A note on ranked-set sampling using a covariate

Ranked-set sampling (RSS) and judgment post-stratification (JPS) use ranking information to obtain more efficient inference than is possible using simple random sampling. Both methods were developed with subjective, judgment-based rankings in mind, but the idea of ranking using a covariate has received a lot of attention. We provide evidence here that when rankings are done using a covariate, the standard RSS and JPS mean estimators no longer make efficient use of the available information. We first show that when rankings are done using a covariate, the standard nonparametric mean estimators in JPS and unbalanced RSS are inadmissible under squared error loss. We then show that when rankings are done using a covariate, nonparametric regression techniques yield mean estimators that tend to be significantly more efficient than the standard RSS and JPS mean estimators. We conclude that the standard estimators are best reserved for settings where only subjective, judgment-based rankings are available.     

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 multiple comparison procedures for unbalanced one-way factorial designs

In this paper, we present several nonparametric multiple comparison (MC) procedures for unbalanced one-way factorial designs. The nonparametric hypotheses are formulated by using normalized distribution functions and the comparisons are carried out on the basis of the relative treatment effects. The proposed test statistics take the form of linear pseudo rank statistics and the asymptotic joint distribution of the pseudo rank statistics for testing treatments versus control satisfies the multivariate totally positive of order two condition irrespective of the correlations among the rank statistics. Therefore, in the context of MCs of treatments versus control, the nonparametric Simes test is validated for the global testing of the intersection hypothesis. For simultaneous testing of individual hypotheses, the nonparametric Hochberg stepup procedure strongly controls the familywise type I error rate asymptotically. With regard to all pairwise comparisons, we generalize various single-step and stagewise procedures to perform comparisons on the relative treatment effects. To further compare with normal theory counterparts, the asymptotic relative efficiencies of the nonparametric MC procedures with respect to the parametric MC procedures are derived under a sequence of Pitman alternatives in a nonparametric location shift model for unbalanced one-way layouts. Monte Carlo simulations are conducted to demonstrate the validity and power of the proposed nonparametric MC procedures.     

SiZer Map for inference with additive models

Sizer Map is proposed as a graphical tool for assistance in nonparametric additive regression testing problems. Four problems have been analyzed by using SiZer Map: testing for additivity, testing the components significance, testing parametric models for the components and testing for interactions. The simplicity and flexibility of SiZer Map for our purposes are highlighted from the performed empirical study with several real datasets. With these data, we compare the conclusions derived from SiZer analysis with the global results derived from standard tests, previously proposed in the literature.     

Universal codes as a basis for time series testing

We suggest a new approach to hypothesis testing for ergodic and stationary processes. In contrast to standard methods, the suggested approach gives a possibility to make tests, based on any lossless data compression method even if the distribution law of the codeword lengths is not known. We apply this approach to the following four problems: goodness-of-fit testing (or identity testing), testing for independence, testing of serial independence and homogeneity testing and suggest nonparametric statistical tests for these problems. It is important to note that practically used so-called archivers can be used for suggested testing.     

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     

Parameterization and inference for nonparametric regression problems

We consider local likelihood or local estimating equations, in which a multivariate function () is estimated but a derived function () of () is of interest. In many applications, when most naturally formulated the derived function is a non-linear function of (). In trying to understand whether the derived non-linear function is constant or linear, a problem arises with this approach: when the function is actually constant or linear, the expectation of the function estimate need not be constant or linear, at least to second order. In such circumstances, the simplest standard methods in nonparametric regression for testing whether a function is constant or linear cannot be applied. We develop a simple general solution which is applicable to nonparametric regression, varying-coefficient models, nonparametric generalized linear models, etc. We show that, in local linear kernel regression, inference about the derived function () is facilitated without a loss of power by reparameterization so that () is itself a component of (). Our approach is in contrast with the standard practice of choosing () for convenience and allowing ()> to be a non-linear function of (). The methods are applied to an important data set in nutritional epidemiology.     

