Olcay Arslan, Yesim Güney, Jana Jureckova, Yetkin Tuac
In the linear regression model with possibly autoregressive errors, we propose a family of nonparametric tests for regression under a nuisance autoregression. The tests avoid the estimation of nuisance parameters, in contrast to the tests proposed in the literature.
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The paper deals with the nonparametric estimation problem at a given fixed point for an autoregressive model with unknown distributed noise. Kernel estimate modifications are proposed. Asymptotic minimax and efficiency properties for proposed estimators are shown.
We constuct a sequential adaptive procedure for estimating the autoregressive function at a given point in nonparametric autoregression models with Gaussian noise. We make use of the sequential kernel estimators. The optimal adaptive convergence rate is given as well as the upper bound for the minimax risk.
We construct efficient robust truncated sequential estimators for the pointwise estimation problem in nonparametric autoregression models with smooth coefficients. For Gaussian models we propose an adaptive procedure based on the constructed sequential estimators. The minimax nonadaptive and adaptive convergence rates are established. It turns out that in this case these rates are the same as for regression models.
We discuss nonparametric estimation of the distribution function $G(x)$ of the autoregressive coefficient $a \in (-1,1)$ from a panel of $N$ random-coefficient AR(1) data, each of length $n$, by the empirical distribution function of lag 1 sample autocorrelations of individual AR(1) processes. Consistency and asymptotic normality of the empirical distribution function and a class of kernel density estimators is established under some regularity conditions on $G(x)$ as $N$ and $n$ increase to infinity. The Kolmogorov-Smirnov goodness-of-fit test for simple and composite hypotheses of Beta distr…
In this paper an autoregressive time series model with conditional heteroscedasticity is considered, where both conditional mean and conditional variance function are modeled nonparametrically. A test for the model assumption of independence of innovations from past time series values is suggested. The test is based on an weighted $L^2$-distance of empirical characteristic functions. The asymptotic distribution under the null hypothesis of independence is derived and consistency against fixed alternatives is shown. A smooth autoregressive residual bootstrap procedure is suggested and its perfo…
We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered only nearly integrated and stationary autoregressive processes. These results have applications to nonparametric predictive regression models. In particular, we show that the null rejection probability of a nonparametric t test is controlled uniformly in the degree of persistence of the regressor. This provides a rigorous justification for the validity of the …
Nonparametric Vector Quantile Autoregression
针对现有分位数自回归方法受限于单变量时间序列的问题，本文基于测度传输型多元分位数概念，提出非参数向量分位数自回归方法，可实现多元时间序列的全分布预测。
In this work we deal with the problem of fitting an error density to the goodness-of-fit test of the errors in nonlinear autoregressive time series models with stationary $α$-mixing error terms. The test statistic is based on the integrated squared error of the nonparametric error density estimate and the null error density. By deriving the asymptotic normality of test statistics in these models, we extend the result of Cheng and Sun (Statist. Probab. Lett. \textbf{78}, 1(2008), 50-59) in the model with i.i.d error terms to the more general case.
In this paper, we study nonparametric models allowing for locally stationary regressors and a regression function that changes smoothly over time. These models are a natural extension of time series models with time-varying coefficients. We introduce a kernel-based method to estimate the time-varying regression function and provide asymptotic theory for our estimates. Moreover, we show that the main conditions of the theory are satisfied for a large class of nonlinear autoregressive processes with a time-varying regression function. Finally, we examine structured models where the regression fu…
We point out necessary and sufficient conditions of uniform consistency of nonparametric sets of alternatives for widespread nonparametric tests. Nonparametric sets of alternatives can be defined both in terms of distribution function and in terms of density (or signals in the problem of signal detection in Gaussian white noise). In this part of paper such conditions are provided for $χ^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients.
We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requi…
For Kolmogorov test we find natural conditions of uniform consistency of sets of alternatives approaching to hypothesis. Sets of alternatives can be defined both in terms of distribution functions and in terms of densities.
