Test du rapport de vraisemblance pour le rang de cointégration d'un VAR avec des erreurs dépendantes

In this Note we consider the estimation and the test of the co-integration rank of multiple autoregressive time series model with nonindependent innovations. We show the consistency of the estimators and the validity of the likelihood ratio test in the presence of error terms with heteroscedasticity or other forms of dependence. To cite this article: H. Raïssi, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the cusum of squares test for variance change in nonstationary and nonparametric time series models

In this paper we consider the problem of testing for a variance change in nonstationary and nonparametric time series models. The models under consideration are the unstable AR(q) model and the fixed design nonparametric regression model with a strong mixing error process. In order to perform a test, we employ the cusum of squares test introduced by Inclán and Tiao (1994,J. Amer. Statist. Assoc.,89, 913–923). It is shown that the limiting distribution of the test statistic is the sup of a standard Brownian bridge as seen in iid random samples. Simulation results are provided for illustration.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

Influence Analysis for Linear Measurement Error Models

In this paper, we present a unified diagnostic method for linear measurement error models based upon the corrected likelihood of Nakamura (1990, Biometrika, 77, 127–137). Both global influence and local influence are discussed. The case-deletion model and mean-shift outlier model are considered, and they are shown to be approximately equivalent. Several diagnostic measures are derived and discussed. It is found that they can be written in terms of the residual and leverage measure. Some existing results are improved. Numerical example illustrates that our method is useful for diagnosing influential observations.     

Tests portmanteau multivariés d'adéquation de modèles VARMA faibles

In this Note, we consider portmanteau tests for testing the adequacy of vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. We relax the standard independence assumption to extend the range of application of the VARMA models, allowing us to treat linear representations of general nonlinear processes. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We thus obtain the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung–Box (or Box–Pierce) portmanteau statistics for VARMA models with nonindependent innovations. We propose a method to adjust the critical values of the portmanteau tests.     

Local Whittle likelihood estimators and tests for non-Gaussian stationary processes

In this paper, we propose a local Whittle likelihood estimator for spectral densities of non-Gaussian processes and a local Whittle likelihood ratio test statistic for the problem of testing whether the spectral density of a non-Gaussian stationary process belongs to a parametric family or not. Introducing a local Whittle likelihood of a spectral density f _θ (λ) around λ, we propose a local estimator [^(q)] = [^(q)] (l){\hat{\theta } = \hat{\theta } (\lambda ) } of θ which maximizes the local Whittle likelihood around λ, and use f_{[^(q)] (l)} (l){f_{\hat{\theta } (\lambda )} (\lambda )} as an estimator of the true spectral density. For the testing problem, we use a local Whittle likelihood ratio test statistic based on the local Whittle likelihood estimator. The asymptotics of these statistics are elucidated. It is shown that their asymptotic distributions do not depend on non-Gaussianity of the processes. Because our models include nonlinear stationary time series models, we can apply the results to stationary GARCH processes. Advantage of the proposed estimator is demonstrated by a few simulated numerical examples.     

Testing a sub-hypothesis in linear regression models with long memory covariates and errors

This paper considers the problem of testing a sub-hypothesis in homoscedastic linear regression models when the covariate and error processes form independent long memory moving averages. The asymptotic null distribution of the likelihood ratio type test based on Whittle quadratic forms is shown to be a chi-square distribution. Additionally, the estimators of the slope parameters obtained by minimizing the Whittle dispersion is seen to be n ^1/2-consistent for all values of the long memory parameters of the design and error processes. Research of the first author was partly supported by the NSF DMS Grant 0701430. Research of the second author was partly supported by the bilateral France-Lithuania scientific project Gilibert and the Lithuanian State Science and Studies Foundation grant T-15/07.     

