Asymptotic equivalence of the harrell-davis median estimator and the sample median

This paper presents an asymptotic equivalence result with a sharp rate of convergence forthe sample median and the Harrell-Davis median estimator. The consequences of this result are discussed.     

A comparison of methods for studentizing the sample median

Various methods for “Studentizing” the sample median are com-pared on the basis of a Monte Carlo study. Several of the methods do rather poorly while two, the bootstrap and the standardized length of a distribution free confidence interval, behave accept-ably acrors a wide range of sample sizes and several distributions of varying tail length. These two methods seem to agree closely with the distribution free confidence intervals and moreover, un-like these intervals, the methods can be extended to a method of accurate inference for λ₁ regreasion.     

Bootstrapping analogs of the one way MANOVA test

Abstract

Analogs of the classical one way MANOVA model have recently been suggested that do not assume that population covariance matrices are equal or that the error vector distribution is known. These tests are based on the sample mean and sample covariance matrix corresponding to each of the p populations. We show how to extend these tests using other measures of location such as the trimmed mean or coordinatewise median. These new bootstrap tests can have some outlier resistance, and can perform better than the tests based on the sample mean if the error vector distribution is heavy tailed.     

Estimating the variance of the sample median

The small-sample bias and root mean squared error of several distribution-free estimators of the variance of the sample median are examined. A new estimator is proposed that is easy to compute and tends to have the smallest bias and root mean squared error.     

Bootstrapping probability-proportional-to-size samples via calibrated empirical population

A collection of six novel bootstrap algorithms, applied to probability-proportional-to-size samples, is explored for variance estimation, confidence interval and p-value production. Developed according to bootstrap fundamentals such as the mimicking principle and the plug-in rule, these algorithms make use of an empirical bootstrap population informed by sampled units each with assigned weight. Starting from the natural choice of Horvitz–Thompson (HT)-type weights, improvements based on calibration to known population features are fostered. Focusing on the population total as the parameter to be estimated and on the distribution of the HT estimator as the target of bootstrap estimation, simulation results are presented with the twofold objective of checking practical implementation and of investigating the statistical properties of the bootstrap estimates supplied by the algorithms explored.     

Bootstrapping in least absolute value regression: an application to hypothesis testing

A Monte Carlo simulation is used to study the performance of hypothesis tests for regression coefficients when least absolute value regression methods are used. In small samples, the results of the simulation suggest that using the bootstrap method to compute standard errors will provide improved test performance     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

An approximation to the distribution of the sample variance

A simple three-moment approximation is introduced for the distribution of the sample variance. Comparisons are given with other approximations discussed by Tan and Wong (1977) and with an approximation developed very recently by Mudholkar and Trivedi (1981).     

New simulation results for the calibration and inverse median estimation problems

This paper gives the results of a new simulation study for the familiar calibration problem and the less familiar inverse median estimation problem. The latter arises when one wishes to estimate from a linear regression analysis the value of the independent variable corresponding to a specified value of the median of the dependent variable. For example, from the results of a regression analysis between stress and time to failure, one might wish to estimate the stress at which the median time to failure is 10,000 hours. In the study, the mean square error, Pitman closeness, and probability of overestimation are compared for both the calibration problem and the inverse median estimation problem for (1) the classical estimator, (2) the inverse estimator, and (3) a modified version of an estimator proposed by Naszodi (1978) for both a small sample and a moderately large sample situation.     

Bootstrapping moving average models

Summary In recent years, the bootstrap method has been extended to time series analysis where the observations are serially correlated. Contributions have focused on the autoregressive model producing alternative resampling procedures. In contrast, apart from some empirical applications, very little attention has been paid to the possibility of extending the use of the bootstrap method to pure moving average (MA) or mixed ARMA models. In this paper, we present a new bootstrap procedure which can be applied to assess the distributional properties of the moving average parameters estimates obtained by a least square approach. We discuss the methodology and the limits of its usage. Finally, the performance of the bootstrap approach is compared with that of the competing alternative given by the Monte Carlo simulation. Research partially supported by CNR and MURST.     

