Efficient weighted bootstraps for the mean

The weighted bootstrap due to Mason and Newton (1992. Ann. Statist. 20, 1611–1624.) is applied to Studentized statistics in view of deriving efficient confidence intervals for the mean. First, we give conditions on the moments of the weights to ensure that the weighted bootstrap approximation leads to uniformly correct two-sided confidence intervals up to the rate O(n^−3/2). Then, we discuss the practical choice of the random weights in order to construct one-sided confidence intervals accurate up to O(n^−3/2) and two-sided confidence intervals up to higher orders. Simulations are given to illustrate the practical efficiency of our approach.     

On the accuracy of the bootstrap approximation of the joint distribution of sample quantiles

It is proved that the accuracy of the bootstrap approximation of the joint distribution of sample quantiles lies between O(n^?1/4) and O(n^?1/4 a_n), where (log(n))1/2=O(a_n). As an application, we investigated confidence intervals based on the bootstrap.     

Jackknifing two-sample statistics

In this paper, a new simple method for jackknifing two-sample statistics is proposed. The method is based on a two-step procedure. In the first step, the point estimator is calculated by leaving one X (or Y) out at a time. At the second step, the point estimator obtained in the first step is further jackknifed, leaving one Y (or X) out at a time, resulting in a simple formula for the proposed point estimator. It is shown that by using the two-step procedure, the bias of the point estimator is reduced in terms of asymptotic order, from O(n⁻¹) up to O(n⁻²), under certain regularity conditions. This conclusion is also confirmed empirically in terms of finite sample numerical examples via a small-scale simulation study. We also discuss the idea of asymptotic bias to obtain parallel results without imposing some conditions that may be difficult to check or too restrictive in practice.     

Bias-corrected estimators for dispersion models with dispersion covariates

In this paper we discuss bias-corrected estimators for the regression and the dispersion parameters in an extended class of dispersion models (Jørgensen, 1997b). This class extends the regular dispersion models by letting the dispersion parameter vary throughout the observations, and contains the dispersion models as particular case. General formulae for the O(n⁻¹) bias are obtained explicitly in dispersion models with dispersion covariates, which generalize previous results obtained by Botter and Cordeiro (1998), Cordeiro and McCullagh (1991), Cordeiro and Vasconcellos (1999), and Paula (1992). The practical use of the formulae is that we can derive closed-form expressions for the O(n⁻¹) biases of the maximum likelihood estimators of the regression and dispersion parameters when the information matrix has a closed-form. Various expressions for the O(n⁻¹) biases are given for special models. The formulae have advantages for numerical purposes because they require only a supplementary weighted linear regression. We also compare these bias-corrected estimators with two different estimators which are also bias-free to order O(n⁻¹) that are based on bootstrap methods. These estimators are compared by simulation.     

2n−m designs with resolution III or IV containing clear two-factor interactions

For a fixed number of runs, not all 2^n−m designs with resolution III or IV have clear two-factor interactions. Therefore, it is highly desirable to know when resolution III or IV designs can have clear two-factor interactions. In this paper, we provide a unified geometrical study of this problem and give a complete classification of the existence of clear two-factor interactions in regular 2^n−m designs with resolution III or IV and reveal the structures of these designs.     

Control variates and importance sampling for efficient bootstrap simulations

Importance sampling and control variates have been used as variance reduction techniques for estimating bootstrap tail quantiles and moments, respectively. We adapt each method to apply to both quantiles and moments, and combine the methods to obtain variance reductions by factors from 4 to 30 in simulation examples.We use two innovations in control variates—interpreting control variates as a re-weighting method, and the implementation of control variates using the saddlepoint; the combination requires only the linear saddlepoint but applies to general statistics, and produces estimates with accuracy of order n ^-1/2 B ^-1, where n is the sample size and B is the bootstrap sample size.We discuss two modifications to classical importance sampling—a weighted average estimate and a mixture design distribution. These modifications make importance sampling robust and allow moments to be estimated from the same bootstrap simulation used to estimate quantiles.     

