Statistics for Spatial Data.

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

Convergence of Probability Measures

The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be equivalent to (or well approximated by) GPs. Putting these two together, we get effective solutions for the practical problems. The basic approach in GP modeling is to attempt to express a practical problem, such as an engineering analysis or design problem, in GP format. In the best case, this formulation is exact; when this is not possible, we settle for an approximate formulation. This tutorial paper collects together in one place the basic background material needed to do GP modeling. We start with the basic definitions and facts, and some methods used to transform problems into GP format. We show how to recognize functions and problems compatible with GP, and how to approximate functions or data in a form compatible with GP (when this is possible). We give some simple and representative examples, and also describe some common extensions of GP, along with methods for solving (or approximately solving) them.

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A tutorial on geometric programming

The Bootstrap and Edgeworth Expansion

If X1 and X2 are random variables with distribution functions F1 and F2, then X1 is said to be stochastically larger than X2 if F1 ≤F2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F1, F2, … , Fk,k ≥ 2, and we assume that F1 ≤ F2 ≤ ˙˙˙ ≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If and are the empirical and our restricted estimators of Fi, then we show that, asymptotically, for all x and all u > 0, with strict inequality in some cases. This clearly show...

Inferences Under a Stochastic Ordering Constraint: The k-Sample Case

Restricted multinomial maximum likelihood estimation based upon Fenchel duality

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely pij = Pji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing Ht,0 against H1 - H0 and H1 against H2 - H 1 when H0:k × i=1 pixji = 0, j = 1,…, s, H1:k × i=1 pixji × 0, j = 1,…, s, and does not impose any restrictions on p. The xji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.



Il y a plusieurs situations dans l'analyse de categories de donnees ou l'on veut tester des hypotheses impliquant un ensemble de contraintes ayant la forme d'inegalites lineaires placees sur les cellules de probabilites. Par exemple il peut ětre interessant de tester la symetrie de tables de contingence de dimension k × k contre des alternatives avec assymetries. Dans ce cas, l'hypothese nulle impose un ensemble d'egalites sur les cellules de probabilites (ou plus precisement pij = pji, ×i > j) alors que les hypotheses alternatives specifies certaines inegalites directionnelles. Une autre application importante (Robertson, Wright and Dykstra, 1988) est de tester pour ou contre l'existence d'un arrangement stochastique des marginales d'une table de contingence k × k lorsque les variables sont de type ordinal et independantes. Ici nous etendons des resultats existants sur les ratio de vraisemblance, pour couvrir des situations plus generales. Pour ětre plus specifique, nous testons H0 contre H1 - H0 et H1 contre H2 - H1 ou H0:k × i=1 pixji = 0, j = 1,…, s, H1:k × i=1 pixji × 0, j = 1,…, s, et H2 n'impose pas de restrictions sur P. Les sont connus constants et s × k - 1. Nous montrons que les distributions asymptotiques des tests de ratio de vraisemblance sont de type Chi-deux, et donnent des expressions pour les valeurs ponderantes.

Testing for and against a set of linear inequality constraints in a multinomial setting

Empirical likelihood ratio test for or against a set of inequality constraints

This paper develops an empirical likelihood approach to testing for the presence of stochastic ordering among univariate distributions based on independent random samples from each distribution. The proposed test statistic is formed by integrating a localized empirical likelihood statistic with respect to the empirical distribution of the pooled sample. The asymptotic null distribution of this test statistic is found to have a simple distribution-free representation in terms of standard Brownian bridge processes. The approach is used to compare the lengths of rule of Roman Emperors over various historical periods, including the “decline and fall” phase of the empire. In a simulation study, the power of the proposed test is found to improve substantially upon that of a competing test due to El Barmi and Mukerjee.

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Hammou El Barmi

Papers

Inferences Under a Stochastic Ordering Constraint: The k-Sample Case

Restricted multinomial maximum likelihood estimation based upon Fenchel duality

Testing for and against a set of linear inequality constraints in a multinomial setting

Empirical likelihood ratio test for or against a set of inequality constraints

Empirical likelihood-based tests for stochastic ordering