Residual Empirical Process Research Articles

Bootstrap specification tests for dynamic conditional distribution models

This paper proposes bootstrap based tests for the specification of a given parametric conditional distribution in autoregressive time series with GARCH-type disturbances. The tests are based on an estimated residual empirical process and are implemented by parametric bootstrap. We show that the proposed tests are asymptotically valid, consistent, and have nontrivial asymptotic power against a large proportion of local alternatives. Our approach relies on non-primitive regularity conditions and certain properties of exponential almost sure convergence. The regularity conditions are shown to be satisfied by GARCH(p,q); this technique of verification is applicable to other models as well. In our Monte Carlo study, the proposed tests performed well and better than several competing tests, including the information matrix test. A real data example illustrates the testing procedure.

Parametric copula adjusted for non- and semiparametric regression

We consider a multivariate response regression model where each coordinate is described by a location-scale non- or semiparametric regression and where the dependence structure of the “noise term” is described by a parametric copula. Our goal is to estimate the associated Euclidean copula parameter, given a sample of the response and the covariate. In the absence of the copula sample, the usual oracle ranks are no longer computable. Instead, we study the normal scores estimator for the Gaussian copula and generalized pseudo-likelihood estimation for general parametric copulas, both based on residual ranks calculated from preliminary non- or semiparametric estimators of the location and scale functions. We show that the residual-based estimators are asymptotically equivalent to their oracle counterparts and provide explicit rate of convergence. Partially to serve this objective, we also study weighted convergence of the residual empirical process under the non- or semiparametric regression model.

Lack-of-fit of a parametric measurement error AR(1) model

This paper proposes an asymptotically distribution free test for fitting a parametric model to the autoregressive function in the AR(1) model in the presence of measurement error. The test is based on a martingale transform of a certain marked residual empirical process. A simulation study assessing the finite sample level and power performance of the proposed test is also included.

Fitting a pth Order Parametric Generalized Linear Autoregressive Multiplicative Error Model

This paper is concerned with the problem of fitting a generalized linear model to the conditional mean function of multiplicative error time series models. These models are particularly suited to model nonnegative time series such as the duration between trades at a stock exchange and volume transactions. The proposed test, based on a marked residual empirical process whose marks are suitably defined residuals and which jumps at the estimated indices, is shown to be asymptotically distribution free.

Bootstrap entropy test for general location-scale time series models with heteroscedasticity

ABSTRACTThis study considers a goodness-of-fit test for location-scale time series models with heteroscedasticity, including a broad class of generalized autoregressive conditional heteroscedastic-type models. In financial time series analysis, the correct identification of model innovations is crucial for further inferences in diverse applications such as risk management analysis. To implement a goodness-of-fit test, we employ the residual-based entropy test generated from the residual empirical process. Since this test often shows size distortions and is affected by parameter estimation, its bootstrap version is considered. It is shown that the bootstrap entropy test is weakly consistent, and thereby its usage is justified. A simulation study and data analysis are conducted by way of an illustration.

On Entropy Test for Conditionally Heteroscedastic Location-Scale Time Series Models

This study considers the goodness of fit test for a class of conditionally heteroscedastic location-scale time series models. For this task, we develop an entropy-type goodness of fit test based on residuals. To examine the asymptotic behavior of the test, we first investigate the asymptotic property of the residual empirical process and then derive the limiting null distribution of the entropy test.

Microarray Analysis Using Rank Order Statistics for ARCH Residual Empirical Process

Statistical two-group comparisons are widely used to identify the significant differentially expressed (DE) signatures against a therapy response for microarray data analysis. We applied a rank order statistics based on an Autoregressive Conditional Heteroskedasticity (ARCH) residual empirical process to DE analysis. This approach was considered for simulation data and publicly available datasets, and was compared with two-group comparison by original data and Auto-regressive (AR) residual. The significant DE genes by the ARCH and AR residuals were reduced by about 20% - 30% to these genes by the original data. Almost 100% of the genes by ARCH are covered by the genes by the original data unlike the genes by AR residuals. GO enrichment and Pathway analyses indicate the consistent biological characteristics between genes by ARCH residuals and original data. ARCH residuals array data might contribute to refining the number of significant DE genes to detect the biological feature as well as ordinal microarray data.

Residual Empirical Processes and Weighted Sums for Time‐Varying Processes with Applications to Testing for Homoscedasticity

In the context of heteroscedastic time‐varying autoregressive (AR)‐process we study the estimation of the error/innovation distributions. Our study reveals that the non‐parametric estimation of the AR parameter functions has a negligible asymptotic effect on the estimation of the empirical distribution of the residuals even though the AR parameter functions are estimated non‐parametrically. The derivation of these results involves the study of both function‐indexed sequential residual empirical processes and weighted sum processes. Exponential inequalities and weak convergence results are derived.As an application of our results we discuss testing for the constancy of the variance function, which in special cases corresponds to testing for stationarity.

A goodness-of-fit test for marginal distribution of linear random fields with long memory

This paper addresses the problem of fitting a known distribution function to the marginal distribution of a stationary long memory moving average random field observed on increasing \(\nu \)-dimensional “cubic” domains when its mean \(\mu \) and scale \(\sigma \) are known or unknown. Using two suitable estimators of \(\mu \) and a classical estimate of \(\sigma \), a modification of the Kolmogorov–Smirnov statistic is defined based on the residual empirical process and having a Cauchy-type limit distribution, independent of \(\mu ,\sigma \) and the long memory parameter d. Based on this result, a simple goodness-of-fit test for the marginal distribution is constructed, which does not require the estimation of d or any other underlying nuisance parameters. The result is new even for the case of time series, i.e., when \(\nu =1\). Findings of a simulation study investigating the finite sample behavior of size and power of the proposed test is also included in this paper.

