An Adaptive Efficient Test for Gumbel Domain of Attraction

We consider n independent observations, generated identically by some distribution function, which belongs to the domain of attraction of an extreme value distribution with unknown shape and scale parameter. We treat the scale parameter as a nuisance parameter and establish an adaptive efficient test sequence, which is based on the k _n largest observations, for the Gumbel domain of attraction. Efficiency is achieved along certain contiguous extreme value alternatives within the concept of local asymptotic normality (LAN). Simulations exemplify the results     

Cressie–Read Power-Divergence Statistics for Non-Gaussian Vector Stationary Processes

Abstract.  For a class of vector-valued non-Gaussian stationary processes, we develop the Cressie–Read power-divergence (CR) statistic approach which has been proposed for the i.i.d. case. The CR statistic includes empirical likelihood as a special case. Therefore, by adopting this CR statistic approach, the theory of estimation and testing based on empirical likelihood is greatly extended. We use an extended Whittle likelihood as score function and derive the asymptotic distribution of the CR statistic. We apply this result to estimation of autocorrelation and the AR coefficient, and get narrower confidence intervals than those obtained by existing methods. We also consider the power properties of the test based on asymptotic theory. Under a sequence of contiguous local alternatives, we derive the asymptotic distribution of the CR statistic. The problem of testing autocorrelation is discussed and we introduce some interesting properties of the local power.     

Conditional Score Tests for Heteroscedasticity in the Two-Way Error Components Model

This paper extends the one-way heteroskedasticity score test of Holly and Gardiol (2000, In: Krishnakumar, J, Ronchetti, E (Eds.), Panel Data Econometrics: Future Directions, North-Holland, Amsterdam, pp. 199–211) to two conditional Lagrange Multiplier (LM) tests of heteroskedasticity under contiguous alternatives within the two-way error components model framework. In each case, the derivation of Rao's efficient score statistics for testing heteroskedasticity is first obtained. Then, based on a specific set of assumptions, the asymptotic distribution of the score under contiguous alternatives is established. Finally, the expression for the score test statistic in the presence of heteroskedasticity and related asymptotic local powers of these score test statistics are derived and discussed.     

Prequential omnibus goodness-of-fit tests for stochastic processes: A numerical study

This article is a contribution to the study of an omnibus goodness-of-fit (Gof) test based on Rosenblatt Probability Integral Transform (RPIT) within Dawid's prequential framework. This Gof test is easy to use since it has a common test statistic (with apparently the same asymptotic distribution) for a wide range of stochastic models. Intensive Monte-Carlo simulations are presented to investigate the behavior of this test for several stochastic models: renewal, autoregressive (AR, ARMA, ARCH, GARCH) and Poisson processes, generalized linear models... These simulations suggest that the RPIT test could be used to test the fit of a wide range of stochastic models but it may be not powerful when compared to Gof tests specifically designed for the tested processes. It is also conjectured that this test is still appropriate for testing the Gof of any discrete-time stochastic process provided that efficient estimators are used.     

A joint test for arch and bilinearity in the regression model

In this paper we argue that a simultaneous test for ARCH and bilinearity should be used to test for the possible nonlinearity of the error process in the regression model. We suggest such a joint test statistic. An empirical example shows that the individual tests of ARCH and bilinearity may not be conclusive while a joint test clearly rejects the linearity hypothesis. Our results are also applicable to pure time series models.     

Statistical analysis of a class of factor time series models

For a class of factor time series models, which is called a multivariate time series variance component (MTV) models, we consider the problem of testing whether an observed time series belongs to this class. We propose the test statistic, and derive its symptotic null distribution. Asymptotic optimality of the proposed test is discussed in view of the local asymptotic normality. Also, numerical evaluation of the local power illuminates some interesting features of the test.     