A nonparametric estimation procedure for the Hawkes process: comparison with maximum likelihood estimation

In earlier work, Kirchner [An estimation procedure for the Hawkes process. Quant Financ. 2017;17(4):571–595], we introduced a nonparametric estimation method for the Hawkes point process. In this paper, we present a simulation study that compares this specific nonparametric method to maximum-likelihood estimation. We find that the standard deviations of both estimation methods decrease as power-laws in the sample size. Moreover, the standard deviations are proportional. For example, for a specific Hawkes model, the standard deviation of the branching coefficient estimate is roughly 20% larger than for MLE – over all sample sizes considered. This factor becomes smaller when the true underlying branching coefficient becomes larger. In terms of runtime, our method clearly outperforms MLE. The present bias of our method can be well explained and controlled. As an incidental finding, we see that also MLE estimates seem to be significantly biased when the underlying Hawkes model is near criticality. This asks for a more rigorous analysis of the Hawkes likelihood and its optimization.     

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.     

Asymptotocally nonparametric tests with censored paired data

A class of asymptotically nonparametric test with contains a test proposed by Wei(1980), is considered for testing the equality of two continuous distribution funcitons when paired observations are subject to arbitrary right censorship. It is shown that under the null hypothesis each test statistic converges in distribution to the standard normal random variable. Furthermore. the Monte Carlo simulation results indicate that some tests in this class are more powerful than Wei's test. A generalization to incomplete censored paired data is also included.     

O(N 2)-Operation Approximation of Covariance Matrix Inverse in Gaussian Process Regression Based on Quasi-Newton BFGS Method

Gaussian process (GP) is a Bayesian nonparametric regression model, showing good performance in various applications. However, during its model-tuning procedure, the GP implementation suffers from numerous covariance-matrix inversions of expensive O(N³) operations, where N is the matrix dimension. In this article, we propose using the quasi-Newton BFGS O(N²)-operation formula to approximate/replace recursively the inverse of covariance matrix at every iteration. The implementation accuracy is guaranteed carefully by a matrix-trace criterion and by the restarts technique to generate good initial guesses. A number of numerical tests are then performed based on the sinusoidal regression example and the Wiener–Hammerstein identification example. It is shown that by using the proposed implementation, more than 80% O(N³) operations could be eliminated, and a typical speedup of 5–9 could be achieved as compared to the standard maximum-likelihood-estimation (MLE) implementation commonly used in Gaussian process regression.     

Positive quadrant dependence tests for copulas

In this paper the interest is in testing the null hypothesis of positive quadrant dependence (PQD) between two random variables. Such a testing problem is important since prior knowledge of PQD is a qualitative restriction that should be taken into account in further statistical analysis, for example, when choosing an appropriate copula function to model the dependence structure. The key methodology of the proposed testing procedures consists of evaluating a “distance” between a nonparametric estimator of a copula and the independence copula, which serves as a reference case in the whole set of copulas having the PQD property. Choices of appropriate distances and nonparametric estimators of copula are discussed, and the proposed methods are compared with testing procedures based on bootstrap and multiplier techniques. The consistency of the testing procedures is established. In a simulation study the authors investigate the finite sample size and power performances of three types of test statistics, Kolmogorov–Smirnov, Cramér–von‐Mises, and Anderson–Darling statistics, together with several nonparametric estimators of a copula, including recently developed kernel type estimators. Finally, they apply the testing procedures on some real data. The Canadian Journal of Statistics 38: 555–581; 2010 © 2010 Statistical Society of Canada     

An empirical likelihood goodness-of-fit test for time series

Summary. Standard goodness-of-fit tests for a parametric regression model against a series of nonparametric alternatives are based on residuals arising from a fitted model. When a parametric regression model is compared with a nonparametric model, goodness-of-fit testing can be naturally approached by evaluating the likelihood of the parametric model within a nonparametric framework. We employ the empirical likelihood for an α -mixing process to formulate a test statistic that measures the goodness of fit of a parametric regression model. The technique is based on a comparison with kernel smoothing estimators. The empirical likelihood formulation of the test has two attractive features. One is its automatic consideration of the variation that is associated with the nonparametric fit due to empirical likelihood's ability to Studentize internally. The other is that the asymptotic distribution of the test statistic is free of unknown parameters, avoiding plug-in estimation. We apply the test to a discretized diffusion model which has recently been considered in financial market analysis.     