On the residual empirical process based on the ALASSO in high dimensions and its functional oracle property

This paper considers post variable-selection inference in a high dimensional penalized regression model based on the ALASSO method of Zou (2006). It is shown that under suitable sparsity conditions, the residual empirical process based on the ALASSO provides valid inference methodology in very high dimensional regression problems where conventional methods fail. It is also shown that the ALASSO based residual empirical process satisfies a functional oracle property, i.e., in addition to selecting the set of relevant variables with probability tending to one, the ALASSO based residual empirical process converges to the same limiting Gaussian process as the OLS based residual empirical process under the oracle. The functional oracle property is critically exploited to construct asymptotically valid confidence bands for the error distribution function and prediction intervals for unobserved values of the response variable in the high dimensional set up, where traditional non-penalized methods are known to fail. Simulation results are presented illustrating finite sample performance of the proposed methodology.

Entropy test and residual empirical process for autoregressive conditional duration models

In this paper, we study the entropy test for the goodness of fit test in (nonlinear) autoregressive conditional duration (ACD) models. To implement a test, we first explore the null limiting distribution of the residual empirical process from ACD models and verify that it has an asymptotic expansion form that consists of the true empirical process and extra terms yielded by parameter estimation. Then, we show that under regularity conditions, the proposed entropy test approximately follows a distribution that is free from the parameter estimation. For illustration, a simulation study and real data analysis are conducted. In the implementation of the test, a parametric bootstrap method is employed.

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies $L^{p}$ convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.

Residual empirical processes and qualitatively robust GM-tests in autoregression

The local qualitative robustness of GM-tests against outliers in the autoregression model is studied in the paper. A local scheme of data contamination by independent outliers with the intensity O(n−1/2) is considered. The qualitative robustness in terms of power equicontinuity is obtained. The GM-tests asymptotically optimal in the maximin sense are constructed.

Empirical process of residuals for regression models with long memory errors

Abstract We consider the residual empirical process in random design regression with long memory. We establish its limiting behaviour, showing that its rates of convergence are different from the rates of convergence for the empirical process based on (unobserved) errors.

Residual empirical processes and their application to GM-testing for the autoregression order

In the paper a new nonparametric generalized M-test for hypotheses about the order of linear autoregression AR(p) is constructed. We also establish robustness of this test in the model of data contamination by independent additive outliers with intensity O(n−1/2). Robustness is formulated in terms of limiting power equicontinuity. Test statistics are constructed with the help of residual empirical processes. We establish the asymptotic uniform linearity of these processes in the defined contamination model.

Goodness-of-fit tests for long memory moving average marginal density

This paper addresses the problem of fitting a known density to the marginal error density of a stationary long memory moving average process when its mean is known and unknown. In the case of unknown mean, when mean is estimated by the sample mean, the first order difference between the residual empirical and null distribution functions is known to be asymptotically degenerate at zero, and hence can not be used to fit a distribution up to an unknown mean. In this paper we show that by using a suitable class of estimators of the mean, this first order degeneracy does not occur. We also investigate the large sample behavior of tests based on an integrated square difference between kernel type error density estimators and the expected value of the error density estimator based on errors. The asymptotic null distributions of suitably standardized test statistics are shown to be chi-square with one degree of freedom in both cases of the known and unknown mean. In addition, we discuss the consistency and asymptotic power against local alternatives of the density estimator based test in the case of known mean. A finite sample simulation study of the test based on residual empirical process is also included.

A divergence test for autoregressive time series models

In this paper, we study the normality test for the innovations of unstable autoregressive models based on the divergence test. In order to investigate the asymptotic behavior of the tests, we use the link between the divergence test and the residual empirical process. Simulation results are provided for illustration.

Two step estimators of the minimum distance type for parameters of the ARMA(1,1) model

New minimum distance estimators are constructed with the help of a preliminary estimator. The asymptotic normality of the constructed estimator is proved with use of a uniform linear expansion of a randomly weighted residual empirical process in a non-standard neighborhood of the true parameter value. Also the question on asymptotic efficiency of the constructed estimator is discussed.

Correction: Residual empirical processes for long and short memory time series

Remark 3.1. This corollary reflects the effects of the slower convergence rate of the estimated parameter a$n. This fact serves as a reminiscence of the classical Kolmogorov-Smirnov statistics problem when the underlying parame? ters are estimated; see Durbin (1976). When cxq is known, the test statistic (1.5) is still valid, however. As pointed out by the reviewer, when F ? F(x,0) involves an unknown parameter ?, one should consider Kn with F(x) being replaced by F(x,0n). When H < 1/2, it can be shown that the limit distribution of the statistic exists by means of the result of Wu (2003). The closed form of such a limit dis? tribution is rather complicated and does not possess a simple expression, however, and is not presented here.

ON THE GOODNESS OF FIT TEST FOR DISCRETELY OBSERVED SAMPLE FROM DIFFUSION PROCESSES: DIVERGENCE MEASURE APPROACH

In this paper, we study the divergence based goodness of fit test for partially observed sample from diffusion processes. In order to derive the limiting distribution of the test, we study the asymptotic behavior of the residual empirical process based on the observed sample. It is shown that the residual empirical process converges weakly to a Brownian bridge and the associated phi-divergence test has a chi-square limiting null distribution.