Local Power Analyses of Goodness-of-fit Tests for Copulas

Abstract.  The asymptotic behaviour of several goodness-of-fit statistics for copula families is obtained under contiguous alternatives. Many comparisons between a Cramér–von Mises functional of the empirical copula process and new moment-based goodness-of-fit statistics are made by considering their associated asymptotic local power curves. It is shown that the choice of the estimator for the unknown parameter can have a significant influence on the power of the Cramér–von Mises test and that some of the moment-based statistics can provide simple and efficient goodness-of-fit methods.     

Robust Tests Against Smooth Transition Autoregressive Models

In this paper, we consider testing for linearity against a well-known class of regime switching models known as the smooth transition autoregressive (STAR) models. Apart from the model selection issues, one reason for interest in testing for linearity in time-series models is that non-linear models such as the STAR are considerably more difficult to use. This testing problem is non-standard because a nuisance parameter becomes unidentified under the null hypothesis. In this paper, we further explore the class of tests proposed by Luukkonen, Saikonnen and Terasvirta (1988). Luukkonen et al . (1988) proposed LM tests for linearity against STAR models. A potential difficulty here is that the linear approximation introduces high leverage points, and hence outliers are likely to be quite influential. To overcome this difficulty, we use the same approximating linear model of Luukkonen et al . (1988), but we apply Wald and F -tests based on l 1 - and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under H 1 . Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al . (1988) for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.     

Robust Testing Procedures in Heteroscedastic Linear Models

There exist many studies which treat the robust tests in homoscedastic linear models. However, the robust testing procedure in heteroscedastic linear models has not been examined. In this article, three classes of testing procedures for testing subhypothesis in heteroscedastic linear models are developed. These are Wald-type, score-type, and drop-in dispersion tests. The asymptotic distributions of these tests are obtained under the null hypothesis and contiguous alternatives. For a robustness criterion, the maximum asymptotic bias of the level of the test for distributions in a shrinking contamination neighborhood is used and the most-efficient robust test is derived. Finally, the performance of these tests in small sample is studied by Monte Carlo simulation.     

Semiparametric lower bounds for tail index estimation

We consider estimation of the tail index parameter from i.i.d. observations in Pareto and Weibull type models, using a local and asymptotic approach. The slowly varying function describing the non-tail behavior of the distribution is considered as an infinite dimensional nuisance parameter. Without further regularity conditions, we derive a local asymptotic normality (LAN) result for suitably chosen parametric submodels of the full semiparametric model. From this result, we immediately obtain the optimal rate of convergence of tail index parameter estimators for more specific models previously studied. On top of the optimal rate of convergence, our LAN result also gives the minimal limiting variance of estimators (regular for our parametric model) through the convolution theorem. We show that the classical Hill estimator is regular for the submodels introduced with limiting variance equal to the induced convolution theorem bound. We also discuss the Weibull model in this respect.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.     

Local power and size properties of the LR,Wald, score and gradient tests in dispersion models

We derive asymptotic expansions for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the class of dispersion models, under a sequence of Pitman alternatives. The asymptotic distributions of these statistics are obtained for testing a subset of regression parameters and for testing the precision parameter. Based on these nonnull asymptotic expansions, the power of all four tests, which are equivalent to first order, are compared. Furthermore, in order to compare the finite-sample performance of these tests in this class of models, Monte Carlo simulations are presented. An empirical application to a real data set is considered for illustrative purposes.     

Bootstrap Tests of Nonnested Hypotheses: Some Further Results

Abstract

Nonnested models are sometimes tested using a simulated reference distribution for the uncentred log likelihood ratio statistic. This approach has been recommended for the specific problem of testing linear and logarithmic regression models. The general asymptotic validity of the reference distribution test under correct choice of error distributions is questioned. The asymptotic behaviour of the test under incorrect assumptions about error distributions is also examined. In order to complement these analyses, Monte Carlo results for the case of linear and logarithmic regression models are provided. The finite sample properties of several standard tests for testing these alternative functional forms are also studied, under normal and nonnormal error distributions. These regression-based variable-addition tests are implemented using asymptotic and bootstrap critical values.     

