A bootstrap test for time series linearity

A bootstrap algorithm is proposed for testing Gaussianity and linearity in stationary time series, and consistency of the relevant bootstrap approximations is proven rigorously for the first time. Subba Rao and Gabr (1980) and Hinich (1982) have formulated some well-known nonparametric tests for Gaussianity and linearity based on the asymptotic distribution of the normalized bispectrum. The proposed bootstrap procedure gives an alternative way to approximate the finite-sample null distribution of such test statistics. We revisit a modified form of Hinich's test utilizing kernel smoothing, and compare its performance to the bootstrap test on several simulated data sets and two real data sets—the S&P 500 returns and the quarterly US real GNP growth rate. Interestingly, Hinich's test and the proposed bootstrapped version yield substantially different results when testing Gaussianity and linearity of the GNP data.     

Bootstrapping with auxiliary information

In the nonparametric setting, the standard bootstrap method is based on the empirical distribution function of a random sample. The author proposes, by means of the empirical likelihood technique, an alternative bootstrap procedure under a nonparametric model in which one has some auxiliary information about the population distribution. By proving the almost sure weak convergence of the modified bootstrapped empirical process, the validity of the proposed bootstrap procedure is established. This new result is used to obtain bootstrap confidence bands for the population distribution function and to perform the bootstrap Kolmogorov test in the presence of auxiliary information. Other applications include bootstrapping means and variances with auxiliary information. Three simulation studies are presented to demonstrate the performance of the proposed bootstrap procedure for small samples.     

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 nonparametric test for similarity of marginals—With applications to the assessment of population bioequivalence

In this paper we suggest a completely nonparametric test for the assessment of similar marginals of a multivariate distribution function. This test is based on the asymptotic normality of Mallows distance between marginals. It is also shown that the n out of n bootstrap is weakly consistent, thus providing a theoretical justification to the work in Czado, C. and Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280]. The test is extended to cross-over trials and is applied to the problem of population bioequivalence, where two formulations of a drug are shown to be similar up to a tolerable limit. This approach was investigated in small samples using bootstrap techniques in Czado, C., Munk, A. [2001. Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions. J. Statist. Comput. Simulation 68, 243–280], showing that the bias corrected and accelerated bootstrap yields a very accurate and powerful finite sample correction. A data example is discussed.     

Book Review: Introducing Monte Carlo Methods with R

In this article we propose a nonparametric test for poolability in large dimensional semiparametric panel data models with cross-section dependence based on the sieve estimation technique. To construct the test statistic, we only need to estimate the model under the alternative. We establish the asymptotic normal distributions of our test statistic under the null hypothesis of poolability and a sequence of local alternatives, and prove the consistency of our test. We also suggest a bootstrap method as an alternative way to obtain the critical values. A small set of Monte Carlo simulations indicate the test performs reasonably well in finite samples.     

Tests for Symmetric Error Distribution in Linear and Nonparametric Regression Models

In linear and nonparametric regression models, the problem of testing for symmetry of the distribution of errors is considered. We propose a test statistic which utilizes the empirical characteristic function of the corresponding residuals. The asymptotic null distribution of the test statistic as well as its behavior under alternatives is investigated. A simulation study compares bootstrap versions of the proposed test to other more standard procedures.     

Bootstrap methods for the nonparametric assessment of population bioequivalence and similarity of distributions

A completely nonparametric approach to population bioequivalence in crossover trials has been suggested by Munk and Czado (1999). It is based on the Mallows (1972) metric as a nonparametric distance measure which allows the comparison between the entire distribution functions of test and reference formulations. It was shown that a separation between carry-over and period effects is not possible in the nonparametric setting. However when carry-over effects can be excluded, treatment effects can be assessed when period effects are or not. Munk and Czado (1999) proved bootstrap limit laws of the corresponding test statistics because estimation of the limiting variance of the test statistic is very cumbersome. The purpose of this paper is to investigate the small sample behavior of various bootstrap methods and to compare it with the asymptotic test obtained by estimation of the limiting variance. The percentile (PC) and bias correct- ed and accelerated (BCA) bootstrap were compared for multivariate normal and nonnormal populations. From the simulation results presented, the BCA bootstrap is found to be less conservative and provides higher power compared to the PC bootstrap, especially when skewed multivariate populations are present.     

