Testing for the Equality of Two Nonparametric Regression Curves with Long Memory Errors

This article discusses the problem of testing the equality of two nonparametric regression functions against two-sided alternatives for uniform design on [0,1] with long memory moving average errors. The standard deviations and the long memory parameters are possibly different for the two errors. The article adapts the partial sum process idea used in the independent observations settings to construct the tests and derives their asymptotic null distributions. The article also shows that these tests are consistent for general alternatives and obtains their limiting distributions under a sequence of local alternatives. Since the limiting null distributions of these tests are unknown, we first conducted a Monte Carlo simulation study to obtain a few selected critical values of the proposed tests. Then based on these critical values, another Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. The article also contains a simulation study that assesses the effect of estimating the nonparametric regression function on an estimate of the long memory parameter of the errors. It is observed that the estimate based on direct observations is generally preferable over the one based on the estimated nonparametric residuals.     

Nonparametric tests for multivariate multi-sample locations based on data depth

Several nonparametric tests for multivariate multi-sample location problem are proposed in this paper. These tests are based on the notion of data depth, which is used to measure the centrality/outlyingness of a given point with respect to a given distribution or a data cloud. Proposed tests are completely nonparametric and implemented through the idea of permutation tests. Performance of the proposed tests is compared with existing parametric test and nonparametric test based on data depth. An extensive simulation study reveals that proposed tests are superior to the existing tests based on data depth with regard to power. Illustrations with real data are provided.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

A bivariate regression model for matched paired survival data: local influence and residual analysis

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we consider a location scale model for bivariate survival times based on the proposal of a copula to model the dependence of bivariate survival data. For the proposed model, we consider inferential procedures based on maximum likelihood. Gains in efficiency from bivariate models are also examined in the censored data setting. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the bivariate regression model for matched paired survival data. Sensitivity analysis methods such as local and total influence are presented and derived under three perturbation schemes. The martingale marginal and the deviance marginal residual measures are used to check the adequacy of the model. Furthermore, we propose a new measure which we call modified deviance component residual. The methodology in the paper is illustrated on a lifetime data set for kidney patients.     

Adjusting Stepwise p-Values in Generalized Linear Models

Stepwise methods for variable selection are frequently used to determine the predictors of an outcome in generalized linear models. Although it is widely used within the scientific community, it is well known that the tests on the explained deviance of the selected model are biased. This arises from the fact that the traditional test statistics upon which these methods are based were intended for testing pre-specified hypotheses; instead, the tested model is selected through a data-driven procedure. A multiplicity problem therefore arises. In this work, we define and discuss a nonparametric procedure to adjust the p-value of the selected model of any stepwise selection method. The unbiasedness and consistency of the method is also proved. A simulation study shows the validity of this procedure. Theoretical differences with previous works in the same field are also discussed.     

Power comparison of data depth-based nonparametric tests for testing equality of locations

In the recent years, the notion of data depth has been used in nonparametric multivariate data analysis since it gives natural ‘centre-outward’ ordering of multivariate data points with respect to the given data cloud. In the literature, various nonparametric tests are developed for testing equality of location of two multivariate distributions based on data depth. Here, we define two nonparametric tests based on two different test statistic for testing equality of locations of two multivariate distributions. In the present work, we compare the performance of these tests with the tests developed by Li and Liu [New nonparametric tests of multivariate locations and scales using data depth. Statist Sci. 2004;(1):686–696] for testing equality of locations of two multivariate distributions. Comparison in terms of power is done for multivariate symmetric and skewed distributions using simulation for three popular depth functions. Application of tests to real life data is provided. Conclusion and recommendations are also provided.     

Generalized Wald-type tests based on minimum density power divergence estimators

In testing of hypothesis, the robustness of the tests is an important concern. Generally, the maximum likelihood-based tests are most efficient under standard regularity conditions, but they are highly non-robust even under small deviations from the assumed conditions. In this paper, we have proposed generalized Wald-type tests based on minimum density power divergence estimators for parametric hypotheses. This method avoids the use of nonparametric density estimation and the bandwidth selection. The trade-off between efficiency and robustness is controlled by a tuning parameter β. The asymptotic distributions of the test statistics are chi-square with appropriate degrees of freedom. The performance of the proposed tests is explored through simulations and real data analysis.     

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.     

