. Study Inst., Vancouver/Can. 1980, 703-718 (1981).
Summary: [For the entire collection see Zbl 0534.00022.] Various univariate and multivariate concepts of total positivity are set forth. Multivariate total positivity and generalized convexity play an underlying role in determining measures of association and orderings among vector random variables and classes of probability inequalities on coverage of regions. In the analysis of reliability systems multidimensional analogs of the concept of increasing failure rate and bounds on system reliability are related to multivariate total positivity. A host of statistical applications in areas such as the analysis of symmetric sampling schemes from finite populations, multivariate ranking and selection procedures, multivariate matching problems, slippage problems, and various properties of power functions fall in this domain. In the analysis of sampling plans comparisons of various functionals such as variances of estimates under different sampling schemes lead to inequalities which are related to total positivity. Probabilistic inequalities arise in ranking and selection problems in comparisons of probabilities of correct selection (or ranking) for different procedures, or in computing bounds for such probabilities. Concepts of positive dependence related to total positivity are relevant to problems of matching of observations, and are widely used in reliability theory where system-components are assumed to have positively dependendent life-times. This paper surveys some basic ideas and examples, in order to provide the background and motivation to more recent results and problems related to multivariate concepts of total positivity.

90B25Reliability, availability, maintenance, inspection, etc. (optimization)
26B25Convexity and generalizations (several real variables)
90-02Research monographs (optimization)
62N05Reliability and life testing (survival analysis)