Two‐Sample Test Against One‐Sided Alternatives

Abstract. This paper proposes, implements and investigates a new non‐parametric two‐sample test for detecting stochastic dominance. We pose the question of detecting the stochastic dominance in a non‐standard way. This is motivated by existing evidence showing that standard formulations and pertaining procedures may lead to serious errors in inference. The procedure that we introduce matches testing and model selection. More precisely, we reparametrize the testing problem in terms of Fourier coefficients of well‐known comparison densities. Next, the estimated Fourier coefficients are used to form a kind of signed smooth rank statistic. In such a setting, the number of Fourier coefficients incorporated into the statistic is a smoothing parameter. We determine this parameter via some flexible selection rule. We establish the asymptotic properties of the new test under null and alternative hypotheses. The finite sample performance of the new solution is demonstrated through Monte Carlo studies and an application to a set of survival times.     

Model Checks in Inverse Regression Models with Convolution‐Type Operators

Abstract. We consider the problem of testing parametric assumptions in an inverse regression model with a convolution‐type operator. An L ₂ ‐type goodness‐of‐fit test is proposed which compares the distance between a parametric and a non‐parametric estimate of the regression function. Asymptotic normality of the corresponding test statistic is shown under the null hypothesis and under a general non‐parametric alternative with different rates of convergence in both cases. The feasibility of the proposed test is demonstrated by means of a small simulation study. In particular, the power of the test against certain types of alternative is investigated. Finally, an empirical example is provided, in which the proposed methods are applied to the determination of the shape of the luminosity profile of the elliptical galaxy NGC 5017.     

A Non‐parametric ANOVA‐type Test for Regression Curves Based on Characteristic Functions

This article studies a new procedure to test for the equality of k regression curves in a fully non‐parametric context. The test is based on the comparison of empirical estimators of the characteristic functions of the regression residuals in each population. The asymptotic behaviour of the test statistic is studied in detail. It is shown that under the null hypothesis, the distribution of the test statistic converges to a finite combination of independent chi‐squared random variables with one degree of freedom. The coefficients in this linear combination can be consistently estimated. The proposed test is able to detect contiguous alternatives converging to the null at the rate n ^{? 1 ∕ 2}. The practical performance of the test based on the asymptotic null distribution is investigated by means of simulations.     

Comparing Conditional Quantile Curves

Abstract. We consider the problem of testing the equality of J quantile curves from independent samples. A test statistic based on an L²‐distance between non‐crossing non‐parametric estimates of the quantile curves from the individual samples is proposed. Asymptotic normality of this statistic is established under the null hypothesis, local and fixed alternatives, and the finite sample properties of a bootstrap‐based version of this test statistic are investigated by means of a simulation study.     

Statistical inference for restricted partially linear varying coefficient errors-in-variables models

As a useful extension of partially linear models and varying coefficient models, the partially linear varying coefficient model is useful in statistical modelling. This paper considers statistical inference for the semiparametric model when the covariates in the linear part are measured with additive error and some additional linear restrictions on the parametric component are available. We propose a restricted modified profile least-squares estimator for the parametric component, and prove the asymptotic normality of the proposed estimator. To test hypotheses on the parametric component, we propose a test statistic based on the difference between the corrected residual sums of squares under the null and alterative hypotheses, and show that its limiting distribution is a weighted sum of independent chi-square distributions. We also develop an adjusted test statistic, which has an asymptotically standard chi-squared distribution. Some simulation studies are conducted to illustrate our approaches.     

Goodness‐of‐Fit Test for Monotone Functions

Abstract. In this article, we develop a test for the null hypothesis that a real‐valued function belongs to a given parametric set against the non‐parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right‐censoring model with monotone hazard rate. The criterion for testing is an ‐distance between a Grenander‐type non‐parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test.     

