Murray D. Burke

A correction to and improvement of «strong approximations of some biometric estimates under random censorship»

SummaryIn the proof of the main result of the original paper there is an error. Instead of repairing that proof to get just the original result, at the critical spot we improve the proof and obtain a much better result. In particular, we approximate the product-limit and empirical cumulative hazard processes by suitable copies of the corresponding limiting Gaussian processes with rates of approximation that on appropriate fixed half lines reduce to the rates of Komlós, Major and Tusnády for the uncensored empirical process.

Multivariate tests-of-fit and uniform confidence bands using a weighted bootstrap

Many tests of fit procedures use the empirical distribution function (e.d.f.) of the data and have limiting distribution dependent on the datas underlying distribution or family of distribution functions. This paper uses a weighted bootstrap method based on independent random variables instead of sampling from the uniform. A proof of convergence of the weighted bootstrap is given using strong martingales. This approach is applied to the problem of obtaining uniform confidence bands for the distribution function of multivariate data. The multivariate two-sample problem, testing whether two independent random samples come from the same multivariate distribution function, is also discussed.

Approximations for a multivariate hybrid process with applications to change-point detection

We obtain probability inequalities and almost sure rates for the approximations of the hybrids of empirical and partial sums processes in the multivariate case. Applications to weighted bootstrap empirical processes as well as to change-point detection tests for general nonparametric regression models are discussed.

Approximations of some hazard rate estimators in a competing risks model

A general model involving k competing risks is studied and the hazard rates of these risks are simultaneously estimated. The estimators are strongly approximated by Gaussian processes and the limiting distribution of certain statistics are obtained.

On goodness-of-fit and the bootstrap

The empirical process, where unknown parameters of the underlying distribution function are estimated by bootstrap methods, is considered. It is approximated by a sequence of Gaussian process. In the maximum likelihood estimation case it converges to a Brownian Bridge. The asymptotic distribution of Cramer-von Mises, Anderson-Darling and Kolmogorov-Smirnov test statistics are derived.

On the multivariate two-sample problem using strong approximations of the EDF

The recently developed strong approximation methods are discussed and applied to the problem of testing whether two independent multivariate samples come from the same population and whether the components of the observations are independent. The usual Cramer-von Mises statistic, as well as one based on the difference between the sum of the two multivariate EDFs and twice the product of the marginal EDFs of one, are studied. A fairly sensitive integral statistic is also discussed. Consistency and some asymptotic power properties are explored. Emphasis is placed on explication of the strong approximation methodology.

A Gaussian Bootstrap Approach to Estimation and Tests

Publisher Summary Many tests of fit procedures use the empirical distribution function (edf) of the data. The resulting process based on edfs may have limiting distribution dependent on the datas underlying family of distribution functions. The bootstrap method is a widely used technique to approximate these limiting distributions. This paper proposes a smooth Gaussian bootstrap method based on independent standard normal random variables instead of sampling from the uniform. This approach is applied to the problem of obtaining uniform confidence regions for influence functions based on L-statistics. A small simulation study comparing this Gaussian bootstrap and the usual bootstrap is performed. This chapter proposes a Gaussian (or smooth) bootstrap method based on independent standard normal random variables instead of sampling from the uniform. This Gaussian bootstrap has two advantages. Each original observation is included once. Secondly, the Gaussian bootstrapped empirical process is already Gaussian, given the original data. The approach can be used to obtain uniform confidence regions for the distribution function or some other object of interest of a single sample.

Goodness-of-fit tests for the Cox model via bootstrap method

Global goodness-of-fit tests for the Cox model, in which no partitions of the time and covariate spaces into cells are needed, are established. Test statistics are based upon the cumulative hazard process under the censored survival data. Although this process converges to a Gaussian process almost surely, a problem arises from the fact that the asymptotic covariance structure depends upon the underlying distribution which is unspecified. Therefore, a bootstrap method suggested by Efron (1981) is employed. It can be shown that the bootstrapped cumulative hazard process converges weakly to the same Gaussian process. As an illustration, the proposed tests are applied to analyze the Stanford heart transplant data. Finally, a discussion for dealing with discrete covariate is given.

The bootstrapped maximum likelihood estimator with an application

The existence, strong consistency and asymptotic normality of bootstrapped maximum likelihood estimators are shown. A distribution-free goodness-of-fit test involving sample quantiles is given as an application.

AN ALMOST-SURE APPROXIMATION OF A MULTIVARIATE PRODUCT-LIMIT ESTIMATOR UNDER RANDOM CENSORSHIP

An almost-sure approximation by a sequence of Gaussian processes of a nailtivariate product-limit eBtimator is obtained. The estimator is based on a sequence of i.i.d. random vectora subject to right censorship by another sequence of random vectors. Properties of the limiting Gaussian process are discussed.