Local bootstrap

The local bootstrap method draws bootstrap samples from a neighborhood of each data point. Unlike the common bootstrap, it is especially useful for heteroscedastic data. This paper considers local bootstrap of y for a data set {(x _i , y _i )} with the neighborhood defined by the window in a kernel regression model. It shows that for each x, the bootstrap distribution and moments converge almost surely to the true distribution and moments of y respectively, and this convergence is also uniform in x within the data set.     

Permutation tests using least distance estimator in the multivariate regression model

This paper proposes two permutation tests based on the least distance estimator in a multivariate regression model. One is a type of t test statistic using the bootstrap method, and the other is a type of F test statistic using the sum of distances between observed and predicted values under the full and reduced models. We conducted a simulation study to compare the power of the proposed permutation tests with that of the parametric tests based on the least squares estimator for three types of hypotheses in several error distributions. The results indicate that the power of the proposed permutation tests is greater than that of the parametric tests when the error distribution is skewed like the Wishart distribution, has a heavy tail like the Cauchy distribution, or has outliers.     

The Accurate Distribution of the Kolmogorov Statistic with Applications to Bootstrap Approximation

Suppose that x₁,..., x_m are i.i.d. variables with distribution F(·). When F(·) is continuous, the accurate distribution of the Kolmogorov statistic was obtained by Zhang. In this paper, a discrete distribution F(·) which masses value 1/n at each point y_i, I = 1,..., n is considered, the obtained accurate distribution of the Kolmogorov, statistic has the same form as that of Zhang′s. By this result, the bootstrap approximation of the distribution of the Kolmogorov statistic is investigated, and the √n -asymptotic rate is obtained.     

L1 regression estimate and its bootstrap

We study the asymptotic distribution of the L ₁ regression estimator under general conditions with matrix norming and possibly non i.i.d. errors. We then introduce an appropriate bootstrap procedure to estimate the distribution of this estimator and study its asymptotic properties. It is shown that this bootstrap is consistent under suitable conditions and in other situations the bootstrap limit is a random distribution. This work was supported by J.C. Bose National Fellowship, Government of India     

Accurate confidence intervals in regression analyses of non-normal data

A linear model in which random errors are distributed independently and identically according to an arbitrary continuous distribution is assumed. Second- and third-order accurate confidence intervals for regression parameters are constructed from Charlier differential series expansions of approximately pivotal quantities around Student’s t distribution. Simulation verifies that small sample performance of the intervals surpasses that of conventional asymptotic intervals and equals or surpasses that of bootstrap percentile-t and bootstrap percentile-|t| intervals under mild to marked departure from normality.     

Bootstrap approximation of nearest neighbor regression function estimates

Let (X, Y) be a random vector in the plane and denote by m(x) = (Y|X = x) the corresponding regression function. We show that the bootstrap approximation for the distribution of a smoothed nearest neighbor estimate of m(x) is valid. Also we compare, by Monte Carlo, confidence intervals which are obtained from both the normal and the bootstrap approximation.     

A smoothed bootstrap estimator for a studentized sample quantile

If the underlying distribution functionF is smooth it is known that the convergence rate of the standard bootstrap quantile estimator can be improved fromn ^–1/4 ton ^–1/2+, for arbitrary >0, by using a smoothed bootstrap. We show that a further significant improvement of this rate is achieved by studentizing by means of a kernel density estimate. As a consequence, it turns out that the smoothed bootstrap percentile-t method produces confidence intervals with critical points being second-order correct and having smaller length than competitors based on hybrid or on backwards critical points. Moreover, the percentile-t method for constructing one-sided or two-sided confidence intervals leads to coverage accuracies of ordern ^–1+, for arbitrary >0, in the case of analytic distribution functions.     

On the asymptotic accuracy of the bootstrap under arbitrary resampling size

We study the order of convergence of the Kolmogorov-Smirnov distance for the bootstrap of the mean and the bootstrap of quantiles when an arbitrary bootstrap sample size is used. We see that for the bootstrap of the mean, the best order of the bootstrap sample is of the order ofn, wheren is the sample size. In the case of non-lattice distributions and the bootstrap of the sample mean; the bootstrap removes the effect of the skewness of the distribution only when the bootstrap sample equals the sample size. However, for the bootstrap of quantiles, the preferred order of the bootstrap sample isn ^2/3. For the bootstrap of quantiles, if the bootstrap sample is of ordern ² or bigger, the bootstrap is not consistent.     

BOOTSTRAP CRITICAL POINT FOR CIRCULAR MEAN DIRECTION AND ITS APPLICATIONS

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Accuracy of <Emphasis Type="Italic">p</Emphasis>-values of approximate tests in testing for equality of means under unequal variances

Seemingly, testing for fixed effects in linear models with variance-covariance components has been solved for decades. However, even in simple situations such as in fixed one-way model with heteroscedastic variances (a multiple means case of the Behrens-Fisher problem) the questions of statistical properties of various approximations of test statistics are still alive. Here we present a brief overview of several approaches suggested in the literature as well as those available in statistical software, accompanied by a simulation study in which the accuracy of p-values is studied. Our interest is limited here to the Welch’s test, the Satterthwaite-Fai-Cornelius test, the Kenward-Roger test, the simple ANOVA F-test, and the parametric bootstrap test. We conclude that for small sample sizes, regardless the number of compared means and the heterogeneity of variance, the ANOVA F-test p-value performs the best. For higher sample sizes (at least 5 per group), the parametric bootstrap performs well, and the Kenward-Roger test also performs well.     