Tests based on weighted rankings in complete blocks: exact distribution and monte carlo simulation

Quade (1972, 1979) proposed a family of nonparametric tests based on weighted within-block rankings, for testing the hypothesis of no treatment effects in a complete randomized blocks layout. In this paper we give a table of the exact null distribution of these tests when the number of treatments is 3, the number of blocks is less than or equal to 14 and the block scores are linear. Moreover, a Monte Carlo study was performed to compare the powers of these tests with parametric and nonparametric competitors     

Assessing additivity in nonparametric models —A kernel‐based method

In this article, we address the testing problem for additivity in nonparametric regression models. We develop a kernel‐based consistent test of a hypothesis of additivity in nonparametric regression, and establish its asymptotic distribution under a sequence of local alternatives. Compared to other existing kernel‐based tests, the proposed test is shown to effectively ameliorate the influence from estimation bias of the additive component of the nonparametric regression, and hence increase its efficiency. Most importantly, it avoids the tuning difficulties by using estimation‐based optimal criteria, while there is no direct tuning strategy for other existing kernel‐based testing methods. We discuss the usage of the new test and give numerical examples to demonstrate the practical performance of the test. The Canadian Journal of Statistics 39: 632–655; 2011. © 2011 Statistical Society of Canada     

Influence Diagnostics of Semiparametric Nonlinear Reproductive Dispersion Models

This article proposes a semiparametric nonlinear reproductive dispersion model (SNRDM) which is an extension of nonlinear reproductive dispersion model and semiparametric regression model. Maximum penalized likelihood estimators (MPLEs) of unknown parameters and nonparametric functions in SNRDMs are presented. Some novel diagnostic statistics such as Cook distance and difference deviance for parametric and nonparametric parts are developed to identify influence observations in SNRDMs on the basis of case-deletion method, and some formulae readily computed with the MPLEs algorithm for diagnostic measures are given. The equivalency of case-deletion models and mean-shift outlier models in SNRDM is investigated. A simulation study and a real example are used to illustrate the proposed diagnostic measures.     

Some partially sequential nonparametric tests for detecting linear trend

In the present study, we develop two nonparametric partially sequential tests for detecting possible presence of linear trend among the incoming series of observations. We assume that a sample of fixed size is available a priori from some unknown univariate continuous population and there is no sign of trend among these historical observations. Our proposed tests can be viewed as the sequential type tests for monitoring structural changes. We use partial sequential sampling schemes based on usual ranks as well as on sequential ranks. We provide detailed discussion on asymptotic studies related to the proposed tests. We compare the two tests under various situations. We also present some numerical results based on simulation studies. Proposed tests are extremely important in profit making in volatile market through Margin Trading. We illustrate the mechanism with a detailed analysis of a stock price data.     

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.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.     

Local influence on bandwidth estimation for kernel smoothing

This paper examines the problem of assessing local influence on the optimal bandwidth estimation in kernel smoothing based on cross validation. The bandwidth for kernel smoothing plays an important role in the model fitting and is often estimated using the cross-validation criterion. Following the argument of the second-order approach to local influence suggested by Wu and Luo (1993), we develop a new diagnostic statistic to examine the local influence of the observations on the estimation of the optimal bandwidth, where the perturbation may belong to one of three schemes. These are the response perturbation, the perturbation in the explanatory variable, and the case-weight

perturbation. The proposed diagnostic is nonparametric and is capable of identifying influential observations with strong influence on the bandwidth estimation. An example is presented to illustrate the application of the proposed diagnostic, and the usefulness of the nonparametric approach is illustrated in comparison with some other approaches to the assessment of local influence     

Some count-based nonparametric tests for circular symmetry of a bivariate distribution

In this paper, we revisit the problem of testing of the hypothesis of circular symmetry of a bivariate distribution. We propose some nonparametric tests based on sector counts. These include tests based on chi-square goodness-of-fit test, the classical likelihood ratio, mean deviation, and the range. The proposed tests are easy to implement and the exact null distributions for small sample sizes of the test statistics are obtained. Two examples with small and large data sets are given to illustrate the application of the tests proposed. For small and moderate sample sizes, the performances of the proposed tests are evaluated using empirical powers (empirical sizes are also reported). Also, we evaluate the performance of these count-based tests with adaptations of several well-known tests such as the Kolmogorov–Smirnov-type tests, tests based on kernel density estimator, and the Wilcoxon-type tests. It is observed that among the count-based tests the likelihood ratio test performs better.     

Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.     

Estimation and variable selection in single-index composite quantile regression

In this article, a new composite quantile regression estimation approach is proposed for estimating the parametric part of single-index model. We use local linear composite quantile regression (CQR) for estimating the nonparametric part of single-index model (SIM) when the error distribution is symmetrical. The weighted local linear CQR is proposed for estimating the nonparametric part of SIM when the error distribution is asymmetrical. Moreover, a new variable selection procedure is proposed for SIM. Under some regularity conditions, we establish the large sample properties of the proposed estimators. Simulation studies and a real data analysis are presented to illustrate the behavior of the proposed estimators.