Modified Schwarz and Hannan–Quinn information criteria for weak VARMA models

Numerous multivariate time series admit weak vector autoregressive moving-average (VARMA) representations, in which the errors are uncorrelated but not necessarily independent nor martingale differences. These models are called weak VARMA by opposition to the standard VARMA models, also called strong VARMA models, in which the error terms are supposed to be independent and identically distributed (iid). This article considers the problem of order selection of the weak VARMA models by using the information criteria. It is shown that the use of the standard information criteria are often not justified when the iid assumption on the noise is relaxed. As a consequence, we propose the modified versions of the Schwarz or Bayesian information criterion and of the Hannan and Quinn criterion for identifying the orders of weak VARMA models. Monte Carlo experiments show that the proposed modified criteria estimate the model orders more accurately than the standard ones. An illustrative application using the squared daily returns of financial series is presented.     

Spatial Regression Models,Response Surfaces,and Process Optimization

Abstract

Spatial regression models are developed as a complementary alternative to second-order polynomial response surfaces in the context of process optimization. These models provide estimates of design variable effects and smooth, data-faithful approximations to the unknown response function over the design space. The predicted response surfaces are driven by the covariance structures of the models. Several structures, isotropic and anisotropic, are considered and connections with thin plate splines are reviewed. Estimation of covariance parameters is achieved via maximum likelihood and residual maximum likelihood. A feature of the spatial regression approach is the visually appealing graphical summaries that are produced. These allow rapid and intuitive identification of process windows on the design space for which the response achieves target performance. Relevant design issues are briefly discussed and spatial designs, such as the packing designs available in Gosset, are suggested as a suitable design complement. The spatial regression models also perform well with no global design, for example with data obtained from series of designs on the same space of design variables. The approach is illustrated with an example involving the optimization of components in a DNA amplification assay. A Monte Carlo comparison of the spatial models with both thin plate splines and second-order polynomial response surfaces for a scenario motivated by the example is also given. This shows superior performance of the spatial models to the second-order polynomials with respect to both prediction over the complete design space and for cross-validation prediction error in the region of the optimum. An anisotropic spatial regression model performs best for a high noise case and both this model and the thin plate spline for a low noise case. Spatial regression is recommended for construction of response surfaces in all process optimization applications.     

Modified likelihood ratio test for homogeneity in bivariate normal mixtures with presence of a structural parameter

This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.     

Empirical likelihood for conditional quantile with left-truncated and dependent data

In this paper, we employ the method of empirical likelihood to construct confidence intervals for a conditional quantile in the presence and absence of auxiliary information, respectively, for the left-truncation model. It is proved that the empirical likelihood ratio admits a limiting chi-square distribution with one degree of freedom when the lifetime observations with multivariate covariates form a stationary α-mixing sequence. For the problem of testing a hypothesis on the conditional quantile, it is shown that the asymptotic power of the test statistic based on the empirical likelihood ratio with the auxiliary information is larger than that of the one based on the standard empirical likelihood ratio. The finite sample performance of the empirical likelihood confidence intervals in the presence and absence of auxiliary information is investigated through simulations.     

Empirical likelihood method for multivariate Cox regression

A unified empirical likelihood approach for three Cox-type marginal models dealing with multiple event times, recurrent event times and clustered event times is proposed. The resulting log-empirical likelihood ratio test statistics are shown to possess chi-squared limiting distributions. When making inferences, there is no need to solve estimating equations nor to estimate limiting covariance matrices. The optimal linear combination property for over-identified empirical likelihood is preserved by the proposed method and the property can be used to improve estimation efficiency. In addition, an adjusted empirical likelihood approach is applied to reduce the error rates of the proposed empirical likelihood ratio tests. The adjusted empirical likelihood tests could outperform the existing Wald tests for small to moderate sample sizes. The proposed approach is illustrated by extensive simulation studies and two real examples.     

NM-QELE for ARMA-GARCH models with non-Gaussian innovations

Although the quasi maximum likelihood estimator based on Gaussian density (Gaussian-QMLE) is widely used to estimate parameters in ARMA models with GARCH innovations (ARMA-GARCH models), it does not perform successfully when error distribution of ARMA-GARCH models is either skewed or leptokurtic. In order to circumvent such defects, Lee and Lee (submitted for publication) proposed the quasi maximum estimated-likelihood estimator using Gaussian mixture-based likelihood (NM-QELE) for GARCH models. In this paper, we adopt the NM-QELE method for estimating parameters in ARMA-GARCH models and demonstrate the validity of NM-QELE by verifying its consistency.     