In this paper, we are concerned with nonparametric estimation of the multivariate regression function in the presence of right censored data. More precisely, we propose a statistic that is shown to be asymptotically normally distributed under the additive assumption, and that could be used to test for additivity in the censored regression setting.
The Bayesian nonparametric inference and Dirichlet process are popular tools in statistical methodologies. In this paper, we employ the Dirichlet process in hypothesis testing to propose a Bayesian nonparametric chi-squared goodness-of-fit test. In our Bayesian nonparametric approach, we consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the updated Dirichlet process and the hypothesized distribution F0. We prove that this distance asymptotically converges to the same chi-squared distribution as the chi…
Nonparametric two sample test of spectral densities
本文提出一种非参数方法，用于检验两个高斯平稳过程的谱密度相等性，并在模拟和真实数据中验证了其有效性。
This paper considers the problem of comparing two processes with panel data. A nonparametric test is proposed for detecting a monotone change in the link between the two process distributions. The test statistic is of CUSUM type, based on the empirical distribution functions. The asymptotic distribution of the proposed statistic is derived and its finite sample property is examined by bootstrap procedures through Monte Carlo simulations.
In this paper we introduce the idea of partially sorting data to design nonparametric tests. This approach gives rise to tests that are sensitive to both the order and the underlying distribution of the data. We focus in particular on a test that uses the bubble sort algorithm to partially sort the data. We show that a function of the data, referred to as the empirical bubble sort curve, converges uniformly to a limiting curve. We define a goodness-of-fit test based on the distance between the empirical curve and its limit. The asymptotic distribution of the test statistic is a generalization …
We consider nonparametric testing in a non-asymptotic framework. Our statistical guarantees are exact in the sense that Type I and II errors are controlled for any finite sample size. Meanwhile, one proposed test is shown to achieve minimax optimality in the asymptotic sense. An important consequence of this non-asymptotic theory is a new and practically useful formula for selecting the optimal smoothing parameter in nonparametric testing. The leading example in this paper is smoothing spline models under Gaussian errors. The results obtained therein can be further generalized to the kernel ri…
Smoothed nonparametric two-sample tests
针对非参数离散检验显著性概率问题，提出平滑中位数与Wilcoxon秩和检验，避免偏差并保留良好性质。
We consider testing the significance of a subset of covariates in a nonparametric regression. These covariates can be continuous and/or discrete. We propose a new kernel-based test that smoothes only over the covariates appearing under the null hypothesis, so that the curse of dimensionality is mitigated. The test statistic is asymptotically pivotal and the rate of which the test detects local alternatives depends only on the dimension of the covariates under the null hypothesis. We show the validity of wild bootstrap for the test. In small samples, our test is competitive compared to existing…
Principled nonparametric tests for regression curvature in $\mathbb{R}^{d}$ are often statistically and computationally challenging. This paper introduces the stratified incomplete local simplex (SILS) tests for joint concavity of nonparametric multiple regression. The SILS tests with suitable bootstrap calibration are shown to achieve simultaneous guarantees on dimension-free computational complexity, polynomial decay of the uniform error-in-size, and power consistency for general (global and local) alternatives. To establish these results, a general theory for incomplete $U$-processes with s…
For the problems of nonparametric hypothesis testing we introduce the notion of maxisets and maxispace. We point out the maxisets of $χ^2-$tests, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients. For these tests we show that, if sequence of alternatives having given rates of convergence to hypothesis is consistent, then each altehrnative can be broken down into the sum of two parts: a function belonging to maxiset and orthogonal function. Sequence of functions belonging to maxiset is con…
This paper presents uniform convergence rates for kernel regression estimators, in the setting of a structural nonlinear cointegrating regression model. We generalise the existing literature in three ways. First, the domain to which these rates apply is much wider than has been previously considered, and can be chosen so as to contain as large a fraction of the sample as desired in the limit. Second, our results allow the regression disturbance to be serially correlated, and cross-correlated with the regressor; previous work on this problem (of obtaining uniform rates) having been confined ent…