Detection and Estimation of Jump Points in Non parametric Regression Function with AR(1) Noise

In this paper, we propose a method based on wavelet analysis to detect and estimate jump points in non parametric regression function. This method is applied to AR(1) noise process under random design. First, the test statistics are constructed on the empirical wavelet coefficients. Then, under the null hypothesis, the critical values of test statistics are obtained. Under the alternative, the consistency of the test is proved. Afterward, the rate of convergence, the estimators of the number, and locations of change points are given theoretically. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real datasets.     

Non‐parametric Regression Tests Using Dimension Reduction Techniques

Abstract. Testing for parametric structure is an important issue in non‐parametric regression analysis. A standard approach is to measure the distance between a parametric and a non‐parametric fit with a squared deviation measure. These tests inherit the curse of dimensionality from the non‐parametric estimator. This results in a loss of power in finite samples and against local alternatives. This article proposes to circumvent the curse of dimensionality by projecting the residuals under the null hypothesis onto the space of additive functions. To estimate this projection, the smooth backfitting estimator is used. The asymptotic behaviour of the test statistic is derived and the consistency of a wild bootstrap procedure is established. The finite sample properties are investigated in a simulation study.     

Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.     

A Lack-of-Fit Test for Heteroscedastic Regression Models via Cosine-Series Smoothers

In this paper, a test is derived to assess the validity of heteroscedastic nonlinear regression models by a non‐parametric cosine regression method. For order selection, the paper proposes a data‐driven method that uses the parametric null model optimal order. This method yields a test that is asymptotically normally distributed under the null hypothesis and is consistent against any fixed alternative. Simulation studies that test the lack of fit of a generalized linear model are conducted to compare the performance of the proposed test with that of an existing non‐parametric kernel test. A dataset of esterase levels is used to demonstrate the proposed method in practice.     

ON TESTING THE GOODNESS-OF-FIT OF NONLINEAR HETEROSCEDASTIC REGRESSION MODELS

For testing the adequacy of a parametric model in regression, various test statistics can be constructed on the basis of a marked empirical process of residuals. By using a discretized version of the decomposition of the corresponding Gaussian limiting process into its principal components, we obtain a test statistic with an asymptotic chi-squared distribution under the null hypothesis. We investigate the consistency of this test statistic and of the estimators needed to compute it. Numerical experiments indicate that the distributional approximations already work for small to moderate sample sizes and reveal that the test has good power properties against a variety of alternatives. The test has a simple implementation. We present an application to a real-data example for testing the adequacy of a possible heteroscedastic exponential model.     

On empirical likelihood for semiparametric two-sample density ratio models

We consider estimation and test problems for some semiparametric two-sample density ratio models. The profile empirical likelihood (EL) poses an irregularity problem under the null hypothesis that the laws of the two samples are equal. We show that a dual form of the profile EL is well defined even under the null hypothesis. A statistical test, based on the dual form of the EL ratio statistic (ELRS), is then proposed. We give an interpretation for the dual form of the ELRS through φ$\phi$-divergences and duality techniques. The asymptotic properties of the test statistic are presented both under the null and the alternative hypotheses, and approximation of the power function of the test is deduced.     

The power of the circular cone test: A noncentral chi-bar-squared distribution

Pincus (1975) derived the null distribution of the likelihood-ratio test statistic for testing that the mean vector of a multivariate normal distribution is zero against the alternative that the mean vector lies in a circular cone. Under the null hypothesis, the likelihood-ratio test statistic has a chi-bar-squared distribution. We extend the results of Pincus by deriving the distribution of the likelihood-ratio test statistic under the alternative hypothesis. In a special case, the distribution is a “noncentral chi-bar-squared” distribution. To our knowledge, this is the first order-restricted testing problem for which the relationship between the null and alternative distributions of the test statistic is similar to the relationship in the linear-model setting. That is, the distribution of the likelihood-ratio test has a central form of a distribution under the null hypothesis and a noncentral form of the same distribution under the alternative.     

