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

We consider the competing risks problem with two risks and when the data are grouped or discrete. We rstly obtain nonparametric maximum likelihood estimates of the sub-survival functions corresponding to the two risks under the restriction that they are uniformly ordered and then use them to derive the likelihood ratio statistic for testing the null hypothesis of equality of the two sub-survival functions against ordered alternatives. The asymptotic null distribution of the test statistic is seen to be of the chi-bar square ( 2 ) type. A simulation

/pdf/inference-for-sub-survival-functions-under-order-59d4fn93t1.pdf

Inference for Sub-survival Functions Under Order Restrictions

Nous envisageons une inference sur la vraisemblance de profil basee sur la distribution multinomiale pour evaluer la precision d'un test diagnostic. La methode s'applique a des donnees ordinales quand la precision est evaluee au moyen de l'aire sous la courbe caracteristique du receveur (courbe ROC). Des resultats de simulation suggerent que les intervalles de confiance derives ont vies probabilites de couverture acceptables, meme lorsque les tailles d'echantillons sont petites et que le test diagnostic a une haute precision. Les methodes s'etendent a des contextes stratifies, et a des situations ou les evaluations sont correlees. Nous illustrons les methodes sur des donnees provenant d'un essai clinique sur la detection du cancer de l'ovaire.

Profile-likelihood inference for highly accurate diagnostic tests.

Restricted one way analysis of variance using the empirical likelihood ratio test

This paper develops an empirical likelihood approach to testing for the presence of uniform stochastic ordering (or hazard rate 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 motion. The approach is extended to the case of right-censored survival data via multiple imputation. Two applications are discussed: (1) uncensored survival time data of mice exposed to radiation, and (2) right-censored time-to-infection data from a human HIV vaccine trial comparing a placebo group with a vaccine group.

/pdf/testing-for-uniform-stochastic-ordering-via-empirical-38auqikchv.pdf

Testing for uniform stochastic ordering via empirical likelihood

In the competing risks problem, an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of failure in the presence of other risks. In some cases there are reasons to believe that the CIFs due to various types of failure are linearly ordered. El Barmi et al. (3) studied the estimation and inference procedures under this ordering when there are only two causes of failure. In this paper we extend the results to the case of k CIFs, where k ≥ 3. Although the analyses are more challenging, we show that most of the results in the 2-sample case carry over to this k-sample case.

/pdf/restricted-estimation-of-the-cumulative-incidence-functions-52vbboziri.pdf

Hammou El Barmi

Papers

Inference for Sub-survival Functions Under Order Restrictions

Profile-likelihood inference for highly accurate diagnostic tests.

Restricted one way analysis of variance using the empirical likelihood ratio test

Testing for uniform stochastic ordering via empirical likelihood

Restricted estimation of the cumulative incidence functions corresponding to competing risks