An Improvement of the Nonparametric Bootstrap Test for the Comparison of the Coefficient of Variations

In this article, we propose a new test for examining the equality of the coefficient of variation between two different populations. The proposed test is based on the nonparametric bootstrap method. It appears to yield several appreciable advantages over the current tests. The quick and easy implementation of the test can be considered as advantages of the proposed test. The test is examined by the Monte Carlo simulations, and also evaluated using various numerical studies.     

Modelling udder infection data using copula models for quadruples

We study copula models for correlated infection times in the four udder quarters of dairy cows. Both a semi-parametric and a nonparametric approach are considered to estimate the marginal survival functions, taking into account the effect of a binary udder quarter level covariate. We use a two-stage estimation approach and we briefly discuss the asymptotic behaviour of the estimators obtained in the first and the second stage of the estimation. A pseudo-likelihood ratio test is used to select an appropriate copula from the power variance copula family that describes the association between the outcomes in a cluster. We propose a new bootstrap algorithm to obtain the p-value for this test. This bootstrap algorithm also provides estimates for the standard errors of the estimated parameters in the copula. The proposed methods are applied to the udder infection data. A small simulation study for a setting similar to the setting of the udder infection data gives evidence that the proposed method provides a valid approach to select an appropriate copula within the power variance copula family.     

Test for new better than used in convex ordering

Class of life distributions which are new better than used in convex ordering (NBUC) is dealt with. A probabilistic characterization is introduced to measure the degree of NBUC-ness. A nonparametric procedure is also developed to test the exponentiality against the strict NBUC property, therein, the theory of U-statistics and jackknife is utilized to establish the asymptotic normality of the test statistic. Furthermore, Edgeworth expansion and bootstrap are employed to improve the accuracy of the approximation. Some numerical simulations on the power are presented as a demonstration for the proposed procedure.     

A NONPARAMETRIC TEST OF CHANGING CONDITIONAL VARIANCES IN AUTOREGRESSIVE TIME SERIES

A nonparametric test for detecting changing conditional variances in stationary AR(p) time series is proposed in this paper. For AR(1) models, the test statistic is a Kolmogorov-Smirnov type statistic and the asymptotic theory is developed under both the null and the alternative hypotheses. For AR(p) models (p ≥ 2), an approximate test procedure is proposed. The empirical upper percentage points for our test are tabulated for both p = 1 and p = 2 cases and a bootstrap procedure is suggested for the p ≥ 3 case. Monte Carlo simulations demonstrate that the test has very good powers for finite samples under both normal and non-normal errors.     

Test for the existence of finite moments via bootstrap

This paper develops a bootstrap hypothesis test for the existence of finite moments of a random variable, which is nonparametric and applicable to both independent and dependent data. The test is based on a property in bootstrap asymptotic theory, in which the m out of n bootstrap sample mean is asymptotically normal when the variance of the observations is finite. Consistency of the test is established. Monte Carlo simulations are conducted to illustrate the finite sample performance and compare it with alternative methods available in the literature. Applications to financial data are performed for illustration.     

Estimation and inference for varying coefficient partially nonlinear errors-in-variables models

In this article, we study the varying coefficient partially nonlinear model with measurement errors in the nonparametric part. A local corrected profile nonlinear least-square estimation procedure is proposed and the asymptotic properties of the resulting estimators are established. Further, a generalized likelihood ratio (GLR) statistic is proposed to test whether the varying coefficients are constant. The asymptotic null distribution of the statistic is obtained and a residual-based bootstrap procedure is employed to compute the p-value of the statistic. Some simulations are conducted to evaluate the performance of the proposed methods. The results show that the estimating and testing procedures work well in finite samples.     