Validation of Nonparametric Two-sample Bootstrap in ROC Analysis on Large Datasets

The nonparametric two-sample bootstrap is applied to computing uncertainties of measures in receiver operating characteristic (ROC) analysis on large datasets in areas such as biometrics, speaker recognition, etc. when the analytical method cannot be used. Its validation was studied by computing the standard errors of the area under ROC curve using the well-established analytical Mann–Whitney statistic method and also using the bootstrap. The analytical result is unique. The bootstrap results are expressed as a probability distribution due to its stochastic nature. The comparisons were carried out using relative errors and hypothesis testing. These match very well. This validation provides a sound foundation for such computations.     

Nonparametric Bayes factors based on empirical likelihood ratios

Bayes methodology provides posterior distribution functions based on parametric likelihoods adjusted for prior distributions. A distribution-free alternative to the parametric likelihood is use of empirical likelihood (EL) techniques, well known in the context of nonparametric testing of statistical hypotheses. Empirical likelihoods have been shown to exhibit many of the properties of conventional parametric likelihoods. In this paper, we propose and examine Bayes factors (BF) methods that are derived via the EL ratio approach. Following Kass and Wasserman (1995), we consider Bayes factors type decision rules in the context of standard statistical testing techniques. We show that the asymptotic properties of the proposed procedure are similar to the classical BF's asymptotic operating characteristics. Although we focus on hypothesis testing, the proposed approach also yields confidence interval estimators of unknown parameters. Monte Carlo simulations were conducted to evaluate the theoretical results as well as to demonstrate the power of the proposed test.     

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     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

A review and some new results on permutation testing for multivariate problems

In recent years permutation testing methods have increased both in number of applications and in solving complex multivariate problems. When available permutation tests are essentially of an exact nonparametric nature in a conditional context, where conditioning is on the pooled observed data set which is often a set of sufficient statistics in the null hypothesis. Whereas, the reference null distribution of most parametric tests is only known asymptotically. Thus, for most sample sizes of practical interest, the possible lack of efficiency of permutation solutions may be compensated by the lack of approximation of parametric counterparts. There are many complex multivariate problems, quite common in empirical sciences, which are difficult to solve outside the conditional framework and in particular outside the method of nonparametric combination (NPC) of dependent permutation tests. In this paper we review such a method and its main properties along with some new results in experimental and observational situations (robust testing, multi-sided alternatives and testing for survival functions).     

An extension of the median test to analysis of variance

In a k-way analysis of variance model, the major concern is testing for main effects and for the presence of interaction between the factors. When the assumptions of normality and equal variances are satisfied, the appropriate test to use is the usual F-test for ANOVA. However, when the normality assumption is not satisfied then a robust or nonparametric test is needed to conduct the analysis. In this paper a nonparametric method based on cell counts is proposed. Each cell is divided into L subcells based on predetermined outpoints and the resulting frequencies are laid out in a contingency table. Then the Pearson x² and tne likelihood ratio tests are performed. A comparison with the classical ANOVA F-test indicates that the proposed method is preferable when the data comes from a thick-tailed highly skewed distribution.     

Smoothing for small samples with model misspecification: nonparametric and semiparametric concerns

Our goal is to find a regression technique that can be used in a small-sample situation with possible model misspecification. The development of a new bandwidth selector allows nonparametric regression (in conjunction with least squares) to be used in this small-sample problem, where nonparametric procedures have previously proven to be inadequate. Considered here are two new semiparametric (model-robust) regression techniques that combine parametric and nonparametric techniques when there is partial information present about the underlying model. A general overview is given of how typical concerns for bandwidth selection in nonparametric regression extend to the model-robust procedures. A new penalized PRESS criterion (with a graphical selection strategy for applications) is developed that overcomes these concerns and is able to maintain the beneficial mean squared error properties of the new model-robust methods. It is shown that this new selector outperforms standard and recently improved bandwidth selectors. Comparisons of the selectors are made via numerous generated data examples and a small simulation study.