Testing regions with nonsmooth boundaries via multiscale bootstrap

A new class of approximately unbiased tests based on bootstrap probabilities is obtained for the multivariate normal model with unknown expectation parameter vector. The null hypothesis is represented as an arbitrary-shaped region with possibly nonsmooth boundary surfaces such as cones, which appear in, for example, multiple comparisons and hierarchical clustering. The size n^′$n^{\prime }$ of bootstrap samples is intentionally altered from the size n of the data. A scaling-law of the bootstrap probability leads to our bias corrected p  -values which are calculated by extrapolating the bootstrap probability back to n^′=-n$n^{\prime }=-n$. The new method approximates the bootstrap iteration applied to the bootstrap probability.     

On empirical Bayes simultaneous selection procedures for comparing normal populations with a standard

In this paper, we derive statistical selection procedures to partition k normal populations into ‘good’ or ‘bad’ ones, respectively, using the nonparametric empirical Bayes approach. The relative regret risk of a selection procedure is used as a measure of its performance. We establish the asymptotic optimality of the proposed empirical Bayes selection procedures and investigate the associated rates of convergence. Under a very mild condition, the proposed empirical Bayes selection procedures are shown to have rates of convergence of order close to O(k^−1/2) where k is the number of populations involved in the selection problem. With further strong assumptions, the empirical Bayes selection procedures have rates of convergence of order O(k^{−α(r−1)/(2r+1)}), where 1<α<2 and r is an integer greater than 2.     

Asymptotic expansions of the distributions of the chi-square statistic based on the asymptotically distribution-free theory in covariance structures

An asymptotic expansion of the null distribution of the chi-square statistic based on the asymptotically distribution-free theory for general covariance structures is derived under non-normality. The added higher-order term in the approximate density is given by a weighted sum of those of the chi-square distributed variables with different degrees of freedom. A formula for the corresponding Bartlett correction is also shown without using the above asymptotic expansion. Under a fixed alternative hypothesis, the Edgeworth expansion of the distribution of the standardized chi-square statistic is given up to order O(1/n). From the intermediate results of the asymptotic expansions for the chi-square statistics, asymptotic expansions of the joint distributions of the parameter estimators both under the null and fixed alternative hypotheses are derived up to order O(1/n).     

Weighted A-optimality for fractional 2m factorial designs of resolution V

A weighted A-optimality (WA-optimality) criterion is discussed for selecting a fractional 2^m factorial design of resolution V. A WA-optimality criterion having one weight may be considered for designs. It is shown that designs derived from orthogonal arrays are WA-optimal for any weight. From a WA-optimal design, a procedure for finding WA-optimal designs for various weights is given. WA-optimal balanced designs are presented for 4 ⩽ m ⩽ 7 and for the values of n assemblies in certain ranges. It is pointed out that designs for m = 7 and for n = 41, 42 given in Chopra and Srivastava (1973a) or in the corrected paper by Chopra et al. (1986), are not A-optimal.     

New bootstrap confidence intervals for means of positively skewed distributions

ABSTRACT

In this paper, we consider the problem of constructing non parametric confidence intervals for the mean of a positively skewed distribution. We suggest calibrated, smoothed bootstrap upper and lower percentile confidence intervals. For the theoretical properties, we show that the proposed one-sided confidence intervals have coverage probability α + O(n^{? 3/2}). This is an improvement upon the traditional bootstrap confidence intervals in terms of coverage probability. A version smoothed approach is also considered for constructing a two-sided confidence interval and its theoretical properties are also studied. A simulation study is performed to illustrate the performance of our confidence interval methods. We then apply the methods to a real data set.     