The use of Kolmogorov-Smirnov-type statistics for testing stochastic dominance goes back to McFadden (1989). In this paper we extend the approach of Barret and Donald (2003) to the bivariate case, without the assumption of absolute continuity for the underlying distributions. Using empirical processes theory and bootstrap techniques we obtain consis- tent nonparametric tests for bivariate first and second order stochastic dominance, over several modularity classes of test functions. This tests are in turn useful tools in applied fields such as multidimensional eco- nomic inequality as shown by…
This paper considers a class of nonparametric autoregressive models with nonstationarity. We propose a nonparametric kernel test for the conditional mean and then establish an asymptotic distribution of the proposed test. Both the setting and the results differ from earlier work on nonparametric autoregression with stationarity. In addition, we develop a new bootstrap simulation scheme for the selection of a suitable bandwidth parameter involved in the kernel test as well as the choice of a simulated critical value. The finite-sample performance of the proposed test is assessed using one simul…
In recent years, Bayesian nonparametric statistics has gathered extraordinary attention. Nonetheless, a relatively little amount of work has been expended on Bayesian nonparametric hypothesis testing. In this paper, a novel Bayesian nonparametric approach to the two-sample problem is established. Precisely, given two samples $\mathbf{X}=X_1,\ldots,X_{m_1}$ $\overset {i.i.d.} \sim F$ and $\mathbf{Y}=Y_1,\ldots,Y_{m_2} \overset {i.i.d.} \sim G$, with $F$ and $G$ being unknown continuous cumulative distribution functions, we wish to test the null hypothesis $\mathcal{H}_0:~F=G$. The method is bas…
In this paper we propose a new test for the hypothesis of a constant coefficient of variation in the common nonparametric regression model. The test is based on an estimate of the $L^2$-distance between the square of the regression function and variance function. We prove asymptotic normality of a standardized estimate of this distance under the null hypothesis and fixed alternatives and the finite sample properties of a corresponding bootstrap test are investigated by means of a simulation study. The results are applicable to stationary processes with the common mixing conditions and are used…
We introduce new test statistic to test the independence of two multi-dimensional random variables. Based on the $L_1$-distance and the historgram density estimation method, the test is compared via Bahadur relative efficiency to several tests available in the literature. It arises that our test reaches better performances than a number of usual tests among whom we cite the Kolmogorov-Smirnov test. Beforehand, large deviation result is stated for the associated statistic. The local asymptotic optimality relative to the test is also studied.
We consider a nonparametric heteroscedastic time series regression model and suggest testing procedures to detect changes in the conditional variance function. The tests are based on a sequential marked empirical process and thus combine classical CUSUM tests with marked empirical process approaches known from goodness-of-fit testing. The tests are consistent against general alternatives of a change in the conditional variance function, a feature that classical CUSUM tests are lacking. We derive a simple limiting distribution and in the case of univariate covariates even obtain asymptotically …
We derive a nonparametric test for constant beta over a fixed time interval from high-frequency observations of a bivariate \Ito semimartingale. Beta is defined as the ratio of the spot continuous covariation between an asset and a risk factor and the spot continuous variation of the latter. The test is based on the asymptotic behavior of the covariation between the risk factor and an estimate of the residual component of the asset, that is orthogonal (in martingale sense) to the risk factor, over blocks with asymptotically shrinking time span. Rate optimality of the test over smoothness class…
We analytically investigate size and power properties of a popular family of procedures for testing linear restrictions on the coefficient vector in a linear regression model with temporally dependent errors. The tests considered are autocorrelation-corrected F-type tests based on prewhitened nonparametric covariance estimators that possibly incorporate a data-dependent bandwidth parameter, e.g., estimators as considered in Andrews and Monahan (1992), Newey and West (1994), or Rho and Shao (2013). For design matrices that are generic in a measure theoretic sense we prove that these tests eithe…
We propose two nonparametric tests for investigating the pathwise properties of a signal modeled as the sum of a Lévy process and a Brownian semimartingale. Using a nonparametric threshold estimator for the continuous component of the quadratic variation, we design a test for the presence of a continuous martingale component in the process and a test for establishing whether the jumps have finite or infinite variation, based on observations on a discrete-time grid. We evaluate the performance of our tests using simulations of various stochastic models and use the tests to investigate the fine …
