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.

/pdf/a-tutorial-on-geometric-programming-47sdznbzxw.pdf

A tutorial on geometric programming

The Bootstrap and Edgeworth Expansion

A life distribution function (DF) F is said to be star-shaped if is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where is increasing using the arg max theorem.

Estimation of a star-shaped distribution function

In this paper, we develop an empirical likelihood-based test for the presence of stochastic ordering under censoring in the k-sample case. The proposed test statistic is formed by taking the supremum of localized empirical likelihood ratio test statistics. Its asymptotic null distribution has a simple representation in terms of a standard Brownian motion process. Through simulations, we show that it outperforms in terms of power existing methods for the same problem at all the distributions that we consider. A real-life example is used to illustrate the applicability of this new test.

A test for the presence of stochastic ordering under censoring: the k-sample case

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 risk in the presence of other risks. Assume that the lifetime distributions of two populations are uniformly stochastically ordered. Since this ordering may not hold for the empiricals due to sampling variability, it is natural to estimate these distributions under this constraint. This will in turn affect the estimation of the CIFs. This article considers this estimation problem. We do not assume that the risk sets in the two populations are related, give consistent estimators of all the CIFs and study the weak convergence of the resulting processes. We also report the results of a simulation study that show that our restricted estimators outperform the unrestricted ones in terms of mean square error. A real life example is used to illustrate our theoretical results.

Hammou El Barmi

Papers

Estimation of a star-shaped distribution function

A test for the presence of stochastic ordering under censoring: the k-sample case

Estimation of cumulative incidence functions when the lifetime distributions are uniformly stochastically ordered.

A linear-quadratic distributional identity