Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This article offers a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function (EDF). Five of the leading statistics are examined—those often labelled D, W 2, V, U 2, A 2—and three important situations: where the hypothesized distribution F(x) is completely specified and where F(x) represents the normal or exponential distribution with one or more parameters to be estimated from the data. EDF statistics are easily calculated, and the tests require only one line of significance points for each situation. They are also shown to be competitive in terms of power.

https://www.jstor.org/stable/pdfplus/2286009.pdf

EDF Statistics for Goodness of Fit and Some Comparisons

Publisher Summary This chapter reviews the supply chain coordination with contracts. Numerous supply chain models are discussed. In each model, the supply chain optimal actions are identified. The chapter extends the newsvendor model by allowing the retailer to choose the retail price in addition to the stocking quantity. Coordination is more complex in this setting because the incentives provided to align one action might cause distortions with the other action. The newsvendor model is also extended by allowing the retailer to exert costly effort to increase demand. Coordination is challenging because the retailer's effort is noncontractible—that is, the firms cannot write contracts based on the effort chosen. The chapter also discusses an infinite horizon stochastic demand model in which the retailer receives replenishments from a supplier after a constant lead time. Coordination requires that the retailer chooses a large basestock level.

Supply Chain Coordination with Contracts

Causal ambiguity inherent in the creation of productive processes is modeled by attaching an irreducible ex ante uncertainty to the level of firm efficiency that is achieved by sequential entrants. Without recourse to scale economies or market power, the model generates equilibria in which there are stable interfirm differences in profitability, an above-normal industry rate of return, and a lack of entry even when firms are atomistic price-takers. The free-entry equilibrium for rational noncollusive firms is characterized for atomistic firms and for firms of fixed size, and some analytic results are obtained for the more realistic case in which firms have an arbitrary cost function. Numerical results for the associations implied between concentration, industry profitability, fixed entry costs, and the dispersion of firm profitabilities are obtained for selected cases.

Uncertain Imitability: An Analysis of Interfirm Differences in Efficiency under Competition

This paper proposes a new methodology for multidimensional poverty measurement consisting of an identification method ρk that extends the traditional intersection and union approaches, and a class of poverty measures Mα. Our identification step employs two forms of cutoff: one within each dimension to determine whether a person is deprived in that dimension, and a second across dimensions that identifies the poor by ‘counting’ the dimensions in which a person is deprived. The aggregation step employs the FGT measures, appropriately adjusted to account for multidimensionality. The axioms are presented as joint restrictions on identification and the measures, and the methodology satisfies a range of desirable properties including decomposability. The identification method is particularly well suited for use with ordinal data, as is the first of our measures, the adjusted headcount ratio. We present some dominance results and an interpretation of the adjusted headcount ratio as a measure of unfreedom. Examples from the US and Indonesia illustrate our methodology.

Counting and Multidimensional Poverty Measurement

Statistical Theory of Reliability and Life Testing Probability Models

: This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.

Statistical Theory of Reliability and Life Testing: Probability Models

Mathematical theory of reliability

Random variables, $X_1, \cdots, X_k$ are said to be negatively associated (NA) if for every pair of disjoint subsets $A_1, A_2$ of $\{1, 2, \cdots, k\}, \operatorname{Cov}\lbrack f(X_1, i \in A_1), g(X_j, j \in A_2) \rbrack \leq 0$, for all nondecreasing functions $f, g$. The basic properties of negative association are derived. Especially useful is the property that nondecreasing functions of mutually exclusive subsets of NA random variables are NA. This property is shown not to hold for several other types of negative dependence recently proposed. One consequence is the inequality $P(X_i \leq x_i, i = 1, \cdots, k) \leq \prod^k_1P(X_i \leq x_i)$ for NA random variables $X_1, \cdots, X_k$, and the dual inequality resulting from reversing the inequalities inside the square brackets. In another application it is shown that negatively correlated normal random variables are NA. Other NA distributions are the (a) multinomial, (b) convolution of unlike multinomials, (c) multivariate hypergeometric, (d) Dirichlet, and (e) Dirichlet compound multinomial. Negative association is shown to arise in situations where the probability measure is permutation invariant. Applications of this are considered for sampling without replacement as well as for certain multiple ranking and selection procedures. In a somewhat striking example, NA and positive association representing quite strong opposing types of dependence, are shown to exist side by side in models of categorical data analysis.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176346079

Negative Association of Random Variables with Applications

Convex Functions. Jensen's and Jensen-Steffensen's Inequalities. Reversals, Refinements, and Converses of Jensen's and Jensen-Steffensen's Inequalities. Applications of Jensen's Inequality to Means and H*adolder's Inequalities. Hermite-Hadamard's and Jensen-Petrovi*aac's Inequalities. Popoviciu's, Burkill's, and Steffensen's Inequalities. *ajCeby*ajsev-Gr*aduss', Favard's, Berwald's, Gauss-Winckler's, and Related Inequalities. Hardy's, Hilbert's, Opial's, Young's, Nanson's, and Related Inequalities. General Linear Inequalities for Convex Sequences and Functions. Orderings and Convexity-Preserving Transformations. Convex Functions and Geometric Inequalities. Convexity, Majorization, and Schur-Convexity. Convexity and Log-Concavity Related Moment and Probability Inequalities. Muirhead's Theorem and Related Inequalities. Arrangement Ordering. Applications of Arrangement Ordering. Multivariate Arrangement Increasing Functions. References. Author Index. Subject Index.

Frank Proschan

Papers

Statistical Theory of Reliability and Life Testing Probability Models

Statistical Theory of Reliability and Life Testing: Probability Models

Mathematical theory of reliability

Negative Association of Random Variables with Applications

Convex Functions, Partial Orderings, and Statistical Applications