We introduce new nonparametric predictors for homogeneous pooled data in the context of group testing for rare abnormalities and show that they achieve optimal rates of convergence. In particular, when the level of pooling is moderate, then despite the cost savings, the method enjoys the same convergence rate as in the case of no pooling. In the setting of "over-pooling" the convergence rate differs from that of an optimal estimator by no more than a logarithmic factor. Our approach improves on the random-pooling nonparametric predictor, which is currently the only nonparametric method availab…
We consider the detection problem of a two-dimensional function from noisy observations of its integrals over lines. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are non-adaptive. We also construct a minimax rate-optimal adaptive test of rather simple structure.
This paper considers the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct two new classes of nonparametric test statistics based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, wherein the nonparametric maximum likelihood approach is used to estimate t…
This paper examines the problem of nonparametric testing for the no-effect of a random covariate (or predictor) on a functional response. This means testing whether the conditional expectation of the response given the covariate is almost surely zero or not, without imposing any model relating response and covariate. The covariate could be univariate, multivariate or functional. Our test statistic is a quadratic form involving univariate nearest neighbor smoothing and the asymptotic critical values are given by the standard normal law. When the covariate is multidimensional or functional, a pr…
We propose a novel nonparametric approach for estimating the location of block boundaries (change-points) of non-overlapping blocks in a random symmetric matrix which consists of random variables having their distribution changing from one block to the other. Our method is based on a nonparametric two-sample homogeneity test for matrices that we extend to the more general case of several groups. We first provide some theoretical results for the two associated test statistics and we explain how to derive change-point location estimators. Then, some numerical experiments are given in order to su…
We consider the problem of testing whether two finite-dimensional random dot product graphs have generating latent positions that are independently drawn from the same distribution, or distributions that are related via scaling or projection. We propose a test statistic that is a kernel-based function of the adjacency spectral embedding for each graph. We obtain a limiting distribution for our test statistic under the null and we show that our test procedure is consistent across a broad range of alternatives.
The problem of curve registration appears in many different areas of applications ranging from neuroscience to road traffic modeling. In the present work, we propose a nonparametric testing framework in which we develop a generalized likelihood ratio test to perform curve registration. We first prove that, under the null hypothesis, the resulting test statistic is asymptotically distributed as a chi-squared random variable. This result, often referred to as Wilks' phenomenon, provides a natural threshold for the test of a prescribed asymptotic significance level and a natural measure of lack-o…
For a continuous random variable $Z$, testing conditional independence $X \perp\!\!\!\perp Y |Z$ is known to be a particularly hard problem. It constitutes a key ingredient of many constraint-based causal discovery algorithms. These algorithms are often applied to datasets containing binary variables, which indicate the 'context' of the observations, e.g. a control or treatment group within an experiment. In these settings, conditional independence testing with $X$ or $Y$ binary (and the other continuous) is paramount to the performance of the causal discovery algorithm. To our knowledge no no…
Nonparametric two-sample hypothesis testing for low-rank random graphs of differing sizes
本文提出了一种非参数两样本检验方法，用于测试不同规模低秩随机图的分布相等性，通过最大均值差异和最优传输估计实现。
The basic motivation and primary goal of this paper is a qualitative evaluation of the performance of a new weighted statistic for a nonparametric test for stochastic dominance based on two samples, which was introduced in Ledwina and Wyłupek (2012). For this purpose, we elaborate a useful variant of Kallenberg's notion of intermediate efficiency. This variant is general enough to be applicable to other nonparametric problems. We provide a formal defnition of the proposed variant of intermediate efficiency, describe the technical tools used in its calculation, and provide proofs of related asy…
We address the issue of lack-of-fit testing for a parametric quantile regression. We propose a simple test that involves one-dimensional kernel smoothing, so that the rate at which it detects local alternatives is independent of the number of covariates. The test has asymptotically gaussian critical values, and wild bootstrap can be applied to obtain more accurate ones in small samples. Our procedure appears to be competitive with existing ones in simulations. We illustrate the usefulness of our test on birthweight data.
