Marked Empirical Process Research Articles

Testing for structural change of AR model to threshold AR model

The purpose of this article is to develop the likelihood ratio test for the structural change of an AR model to a threshold AR model. It is shown that the log-likelihood ratio test converges to the maxima of a two-parameter Gaussian process in distribution. This limiting distribution is novel and we tabulate the critical values. Some simulations are carried out to examine the finite-sample performance of this test statistic. This article also includes a weak convergence of a two-parameter marked empirical process, which is of independent interest.

Data-driven smooth tests for the martingale difference hypothesis

A general method for testing the martingale difference hypothesis is proposed. The new tests are data-driven smooth tests based on the principal components of certain marked empirical processes that are asymptotically distribution-free, with critical values that are already tabulated. The smooth tests are shown to be optimal in a semiparametric sense discussed in the paper, and they are robust to conditional heteroscedasticity of unknown form. A simulation study shows that the data-driven smooth tests perform very well for a wide range of realistic alternatives and have more power than omnibus and other competing tests. Finally, an application to the S&P 500 stock index and some of its components highlights the merits of our approach. The paper also contains a new weak convergence theorem that is of independent interest.

Scan statistics of Lévy noises and marked empirical processes

Let n points be chosen independently and uniformly in the unit cube [0,1]d, and suppose that each point is supplied with a mark, the marks being independent and identically distributed random variables independent of the location of the points. To each cube R contained in [0,1]d we associate its score defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of taken over all cubes R contained in [0,1]d. We show that if the marks are nonlattice random variables with finite exponential moments, having negative mean and assuming positive values with nonzero probability, then the appropriately normalized distribution of the scan statistic converges as n → ∞ to the Gumbel distribution. We also prove a corresponding result for the scan statistic of a Lévy noise with negative mean. The more elementary cases of zero and positive mean are also considered.

Scan statistics of Lévy noises and marked empirical processes

Let n points be chosen independently and uniformly in the unit cube [0,1] d , and suppose that each point is supplied with a mark, the marks being independent and identically distributed random variables independent of the location of the points. To each cube R contained in [0,1] d we associate its score defined as the sum of marks of all points contained in R. The scan statistic is defined as the maximum of taken over all cubes R contained in [0,1] d . We show that if the marks are nonlattice random variables with finite exponential moments, having negative mean and assuming positive values with nonzero probability, then the appropriately normalized distribution of the scan statistic converges as n → ∞ to the Gumbel distribution. We also prove a corresponding result for the scan statistic of a Lévy noise with negative mean. The more elementary cases of zero and positive mean are also considered.

Argmax-stable marked empirical processes

We consider a marked empirical process corresponding to a sample X 1 , … , X n of independent and identically distributed random variables and exchangeable random marks C 1 , … , C n . If the sum of all marks is equal to zero, then there exists a certain order statistic with random index, which is a maximizing point of the process. It is shown that for each finite sample size n ∈ N this point of maximum has the same distribution as X 1 (argmax-stability). As an application we derive the exact distribution of an estimator for the discontinuity point in a regression function.

A bootstrap test for the comparison of nonlinear time series

The difference between the regression functions of two stationary conditional heteroskedastic autoregressive time series is tested. The functions can be equal, or shifted, under the null hypothesis. Local linear estimation of the regression function results in observable residuals. Bootstrap residuals lead to a marked empirical process as test statistic and a Kolmogorov–Smirnov version is applied. The simulation study for linear, exponential or trigonometric regression functions with homoskedastic or heteroskedastic errors finds the rejection probability under the null hypothesis to be near the level. Comparing series with different combinations of linear, exponential and trigonometric functions, the rejection probability under the alternative yields mixed results.

The marked empirical process to test a general AR-ARCH against an other general AR-ARCH when the random vectors are nonstationary and absolutely regular

In this Note, we study a procedure on goodness-of-fit testing for nonlinear time-series models against a large class of alternatives under nonstationarity and absolute regularity. For that, we define a marked empirical process based on residuals which converges in distribution to a Gaussian process with respect to the Skorohod topology. This method was first introduced by Stute (1997) and then widely developed by Ngatchou-Wandji (2002, 2005, 2008) [1–3] under more general conditions. Applications to general AR-ARCH models are given. To cite this article: M. Harel, E. Elharfaoui, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

Goodness of fit test for ergodic diffusion processes

A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypothesis and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact rejection regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any simple fixed alternative.