Test-Based Variable Selection via Cross-Validation

Abstract

Test-based variable selection algorithms in regression often are based on sequential comparison of test statistics to cutoff values. A predetermined a level typically is used to determine the cutoffs based on an assumed probability distribution for the test statistic. For example, backward elimination or forward stepwise involve comparisons of test statistics to prespecified t or F cutoffs in Gaussian linear regression, while a likelihood ratio. Wald, or score statistic, is typically used with standard normal or chi square cutoffs in nonlinear settings. Although such algorithms enjoy widespread use, their statistical properties are not well understood, either theoretically or empirically. Two inherent problems with these methods are that (1) as in classical hypothesis testing, the value of α is arbitrary, while (2) unlike hypothesis testing, there is no simple analog of type I error rate corresponding to application of the entire algorithm to a data set. In this article we propose a new method, backward elimination via cross-validation (BECV), for test-based variable selection in regression. It is implemented by first finding the empirical p value α*, which minimizes a cross-validation estimate of squared prediction error, then selecting the model by running backward elimination on the entire data set using α* as the nominal p value for each test. We present results of an extensive computer simulation to evaluate BECV and compare its performance to standard backward elimination and forward stepwise selection.     

A proof of independent Bartlett correctability of nested likelihood ratio tests

It is well known that likelihood ratio statistic is Bartlett correctable. We consider decomposition of a likelihood ratio statistic into 1 degree of freedom components based on sequence of nested hypotheses. We give a proof of the fact that the component likelihood ratio statistics are distributed mutually independently up to the order O(1/n) and each component is independently Bartlett correctable. This was implicit in Lawley (1956, Biometrika, 43, 295–303) and proved in Bickel and Ghosh (1990, Ann. Statist., 18, 1070–1090) using a Bayes method. We present a more direct frequentist proof.     

<Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> linear interpolator for missing values in time series

We propose a minimum mean absolute error linear interpolator (MMAELI), based on theL ₁ approach. A linear functional of the observed time series due to non-normal innovations is derived. The solution equation for the coefficients of this linear functional is established in terms of the innovation series. It is found that information implied in the innovation series is useful for the interpolation of missing values. The MMAELIs of the AR(1) model with innovations following mixed normal andt distributions are studied in detail. The MMAELI also approximates the minimum mean squared error linear interpolator (MMSELI) well in mean squared error but outperforms the MMSELI in mean absolute error. An application to a real series is presented. Extensions to the general ARMA model and other time series models are discussed. This research was supported by a CityU Research Grant and Natural Science Foundation of China.     

Bayesian Computation of Software Reliability

Abstract

Bayesian methods for the Jelinski and Moranda and the Littlewood and Verrall models in software reliability are studied. A Gibbs sampling approach is employed to compute the Bayes estimates. In addition, prediction of future failure times and future reliabilities is examined. Model selection based on the mean squared prediction error and the prequential likelihood of the conditional predictive ordinates is developed.     

Offline and online weighted least squares estimation of nonstationary power ARCH processes

This paper proposes two estimation methods based on a weighted least squares criterion for non-(strictly) stationary power ARCH models. The weights are the squared volatilities evaluated at a known value in the parameter space. The first method is adapted for fixed sample size data while the second one allows for online data available in real time. It will be shown that these methods provide consistent and asymptotically Gaussian estimates having asymptotic variance equal to that of the quasi-maximum likelihood estimate (QMLE) regardless of the value of the weighting parameter. Finite-sample performances of the proposed WLS estimates are shown via a simulation study for various sub-classes of power ARCH models.     

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the Ginibre ensemble of n × n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order n ^?1/3. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy–Widom distributions.