Bootstrapping Regression Parameters in Multivariate Survival Analysis

Bootstrap methods are proposed for estimating sampling distributions and associated statistics for regression parameters in multivariate survival data. We use an Independence Working Model (IWM) approach, fitting margins independently, to obtain consistent estimates of the parameters in the marginal models. Resampling procedures, however, are applied to an appropriate joint distribution to estimate covariance matrices, make bias corrections, and construct confidence intervals. The proposed methods allow for fixed or random explanatory variables, the latter case using extensions of existing resampling schemes (Loughin,1995), and they permit the possibility of random censoring. An application is shown for the viral positivity time data previously analyzed by Wei, Lin, and Weissfeld (1989). A simulation study of small-sample properties shows that the proposed bootstrap procedures provide substantial improvements in variance estimation over the robust variance estimator commonly used with the IWM. This revised version was published online in July 2006 with corrections to the Cover Date.     

Bootstrapping robust regression

The bootstrap principle is justified for. robust M-estimates in regression, (A short proof justifying bootstrapping the empirical process is also given.)     

Moment Consistency of the Exchangeably Weighted Bootstrap for Semiparametric M‐estimation

The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well‐known open problem. In this paper, we provide a first theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter, which immediately implies the consistency of t‐type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the L₁ maximal inequality condition required in the latter to the L_p maximal inequality condition for p≥1. The general L_p multiplier inequality developed in this paper is also of independent interest. These general conclusions hold for the bootstrap methods with exchangeable bootstrap weights, for example, non‐parametric bootstrap and Bayesian bootstrap. Our general theory is illustrated in the celebrated Cox regression model.     

A Monte Carlo comparison of three consistent bootstrap procedures

Since bootstrap samples are simple random samples with replacement from the original sample, the information content of some bootstrap samples can be very low. To avoid this fact, several variants of the classical bootstrap have been proposed. In this paper, we consider two of them: the sequential or Poisson bootstrap and the reduced bootstrap. Both of these, like the ordinary bootstrap, can yield second-order accurate distribution estimators, that is, the three bootstrap procedures are asymptotically equivalent. The question that naturally arises is which of them should be used in a practical situation, in other words, which of them should be used for finite sample sizes. To try to answer this question, we have carried out a simulation study. Although no method was found to exhibit best performance in all the considered situations, some recommendations are given.     

Bootstrapping an inhomogeneous point process

In this paper, we focus on resampling non-stationary weakly dependent point processes in two dimensions to make inference on the inhomogeneous K function ( Baddeley et al., 2000). We provide theoretical results that show a consistency result of the bootstrap estimates of the variance as the observation region and resampling blocks increase in size. We present results of a simulation study that examines the performance of nominal 95% confidence intervals for the inhomogeneous K function obtained via our bootstrap procedure. The procedure is also applied to a rainforest dataset.     

A review and re‐interpretation of a group‐sequential approach to sample size re‐estimation in two‐stage trials

In this paper, we review the adaptive design methodology of Li et al. (Biostatistics 3 :277–287) for two‐stage trials with mid‐trial sample size adjustment. We argue that it is closer in principle to a group sequential design, in spite of its obvious adaptive element. Several extensions are proposed that aim to make it even more attractive and transparent alternative to a standard (fixed sample size) trial for funding bodies to consider. These enable a cap to be put on the maximum sample size and for the trial data to be analysed using standard methods at its conclusion. The regulatory view of trials incorporating unblinded sample size re‐estimation is also discussed. © 2014 The Authors. Pharmaceutical Statistics published by John Wiley & Sons, Ltd.     

A modified bootstrap estimator for the mean of an asymmetric distribution

A modified bootstrap estimator of the population mean is proposed which is a convex combination of the sample mean and sample median, where the weights are random quantities. The estimator is shown to be strongly consistent and asymptotically normally distributed. The small- and moderate-sample-size behavior of the estimator is investigated and compared with that of the sample mean by means of Monte Carlo studies. It is found that the newly proposed estimator has much smaller mean squared errors and also yields significantly shorter confidence intervals for the population mean.