Empirical likelihood based testing for regression

Consider a random vector $(X,Y)$ and let $m(x)=E(Y|X=x)$. We are interested in testing $H_0:m\in {\cal M}_{\Theta,{\cal G}}=\{\gamma(\cdot,\theta,g):\theta \in \Theta,g\in {\cal G}\}$ for some known function $\gamma$, some compact set $\Theta \subset $IR$^p$ and some function set ${\cal G}$ of real valued functions. Specific examples of this general hypothesis include testing for a parametric regression model, a generalized linear model, a partial linear model, a single index model, but also the selection of explanatory variables can be considered as a special case of this hypothesis. To test this null hypothesis, we make use of the so-called marked empirical process introduced by \citeD and studied by \citeSt for the particular case of parametric regression, in combination with the modern technique of empirical likelihood theory in order to obtain a powerful testing procedure. The asymptotic validity of the proposed test is established, and its finite sample performance is compared with other existing tests by means of a simulation study.

Regression model checking with Berkson measurement errors

This paper discusses asymptotically distribution free tests for the lack-of-fit of a parametric regression model in the Berkson measurement error model. These tests are based on a martingale transform of a certain marked empirical process of calibrated residuals. A simulation study is included to assess the effect of measurement error on the proposed test. It is observed that empirical level is more stable across the chosen measurement error variances when fitting a linear model compared to when fitting a nonlinear model, while, in both cases, the empirical power decreases as this error variance increases, against all chosen alternatives.

Weak convergence of non-stationary multivariate marked processes with applications to martingale testing

This paper establishes the weak convergence of a class of marked empirical processes of possibly non-stationary and/or non-ergodic multivariate time series sequences under martingale conditions. The assumptions involved are similar to those in Brown's martingale central limit theorem. In particular, no mixing conditions are imposed. As an application, we propose a test statistic for the martingale hypothesis and we derive its asymptotic null distribution. Finally, a Monte Carlo study shows that the asymptotic results provide good approximations for small and moderate sample sizes. An application to the S&P 500 is also considered.

A CONSISTENT DIAGNOSTIC TEST FOR REGRESSION MODELS USING PROJECTIONS

This paper proposes a consistent test for the goodness-of-fit of parametric regression models that overcomes two important problems of the existing tests, namely, the poor empirical power and size performance of the tests due to the curse of dimensionality and the subjective choice of parameters such as bandwidths, kernels, and integrating measures. We overcome these problems by using a residual marked empirical process based on projections (RMPP). We study the asymptotic null distribution of the test statistic, and we show that our test is able to detect local alternatives converging to the null at the parametric rate. It turns out that the asymptotic null distribution of the test statistic depends on the data generating process, and so a bootstrap procedure is considered. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. For completeness, we propose a new minimum distance estimator constructed through the same RMPP as in the testing procedure. Therefore, the new estimator inherits all the good properties of the new test. We establish the consistency and asymptotic normality of the new minimum distance estimator. Finally, we present some Monte Carlo evidence that our testing procedure can play a valuable role in econometric regression modeling.The author thanks Carlos Velasco and Miguel A. Delgado for useful comments. The paper has also benefited from the comments of two referees and the co-editor. This research was funded by the Spanish Ministry of Education and Science reference number SEJ2004-04583/ECON and by the Universidad de Navarra reference number 16037001.

Bootstrap Approximations in Model Checks for Binary Data

Consider a binary regression model in which the conditional expectation of a binary variable given an explanatory variable belongs to a parametric family. To check whether a sequence of independent and identically distributed observations belongs to such a parametric family, we use Kolmogorov–Smirnov and Cramér–von Mises type tests based on a marked empirical process introduced by Stute. We propose and study a new resampling scheme for a bootstrap in this setup to approximate critical values for these tests. We also apply this approach to simulated and real data. In the latter case we check some parametric models that are used to analyze right-censored lifetime data under a semiparametric random censorship model.