Testing heteroscedasticity in nonlinear and nonparametric regressions

This article presents new nonparametric tests for heteroscedasticity in nonlinear and nonparametric regression models. The tests have an asymptotic standard normal distribution under the null hypothesis of homoscedasticity and are robust against any form of heteroscedasticity. A Monte Carlo simulation with critical values obtained from the wild bootstrap procedure is provided to asses the finite sample performances of the tests. A real application of testing interest rate volatility functions illustrates the usefulness of the tests proposed. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Nonparametric confidence regions via the analytic wild bootstrap

The wild bootstrap is a nonparametric tool that can be used to estimate a sampling distribution in the presence of heteroscedastic errors. In particular, the wild bootstrap enables us to compute confidence regions for regression parameters under non-i.i.d. models. While the wild bootstrap may perform well in these settings, its obvious drawback is a lack of computational efficiency. The wild bootstrap requires a large number of bootstrap replications, making the use of this tool impractical when dealing with big data. We introduce the analytic wild bootstrap (ANWB), which provides a nonparametric alternative way of constructing confidence regions for regression parameters. The ANWB is superior to the wild bootstrap from a computational standpoint while exhibiting similar finite-sample performance. We report simulation results for both least squares and ridge regression. Additionally, we test the ANWB on a real dataset and compare its performance with that of other standard approaches.     

Two-sample nonparametric stochastic order inference with an application in plant physiology

In this paper, a new nonparametric methodology is developed for testing whether the changing pattern of a response variable over multiple ordered sub-populations from one treatment group differs with the one from another treatment group. The question is formalized into a nonparametric two-sample comparison problem for the stochastic order among subsamples, through U-statistics with accommodations for zero-inflated distributions. A novel bootstrap procedure is proposed to obtain the critical values with given type I error. Following the procedure, bootstrapped p-values are obtained through simulated samples. It is proven that the distribution of the test statistics is independent from the underlying distributions of the subsamples, when certain sufficient statistics provided. Furthermore, this study also develops a feasible framework for power studies to determine sample sizes, which is necessary in real-world applications. Simulation results suggest that the test is consistent. The methodology is illustrated using a biological experiment with a split-plot design, and significant differences in changing patterns of seed weight between treatments are found with relative small subsample sizes.     

A Comparison of Some Methods for the Selection of a Common Eigenvector Model for the Covariance Matrices of Two Groups

Two new nonparametric common principal component model selection procedures based on bootstrap distributions of the vector correlations of all combinations of the eigenvectors from two groups are proposed. The performance of these methods is compared in a simulation study to the two parametric methods previously suggested by Flury in 1988, as well as modified versions of two nonparametric methods proposed by Klingenberg in 1996 and then by Klingenberg and McIntyre in 1998. The proposed bootstrap vector correlation distribution (BVD) method is shown to outperform all of the existing methods in most of the simulated situations considered.     

Nonparametric inference for panel count data with competing risks

In survival and reliability studies, panel count data arise when we investigate a recurrent event process and each study subject is observed only at discrete time points. If recurrent events of several types are possible, we obtain panel count data with competing risks. Such data arise frequently from transversal studies on recurrent events in demography, epidemiology and reliability experiments where the individuals cannot be observed continuously. In the present paper, we propose an isotonic regression estimator for the cause specific mean function of the underlying recurrent event process of a competing risks panel count data. Further, a nonparametric test is proposed to compare the cause specific mean functions of the panel count competing risks data. Asymptotic properties of the proposed estimator and test statistic are studied. A simulation study is conducted to assess the finite sample behaviour of the proposed estimator and test statistic. Finally, the procedures developed are applied to a real data arising from skin cancer chemo prevention trial.     

Non parametric resampling for stationary Markov processes: The local grid bootstrap approach

A new resampling technique, referred as “local grid bootstrap” (LGB), based on nonparametric local bootstrap and applicable to a wide range of stationary general space Markov processes is proposed. This nonparametric technique resamples local neighborhoods defined around the true samples of the observed multivariate time serie. The asymptotic behavior of this resampling procedure is studied in detail. Applications to linear and nonlinear (in particular chaotic) simulated time series are presented, and compared to Paparoditis and Politis [2002. J. Statist. Plan. Inf. 108, 301–328] approach, referred as “local bootstrap” (LB) and developed in earlier similar works. The method shows to be efficient and robust even when the length of the observed time series is reasonably small.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.