The total bootstrap median: a robust and efficient estimator of location and scale for small samples

We propose the total bootstrap median (TBM) as a robust and efficient estimator of location and scale for small samples. We demonstrate its performance by estimating the mean and variance of a variety of distributions. We also show that, if the underlying distribution is unknown and there is either no contamination or low to moderate contamination, the TBM provides a better estimate of the mean, in mean square terms, than the sample mean or the sample median. In addition, the TBM is a better estimator of the variance of the underlying distribution than the sample variance or the square of the bias-corrected median absolute deviation from the median estimator. We also show that the TBM is an explicit L-estimator, which allows a direct study of its properties.     

Kernel estimations for multivariate density functional with bootstrap

In this article the bootstrap method is discussed for the kernel estimation of the multivariate density function. We have considered sample mean functional and constructed its consistency and asymptotic normality by bootstrap estimator. It has been shown that the bootstrap works for kernel estimates of multivariate density functional. The convergence rate with bootstrap for density has been proved. Finally, two simulations of application are given.     

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.     

A weighted spatial median for clustered data

A weighted spatial median is proposed for the multivariate one-sample location problem with clustered data. Its limiting distribution is derived under mild conditions (no moment assumptions) and it is shown to be multivariate normal. Asymptotic as well as finite sample efficiencies and breakdown properties are considered, and the theoretical results are supplied with illustrative examples. It turns out that there is a potential for meaningful gains in estimation efficiency: the weighted spatial median has superior efficiency to the unweighted spatial median particularly when the cluster sizes are widely disparate and in the presence of strong intracluster correlation. The unweighted spatial median for clustered data was considered earlier by Nevalainen et al. (Can J Statist, in press, 2007). The proposed weighted estimators provide companion estimates to the weighted affine invariant sign test proposed recently by Larocque et al. (Biometrika, in press, 2007). An affine equivariant weighted spatial median is discussed in parallel.     

Symmetric maximum kernel likelihood estimation

We introduce an estimator for the population mean based on maximizing likelihoods formed from a symmetric kernel density estimate. Due to these origins, we have dubbed the estimator the symmetric maximum kernel likelihood estimate (smkle). A speedy computational method to compute the smkle based on binning is implemented in a simulation study which shows that the smkle at an optimal bandwidth is decidedly superior in terms of efficiency to the sample mean and other measures of location for heavy-tailed symmetric distributions. An empirical rule and a computational method to estimate this optimal bandwidth are developed and used to construct bootstrap confidence intervals for the population mean. We show that the intervals have approximately nominal coverage and have significantly smaller average width than the corresponding intervals for other measures of location.     

Balanced Confidence Regions Based on Tukey's Depth and the Bootstrap

We propose and study the bootstrap confidence regions for multivariate parameters based on Tukey's depth. The bootstrap is based on the normalized or Studentized statistic formed from an independent and identically distributed random sample obtained from some unknown distribution in R ^q . The bootstrap points are deleted on the basis of Tukey's depth until the desired confidence level is reached. The proposed confidence regions are shown to be second order balanced in the context discussed by Beran. We also study the asymptotic consistency of Tukey's depth-based bootstrap confidence regions. The applicability of the method proposed is demonstrated in a simulation study.     

Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest, which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem, and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain multivariate confidence corridors for the regression function in the classical mean regression. To deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite-sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials online.     

On the multivariate spatial median for clustered data

The authors consider the multivariate one-sample location problem with clustered data from a nonparametric viewpoint. They propose the spatial median and its affine equivariant version as companion estimators to the affine invariant sign test of Larocque (2003). They extend the asymptotics of the proposed estimators to cluster dependent data and explore the limiting as well as finite-sample efficiencies for multivariate Student distributions. They demonstrate that the efficiency of the spatial median suffers less from intracluster correlation than the mean vector. They use data on the well-being of pupils in Finnish schools to illustrate their work.     

Poverty and inequality in European regions

The European Union Statistics on Income and Living Conditions (EU-SILC) is the main source of information about poverty and economic inequality in the member states of the European Union. The sample sizes of its annual national surveys are sufficient for reliable estimation at the national level but not for inferences at the sub-national level, failing to respond to a rising demand from policy-makers and local authorities. We provide a comprehensive map of median income, inequality (Gini coefficient and Lorenz curve) and poverty (poverty rates) based on the equivalised household income in the countries in which the EU-SILC is conducted. We study the distribution of income of households (pro-rated to its members), not merely its median (or mean), because we regard its dispersion and frequency of lower extremes (relative poverty) as important characteristics. The estimation for the regions with small sample sizes is improved by the small-area methods. The uncertainty of complex nonlinear statistics is assessed by bootstrap. Household-level sampling weights are taken into account in both the estimates and the associated bootstrap standard errors.     

On an adaptive transformation–retransformation estimate of multivariate location

An affine equivariant estimate of multivariate location based on an adaptive transformation and retransformation approach is studied. The work is primarily motivated by earlier work on different versions of the multivariate median and their properties. We explore an issue related to efficiency and equivariance that was originally raised by Bickel and subsequently investigated by Brown and Hettmansperger. Our estimate has better asymptotic performance than the vector of co-ordinatewise medians when the variables are substantially correlated. The finite sample performance of the estimate is investigated by using Monte Carlo simulations. Some examples are presented to demonstrate the effect of the adaptive transformation–retransformation strategy in the construction of multivariate location estimates for real data.     