A new confidence interval for the difference between two binomial proportions of paired data

Motivated by a study on comparing sensitivities and specificities of two diagnostic tests in a paired design when the sample size is small, we first derived an Edgeworth expansion for the studentized difference between two binomial proportions of paired data. The Edgeworth expansion can help us understand why the usual Wald interval for the difference has poor coverage performance in the small sample size. Based on the Edgeworth expansion, we then derived a transformation based confidence interval for the difference. The new interval removes the skewness in the Edgeworth expansion; the new interval is easy to compute, and its coverage probability converges to the nominal level at a rate of O(n^−1/2). Numerical results indicate that the new interval has the average coverage probability that is very close to the nominal level on average even for sample sizes as small as 10. Numerical results also indicate this new interval has better average coverage accuracy than the best existing intervals in finite sample sizes.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

Convergence rates of empirical Bayes estimation in exponential family

The convergence rates of empirical Bayes estimation in the exponential family are studied in this paper. We first develop an approach for obtaining the lower bound of empirical Bayes estimators. As an application of the approach, we demonstrate that O(n⁻¹) is the lower bound rate for priors with bounded compact support. Second, we construct an empirical Bayes estimator using kernel sequence method and show that it has a rate of convergence of $\text{O}\text{(n}{}^{\mathrm{-1}}\text{(}\text{ln}\text{n)}{}^8\text{)}$. This upper bound rate is much faster compared to the earlier results published in the literature under the same assumption.     

A recentred bootstrap procedure for constructing uniformly correct confidence sets under smooth function models

It has been found, under a smooth function model setting, that the n out of n bootstrap is inconsistent at stationary points of the smooth function, but that the m out of n bootstrap is consistent, provided that a correct convergence rate is specified of the plug-in smooth function estimator. By considering a more general moving-parameter framework, we show that neither of the above bootstrap methods is consistent uniformly over neighbourhoods of stationary points, so that anomalies often arise of coverages of bootstrap sets over certain subsets of parameter values. We propose a recentred bootstrap procedure for constructing confidence sets with uniformly correct coverages over compact sets containing stationary points. A weighted bootstrap procedure is also proposed as an alternative under more general circumstances. Unlike the m out of n bootstrap, both procedures do not require knowledge of the convergence rate of the smooth function estimator. Empirical performance of our procedures is illustrated with numerical examples.     

On histograms for linear processes

Sharp rates of convergence of histogram estimates of the marginal density of a linear process are obtained. Histograms can achieve optimal rates of convergence (n⁻¹ log n)^1·3 under general conditions. The assumptions involved are easily verifiable. Histograms appear to be very good estimators from the point of view of uniform convergence.     

Minimum distance lack-of-fit tests for fixed design

This paper discusses a class of tests of lack-of-fit of a parametric regression model when design is non-random and uniform on [0,1]. These tests are based on certain minimized distances between a nonparametric regression function estimator and the parametric model being fitted. We investigate asymptotic null distributions of the proposed tests, their consistency and asymptotic power against a large class of fixed and sequences of local nonparametric alternatives, respectively. The best fitted parameter estimate is seen to be n^1/2-consistent and asymptotically normal. A crucial result needed for proving these results is a central limit lemma for weighted degenerate U statistics where the weights are arrays of some non-random real numbers. This result is of an independent interest and an extension of a result of Hall for non-weighted degenerate U statistics.     

Asymptotic mean squared error of constrained James–Stein estimators

In this paper, we consider the asymptotic expansion of the MSE of constrained James–Stein estimators. We provide an estimator of the MSE which is asymptotically valid upto O(m⁻¹). A simulation study is undertaken to evaluate the performance of these estimators.     

Higher-order Bartlett-type adjustment

This paper deals with Bartlett-type adjustment which makes all the terms up to order n^−k in the asymptotic expansion vanish, where k is an integer k ⩾ 1 and n depends on the sample size. Extending Cordeiro and Ferrari (1991, Biometrika, 78, 573–582) for the case of k = 1, we derive a general formula of the kth-order Bartlett-type adjustment for the test statistic whose kth-order asymptotic expansion of the distribution is given by a finite linear combination of chi-squared distribution with suitable degrees of freedom. Two examples of the second-order Bartlett-type adjustment are given. We also elucidate the connection between Bartlett-type adjustment and Cornish-Fisher expansion.