For $χ^2-$tests with increasing number of cells, Cramer-von Mises tests, tests generated $\mathbb{L}_2$- norms of kernel estimators and tests generated quadratic forms of estimators of Fourier coefficients, we find necessary and sufficient conditions of consistency and inconsistency for sequences of alternatives having a given rate of convergence to hypothesis in $\mathbb{L}_2$-norm. We provide transparent interpretations of these conditions allowing to understand the structure of such consistent sequences. For problem of signal detection in Gaussian white noise we show that, if set of alterna…
In a functional setting, we propose two test statistics to highlight the Poisson nature of a Cox process when n copies of the process are available. Our approach involves a comparison of the empirical mean and the empirical variance of the functional data and can be seen as an extended version of a classical overdispersion test for counting data. The limiting distributions of our statistics are derived using a functional central limit theorem for c`adl`ag martingales. We also study the asymptotic power of our tests under some local alternatives. Our procedure is easily implementable and does n…
We propose several statistics to test the Markov hypothesis for $β$-mixing stationary processes sampled at discrete time intervals. Our tests are based on the Chapman--Kolmogorov equation. We establish the asymptotic null distributions of the proposed test statistics, showing that Wilks's phenomenon holds. We compute the power of the test and provide simulations to investigate the finite sample performance of the test statistics when the null model is a diffusion process, with alternatives consisting of models with a stochastic mean reversion level, stochastic volatility and jumps.
This paper studies the problem of nonparametric testing for the effect of a random functional covariate on a real-valued error term. The covariate takes values in $L^2[0,1]$, the Hilbert space of the square-integrable real-valued functions on the unit interval. The error term could be directly observed as a response or \emph{estimated} from a functional parametric model, like for instance the functional linear regression. Our test is based on the remark that checking the no-effect of the functional covariate is equivalent to checking the nullity of the conditional expectation of the error term…
A note on the distribution of the partial correlation coefficient with nonparametrically estimated marginal regressions
本文提出一种基于样本偏相关系数并采用非参数边际回归的置换检验，证明其在大样本和小样本下均表现良好，用于条件独立性的非参数检验。
In this paper, we propose novel, fully Bayesian non-parametric tests for one-sample and two-sample multivariate location problems. We model the underlying distribution using a Dirichlet process prior, and develop a testing procedure based on the posterior credible region for the spatial median functional of the distribution. For the one-sample problem, we fail to reject the null hypothesis if the credible set contains the null value. For the two-sample problem, we form a credible set for the difference of the spatial medians of the two samples and we fail to reject the null hypothesis of equal…
Efficiency of nonparametric e-tests
本文针对对称性非参数假设，构造三种常用非参数检验的 e-value 版本，并提出对应于 Pitman 渐近相对效率的评价概念以分析其效率。
We formulate nonparametric and semiparametric hypothesis testing of multivariate stationary linear time series in a unified fashion and propose new test statistics based on estimators of the spectral density matrix. The limiting distributions of these test statistics under null hypotheses are always normal distributions, and they can be implemented easily for practical use. If null hypotheses are false, as the sample size goes to infinity, they diverge to infinity and consequently are consistent tests for any alternative. The approach can be applied to various null hypotheses such as the indep…
The purpose of this paper is to investigate the asymptotic behavior of the Durbin-Watson statistic for the stable $p-$order autoregressive process when the driven noise is given by a first-order autoregressive process. It is an extension of the previous work of Bercu and Proïa devoted to the particular case $p=1$. We establish the almost sure convergence and the asymptotic normality for both the least squares estimator of the unknown vector parameter of the autoregressive process as well as for the serial correlation estimator associated with the driven noise. In addition, the almost sure rate…