A Joint Score Test for Heteroscedasticity in the Two Way Error Components Model

This article extends the work by Holly and Gardiol (2000) (A score test for individual heteroscedasticity in a one-way error component model. In: Krishnakumar, J., Ronchetti, E., Eds. Panel Data Econometrics: Future Directions. Elsevier, North-Holland, Amsterdam, pp. 199–211, Ch. 10) to the two-way error components model. It deals exclusively with a joint heteroscedasticity test by first deriving Rao's efficient score statistics. Then, based on appropriate set of assumptions, we deduce the asymptotic distribution of the score under contiguous alternatives. Finally, we provide the expression for the score test statistic in the presence of heteroscedasticity and discuss its asymptotic local power.     

Empirical Likelihood Method for General Additive-Multiplicative Hazard Models

An empirical likelihood-based inferential procedure is developed for a class of general additive-multiplicative hazard models. The proposed log-empirical likelihood ratio test statistic for the parameter vector is shown to have a chi-squared limiting distribution. The result can be used to make inference about the entire parameter vector as well as any linear combination of it. The asymptotic power of the proposed test statistic under contiguous alternatives is discussed. The method is illustrated by extensive simulation studies and a real example.     

Limits of experiments associated with sampling plans

Starting from Milbrodt (1985), the asymptotic behaviour of experiments associated with Poisson sampling, Rejective sampling and its Sampford-Durbin modification is investigated. As superpopulation models so-called L^r-generated regression parameter families (1⩽r⩽2) are considered, allowing also the presence of nuisance parameters. Under some assumptions on the first order probabilities of inclusion it can be shown that the sampling experiments converge weakly if the underlying shift parameter families do so. In case of convergence the limit of the sampling experiments is characterized in terms of its Hellinger transforms and its Lévy-Khintchine representation, leading to criteria for the limit to be a pure Gaussian or a pure Poisson experiment respectively. These results are then applied to the situation of sampling in the presence of random non-response, and to establish local asymptotic normality (LAN) under more restrictive conditions. Applications also include asymptotic optimality properties of tests based on Horvitz-Thompson-type statistics, and LAM bounds and criteria for adaptivity, when testing or estimating a continuous linear functional in LAN situations. They especially cover the case of sampling from an unknown symmetric distribution, which has been subject to detailed investigations in the i.i.d. case.     

Inference for linear and nonlinear stable error processes via estimating functions

This paper describes an estimating function approach for parameter estimation in linear and nonlinear times series models with infinite variance stable errors. Joint estimates of location and scale parameters are derived for classes of autoregressive (AR) models and random coefficient autoregressive (RCA) models with stable errors, as well as for AR models with stable autoregressive conditionally heteroscedastic (ARCH) errors. Fast, on-line, recursive parametric estimation for the location parameter based on estimating functions is discussed using simulation studies. A real financial time series is also discussed in some detail.     

Estimated Generalized Least Squares for Random Coefficient Regression Models

ABSTRACT. This paper considers a general class of random coefficient regression (RCR) models to represent pooled cross-sectional and time series data. A new method is given to estimate the covariance matrix of the error component in these RCR models. Also, the asymptotic and small sample properties of the estimated generalized least squares estimator of the regression coefficient vector are established. Procedures for testing a linear restriction on the mean vector of the random coefficients are derived. Finally, a test for non-randomness in the RCR model is devised, and the asymptotic distribution of the test statistic is obtained.     

COMPARING TESTS OF AUTOREGRESSIVE VERSUS MOVING AVERAGE ERRORS IN REGRESSION MODELS USING BAHADUR’S ASYMPTOTIC RELATIVE EFFICIENCY

ABSTRACT

The purpose of this paper is to use Bahadur's asymptotic relative efficiency measure to compare the performance of various tests of autoregressive (AR) versus moving average (MA) error processes in regression models. Tests to be examined include non-nested procedures of the models against each other, and classical procedures based upon testing both the AR and MA error processes against the more general autoregressive-moving average model.