One and two sample confidence intervals for estimating the mean of skewed populations: an empirical comparative study

In this paper we consider and propose some confidence intervals for estimating the mean or difference of means of skewed populations. We extend the median t interval to the two sample problem. Further, we suggest using the bootstrap to find the critical points for use in the calculation of median t intervals. A simulation study has been made to compare the performance of the intervals and a real life example has been considered to illustrate the application of the methods.     

Comparison of bootstrap estimation intervals to forecast arithmetic mean and median air passenger demand

The aim of this paper is to compare passenger (pax) demand between airports based on the arithmetic mean (MPD) and the median pax demand (MePD). A three phases approach is applied. First phase, we use bootstrap procedures to estimate the distribution of the arithmetic MPD and the MePD for each block of routes distance; second phase, we use percentile, standard, bias corrected, and bias corrected accelerated methods to calculate bootstrap confidence bands for the MPD and the MePD; and third phase, we implement Monte Carlo (MC) experiments to analyse the finite sample performance of the applied bootstrap. Our results conclude that it is more meaningful to use the estimation of MePD rather than the estimation of MPD in the air transport industry. By carrying out MC experiments, we demonstrate that the bootstrap methods produce coverages close to the nominal for the MPD and the MePD.     

A nonparametric approach to detecting the difference of survival medians

Some nonparametric methods have been proposed to compare survival medians. Most of them are based on the asymptotic null distribution to estimate the p-value. However, for small to moderate sample sizes, those tests may have inflated Type I error rate, which makes their application limited. In this article, we proposed a new nonparametric test that uses bootstrap to estimate the sample mean and variance of the median. Through comprehensive simulation, we show that the proposed approach can control Type I error rates well. A real data application is used to illustrate the use of the new test.     

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.     

Double bootstrap estimation of variance under systematic sampling with probability proportional to size

Variance estimation under systematic sampling with probability proportional to size is known to be a difficult problem. We attempt to tackle this problem by the bootstrap resampling method. It is shown that the usual way to bootstrap fails to give satisfactory variance estimates. As a remedy, we propose a double bootstrap method which is based on certain working models and involves two levels of resampling. Unlike existing methods which deal exclusively with the Horvitz–Thompson estimator, the double bootstrap method can be used to estimate the variance of any statistic. We illustrate this within the context of both mean and median estimation. Empirical results based on five natural populations are encouraging.     

空间经济计量滞后模型Bootstrap Moran检验功效的模拟分析

 当误差项不服从独立同分布时,利用Moran’s I统计量的渐近检验,无法有效判断空间经济计量滞后模型2SLS估计残差间存在空间关系与否。本文采用两种基于残差的Bootstrap方法,诊断空间经济计量滞后模型残差中的空间相关关系。大量Monte Carlo模拟结果显示,从功效角度看,无论误差项服从独立同分布与否,与渐近检验相比,Bootstrap Moran检验都具有更好的有限样本性质,能够更有效地进行空间相关性检验。尤其是,在样本量较小和空间衔接密度较高情况下,Bootstrap Moran检验的功效显著大于渐近检验。     

Simulated Power Curves for some Location Tests

In this paper we compare the power properties of some location tests. The most widely used such test is Student's t. Recently bootstrap-based tests have received much attention in the literature. A bootstrap version of the t-test will be included in our comparison. Finally, the nonparametric tests based on the idea of permuting the signs will be represented in our comparison. Again, we will initially concentrate on a version of that test based on the mean. The permutation tests predate the bootstrap by about fourty years. Theoretical results of Pitman (1937) and Bickel & Freedman (1981) show that these three methods are asymptotically equivalent if the underlying distribution is symmetric and has finite second moment. In the modern literature, the use of the nonparametric techniques is advocated on the grounds that the size of the test would be either exact, or more nearly exact. In this paper we report on a simulation study that compares the power curves and we show that it is not necessary to use resampling tests with a statistic based on the mean of the sample.     

AFFINE‐EQUIVARIANT SPATIAL MEDIAN AND ITS USE IN THE MULTIVARIATE MULTI‐SAMPLE LOCATION PROBLEM

The classical spatial median is not affine‐equivariant, which often turns out to be an unfavourable property. In this paper, the asymptotic properties of an affine‐equivariant modification of the spatial median are investigated. It is shown that under some weak regularity conditions, the modified spatial median computed by means of the sample norming matrix is asymptotically equivalent to the one computed by means of the population norming matrix, which yields its asymptotic normality. A consistent estimate of the asymptotic covariance matrix of the modified spatial median is also presented. These results are implemented in a scheme, where the sample norm is determined by means of the sample Dümbgen scatter matrix. The results are utilized in the construction of affine‐invariant test statistics for testing the multi‐sample hypothesis of equality of location parameters. The performance of the proposed tests is demonstrated through a simulation study.     

Estimation of the variance of an adaptive estimator by bootstrapping

In this paper we propose a bootstrap-based method to estimate the standard error of adaptive estimators. We apply it in the standard problem of location estimation discussed in Randies and Hogg (1973) and in Hogg and Lenth (1984). Our adaptive estimator is based on a choice between the mean the 35% trimmed mean and the median. Finally, we carry out a simulation study to see how well the proposed method performs in small and moderate sample sizes.