A CONSISTENT MODEL SPECIFICATION TEST BASED ON THE KERNEL SUM OF SQUARES OF RESIDUALS

This paper constructs a consistent model specification test based on the difference between the nonparametric kernel sum of squares of residuals and the sum of squares of residuals from a parametric null model. We establish the asymptotic normality of the proposed test statistic under the null hypothesis of correct parametric specification and show that the wild bootstrap method can be used to approximate the null distribution of the test statistic. Results from a small simulation study are reported to examine the finite sample performance of the proposed tests.     

Robust durbin's and portmanteau goodness-of-fit test statistics and their limit distributions

Robust analogue of Durbin's(1970)statistic is derived and its limiting distribution is obtained under both null and alternative hypotheses. Also, robust version of the portmanteau goodness-of-fit test statistic for AR(p)model is given and the asymptotic distribution is derived.     

Testing for the Null Hypothesis of Cointegration with a Structural Break

In this paper we propose residual-based tests for the null hypothesis of cointegration with a structural break against the alternative of no cointegration. The Lagrange Multiplier (LM) test is proposed and its limiting distribution is obtained for the case in which the timing of a structural break is known. Then the test statistic is extended to deal with a structural break of unknown timing. The test statistic, a plug-in version of the test statistic for known timing, replaces the true break point by the estimated one. We show the limiting properties of the test statistic under the null as well as the alternative. Critical values are calculated for the tests by simulation methods. Finite-sample simulations show that the empirical size of the test is close to the nominal one unless the regression error is very persistent and that the test rejects the null when no cointegrating relationship with a structural break is present. We provide empirical examples based on the present-value model, the term structure model, and the money-output relationship model.     

Testing for the Null Hypothesis of Cointegration with a Structural Break

In this paper we propose residual-based tests for the null hypothesis of cointegration with a structural break against the alternative of no cointegration. The Lagrange Multiplier (LM) test is proposed and its limiting distribution is obtained for the case in which the timing of a structural break is known. Then the test statistic is extended to deal with a structural break of unknown timing. The test statistic, a plug-in version of the test statistic for known timing, replaces the true break point by the estimated one. We show the limiting properties of the test statistic under the null as well as the alternative. Critical values are calculated for the tests by simulation methods. Finite-sample simulations show that the empirical size of the test is close to the nominal one unless the regression error is very persistent and that the test rejects the null when no cointegrating relationship with a structural break is present. We provide empirical examples based on the present-value model, the term structure model, and the money-output relationship model.     

Changepoint Estimation in a Segmented Linear Regression via Empirical Likelihood

For a segmented regression system with an unknown changepoint over two domains of a predictor, a new empirical likelihood ratio statistic is proposed to test the null hypothesis of no change. Under the null hypothesis of no change, the proposed test statistic is shown empirically to be Gumbel distributed with robust location and scale estimators against various parameter settings and error distributions. A power analysis is conducted to illustrate the performance of the test. Under the alternative hypothesis with a changepoint, the test statistic is utilized to estimate the changepoint between the two domains. A comparison of the frequency distributions between the proposed estimator and two parametric methods indicates that the proposed method is effective in capturing the true changepoint.     

Statistical inference on partial linear additive models with distortion measurement errors

We consider statistical inference for partial linear additive models (PLAMs) when the linear covariates are measured with errors and distorted by unknown functions of commonly observable confounding variables. A semiparametric profile least squares estimation procedure is proposed to estimate unknown parameter under unrestricted and restricted conditions. Asymptotic properties for the estimators are established. To test a hypothesis on the parametric components, a test statistic based on the difference between the residual sums of squares under the null and alternative hypotheses is proposed, and we further show that its limiting distribution is a weighted sum of independent standard chi-squared distributions. A bootstrap procedure is further proposed to calculate critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analyzed for an illustration.     

A simple test for comparing regression curves versus one-sided alternatives

In this article we present a simple procedure to test for the null hypothesis of equality of two regression curves versus one-sided alternatives in a general nonparametric and heteroscedastic setup. The test is based on the comparison of the sample averages of the estimated residuals in each regression model under the null hypothesis. The test statistic has asymptotic normal distribution and can detect any local alternative of rate n^-1/2. Some simulations and an application to a data set are included.