Consistent Tests of Conditional Moment Restrictions

We propose two classes of consistent tests in parametric econometric models defi ned through multiple conditional moment restrictions. The fi rst type of tests relies on nonparametric estimation, while the second relies on a functional of a marked empirical process. For both tests, a simulation procedure for obtaining critical values is shown to be asymptotically valid. Finite sample performances of the tests are investigated by means of several Monte-Carlo experiments. 234

Testing for a linear MA model against threshold MA models

This paper investigates the (conditional) quasi-likelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has nontrivial asymptotic power. The results are based on a new weak convergence of a linear marked empirical process, which is independently of interest. This paper also gives an invertible expansion of the threshold MA models.

Goodness-of-fit testing in interval censoring case 1

In the interval censoring case 1, an event occurrence time is unobservable, but one observes an inspection time and whether the event has occurred prior to this time or not. The focus here is to provide tests of goodness-of-fit hypothesis pertaining to the distribution of the event occurrence time. The proposed tests are based on certain marked empirical processes for testing a simple hypothesis and the Stute–Thies–Zhu transformation of such a process for fitting a parametric family of distributions. These tests are asymptotically distribution-free, consistent against a large class of fixed alternatives and have nontrivial asymptotic power against a large class of local alternatives. A simulation study is included to exhibit the finite sample level preservation and power behavior.

Nonparametric comparison of regression curves: an empirical process approach

We propose a new test for the comparison of two regression curves that is based on a difference of two marked empirical processes based on residuals. The large sample behavior of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature, the new procedure is applicable in the case of different design points and heteroscedasticity. Moreover, it is demonstrated that the proposed test detects continuous alternatives converging to the null at a rate $N^{-1/2}$ and that, in contrast to all other available procedures based on marked empirical processes, the new test allows the optimal choice of bandwidths for curve estimation (e.g., $N^{-1/5}$ in the case of twice differentiable regression functions). As a by-product we explain the problems of a related test proposed by Kulasekera [J. Amer. Statist. Assoc. 90 (1995) 1085-1093] and Kulasekera and Wang [J. Amer. Statist. Assoc. 92 (1997) 500-511] with respect to accuracy in the approximation of the level. These difficulties mainly originate from the comparison with the quantiles of an inappropriate limit distribution. A simulation study is conducted to investigate the finite sample properties of a wild bootstrap version of the new test and to compare it with the so far available procedures. Finally, heteroscedastic data is analyzed in order to demonstrate the benefits of the new test compared to the so far available procedures which require homoscedasticity.

Une méthode semi-paramétrique pour tester un modèle de régression

The purpose is to test the hypothesis H 0 that a regression model is parametric and belongs to a given family versus the alternative H 1 approaches the hypothesis from a specific direction at the rate n −1/2. For that, we consider an empirical process such that under H 0 this process depends of a parameter θ 0. First, we start by estimating the parameter and we prove that the empirical process converges in distribution to a certain Gaussian process when the parameter is replaced by its estimator θ n . However it is important to check the impact of an alternative approaching H 0 from a specific direction ( at the rate n 1/2). For that, we need tests which are consistent on the whole of H 1. Our idea is to use a marked empirical process based on residuals which converges in distribution to a Gaussian process. To cite this article: M. Harel, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Weak convergence of some marked empirical processes: Application to testing heteroscedasticity

We present a procedure for testing the goodness-of-fit of the conditional variance function of a Markov model of order 1, under stationarity and ergodicity. The autoregressive parameter, the distribution of the noise and the stationary distribution of the observations are assumed to be unknown. Under the null hypothesis H 0 that the conditional variance function belongs to a class of parametric functions, we define an estimator {\ ilde{\ heta}}_{n} of 0 , the assumed true parameter, and we establish its consistency and asymptotic normality. We define a marked empirical process A n (), for which we state and prove a functional limit theorem under H 0 . The asymptotic behavior of this process is studied under fixed alternatives H 1 . Based on the process A n (), a chi-squared test is derived. Simulation experiments show that the test is powerful against some heteroscedastic time series models.

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.