Random Quadratic Forms and the Bootstrap for U-Statistics

We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows′ metrics. We also study the i.i.d. weighted bootstrap [formula] where (X_i) and (ξ_i) are two i.i.d. sequences, independent of each other and where Eξ_i = 0, Var(ξ_i) = 1. It turns out that, conditionally given (X_i), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral ∫¹₀ ∫¹₀h(F⁻¹(x), F⁻¹(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix H_n=((1/n)h(X_i, X_j))_{1 ≤ i, j ≤ n}.     

Weighted bootstrap for U-statistics

In this paper we investigate the weighted bootstrap for U-statistics and its properties. Under very general choices of random weights and certain regularity conditions, we show that the weighted bootstrap method with U-statistics provides second-order accurate approximations to the distribution of U-statistics. We shall prove this via one-term Edgeworth expansions of weighted U-statistics.     

Asymptotic theory for estimators under random censorship

Summary The product limit estimator of an unknown distributionF is represented as aU-statistic plus an error of the ordero(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0,M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0,M], to the mean life time of an individual up to that time pointM. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.Research supported in part by NSA Grant MDA 904-90-H-1001 and by NSF Grant DMS-9007717     

Asymptotics and bootstrap for inverse Gaussian regression

This paper studies regression, where the reciprocal of the mean of a dependent variable is considered to be a linear function of the regressor variables, and the observations on the dependent variable are assumed to have an inverse Gaussian distribution. The large sample theory for the pseudo maximum likelihood estimators is available in the literature, only when the number of replications increase at a fixed rate. This is inadequate for many practical applications. This paper establishes consistency and derives the asymptotic distribution for the pseudo maximum likelihood estimators under very general conditions on the design points. This includes the case where the number of replications do not grow large, as well as the one where there are no replications. The bootstrap procedure for inference on the regression parameters is also investigated.Research supported in part by NSF Grant DMS-9208066.Research supported in part by NSERC of Canada.     

Bootstrap method and empirical process

In this paper we consider the sampling properties of the bootstrap process, that is, the empirical process obtained from a random sample of size n (with replacement) of a fixed sample of size n of a continuous distribution. The cumulants of the bootstrap process are given up to the order n ^–1 and their unbiased estimation is discussed. Furthermore, it is shown that the bootstrap process has an asymptotic minimax property for some class of distributions up to the order n ^–1/2.     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

On the distributions of some test criteria for a covariance matrix under local alternatives and bootstrap approximations

The asymptotic distribution of some test criteria for a covariance matrix are derived under local alternatives. Except for the existence of some higher moments, no assumption as to the form of the distribution function is made. As an illustration, a case of t distribution included normal model is considered and the power of the likelihood ratio test and Nagao's test for sphericity, as described in Srivastava and Khatri and Anderson, is computed. Also, the power is computed using the bootstrap method. In the case of t distribution, the bootstrap approximation does not appear to be as good as the one obtained by the asymptotic expansion method.     

An evaluation of the multivariate dispersion charts with estimated parameters under non‐normality

Various charts such as |S|, W, and G are used for monitoring process dispersion. Most of these charts are based on the normality assumption, while exact distribution of the control statistic is unknown, and thus limiting distribution of control statistic is employed which is applicable for large sample sizes. In practice, the normality assumption of distribution might be violated, while it is not always possible to collect large sample size. Furthermore, to use control charts in practice, the in‐control state usually has to be estimated. Such estimation has a negative effect on the performance of control chart. Non‐parametric bootstrap control charts can be considered as an alternative when the distribution is unknown or a collection of large sample size is not possible or the process parameters are estimated from a Phase I data set. In this paper, non‐parametric bootstrap multivariate control charts |S|, W, and G are introduced, and their performances are compared against Shewhart‐type control charts. The proposed method is based on bootstrapping the data used for estimating the in‐control state. Simulation results show satisfactory performance for the bootstrap control charts. Ultimately, the proposed control charts are applied to a real case study.     

Hypothesis testing for the generalized multivariate modified Bessel model

In this paper, we consider hypothesis testing problems in which the involved samples are drawn from generalized multivariate modified Bessel populations. This is a much more general distribution that includes both the multivariate normal and multivariate-t distributions as special cases. We derive the distribution of the Hotelling's T²-statistic for both the one- and two-sample problems, as well as the distribution of the Scheffe's T²-statistic for the Behrens–Fisher problem. In all cases, the non-null distribution of the corresponding F-statistic follows a new distribution which we introduce as the non-central F-Bessel distribution. Some statistical properties of this distribution are studied. Furthermore, this distribution was utilized to perform some power calculations for tests of means for different models which are special cases of the generalized multivariate modified Bessel distribution, and the results compared with those obtained under the multivariate normal case. Under the null hypothesis, however, the non-central F-Bessel distribution reduces to the central F-distribution obtained under the classical normal model.     

Consistency of general bootstrap methods for degenerate -type and -type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.