Encompassing Tests for Nonparametric Regressions
本文通过L2距离构建非参数回归包容性检验框架，提出完全非参数检验程序，并利用wild bootstrap和数据驱动带宽解决核回归中的带宽选择与小偏差问题，在小样本和中等样本下表现良好。
We develop a nonparametric extension of the sequential generalized likelihood ratio (GLR) test and corresponding time-uniform confidence sequences for the mean of a univariate distribution. By utilizing a geometric interpretation of the GLR statistic, we derive a simple analytic upper bound on the probability that it exceeds any prespecified boundary; these are intractable to approximate via simulations due to infinite horizon of the tests and the composite nonparametric nulls under consideration. Using time-uniform boundary-crossing inequalities, we carry out a unified nonasymptotic analysis …
Several works have been undertaken in the context of proportional reversed hazard rate (PRHR) since last few decades. But any specific statistical methodology for the PRHR hypothesis is absent in the literature. In this paper, a two-sample nonparametric test based on two independent samples is proposed for verifying the PRHR assumption. Based on a consistent U-statistic three statistical methodologies have been developed exploiting U-statistics theory, jackknife empirical likelihood and adjusted jackknife empirical likelihood method. A simulation study has been performed to assess the merit of…
This paper studies the problem of testing whether a function is monotone from a nonparametric Bayesian perspective. Two new families of tests are constructed. The first uses constrained smoothing splines, together with a hierarchical stochastic-process prior that explicitly controls the prior probability of monotonicity. The second uses regression splines, together with two proposals for the prior over the regression coefficients. The finite-sample performance of the tests is shown via simulation to improve upon existing frequentist and Bayesian methods. The asymptotic properties of the Bayes …
Nonparametric two sample or homogeneity testing is a decision theoretic problem that involves identifying differences between two random variables without making parametric assumptions about their underlying distributions. The literature is old and rich, with a wide variety of statistics having being intelligently designed and analyzed, both for the unidimensional and the multivariate setting. Our contribution is to tie together many of these tests, drawing connections between seemingly very different statistics. In this work, our central object is the Wasserstein distance, as we form a chain …
We present a novel family of nonparametric omnibus tests of the hypothesis that two unknown but estimable functions are equal in distribution when applied to the observed data structure. We developed these tests, which represent a generalization of the maximum mean discrepancy tests described in Gretton et al. [2006], using recent developments from the higher-order pathwise differentiability literature. Despite their complex derivation, the associated test statistics can be expressed rather simply as U-statistics. We study the asymptotic behavior of the proposed tests under the null hypothesis…
We investigate hypothesis testing in nonparametric additive models estimated using simplified smooth backfitting (Huang and Yu, Journal of Computational and Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth backfitting achieves oracle properties under regularity conditions and provides closed-form expressions of the estimators that are useful for deriving asymptotic properties. We develop a generalized likelihood ratio (GLR) and a loss function (LF) based testing framework for inference. Under the null hypothesis, both the GLR and LF tests have asymptotically rescaled ch…
Nonparametric goodness of fit tests for Pareto type-I distribution with complete and censored data
基于Steins型恒等式与不动点特征，针对完整和右删失数据提出了两种Pareto I型分布的新非参数拟合优度检验，模拟与真实数据表明其功效优于现有方法。
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