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.     

Rao's score test with nonparametric density estimators

Traditionally, Rao's score (RS) tests are constructed under a parametric specification of the probability density function. We estimate the density function by a non-parametric estimator and consider a semi-parametric Rao's score (SPRS) test for a set of hypotheses concerning the parametric model. The asymptotic distribution of the SPRS test is analyzed. Further, for the regression model, we carry out a set of Monte Carlo experiments to analyze the size and power of the SPRS test in small samples. The robustness of SPRS test to the choice of the density estimator is also analyzed.     

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.     

RAO'S SCORE TESTS IN SURVIVAL ANALYSIS: EXAMPLES AND BARTLETT TYPE ADJUSTMENTS

Score method in hypothesis testing is one of Professor C. R. Rao's great contributions to statistics. It provides a simple and unified way to test some simple and composite hypotheses in many statistical problems. Some popular tests in statistical practice derived with the help of intuitions can be shown as score tests under some statistical models. The subject-years test and log-rank test in survival analysis are two of the examples. In this paper, we first introduce these two examples. After formulating these two tests as score tests, we then review some recent results on the Bartlett type adjustments for these tests.     

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.     

Sensitivity of Rao's score test,the Wald test and the likelihood ratio test to nuisance parameters

The purpose of this paper is to compare the sensitivity of the likelihood ratio test, Rao's score test, and the Wald test to the change of the nuisance parameters. The main result is that, with an error of magnitude O(n⁻¹), the null distributions and the local alternative distributions of these tests are equally sensitive to nuisance parameter. We will also give accurate factorizations of these test statistics as quadratic forms, which are themselves useful for asymptotic analyses.     

On improving the robustness and reliability of Rao's score test

The results of misspecification tests, based on Rao's score principle, are now routinely reported in applied econometric work. This paper draws together some important recent results which are designed to improve: (a) the robustness of standard score tests; and (b) the reliability of the asymptotic approximations used for inferential purposes. The discussion of robustness includes (i) parametric, (ii) distributional, and (iii) higher-order moment robustness. The issue of finite sample reliability focuses on controlling the size of the score test using (i) different variance estimators in conjunction with standard asymptotic theory, (ii) finite sample corrections obtainable from higher-order asymptotic analysis, and (iii) bootstrap procedures.     

MORE INFORMATIVE TESTING FOR BIVARIATE SYMMETRY

In testing for bivariate symmetry against arbitrary alternatives the well‐known test developed by Bowker in 1948 is shown to be a score test, and to have useful components. These components are asymptotically independent and asymptotically have the standard normal distribution. Moreover they assess particular pairs of cells for symmetry. These components can also be used in a data analytic manner to complement a test for bivariate symmetry against ordered alternatives.     

Test of Normality Against Generalized Exponential Power Alternatives

The family of symmetric generalized exponential power (GEP) densities offers a wide range of tail behaviors, which may be exponential, polynomial, and/or logarithmic. In this article, a test of normality based on Rao's score statistic and this family of GEP alternatives is proposed. This test is tailored to detect departures from normality in the tails of the distribution. The main interest of this approach is that it provides a test with a large family of symmetric alternatives having non-normal tails. In addition, the test's statistic consists of a combination of three quantities that can be interpreted as new measures of tail thickness. In a Monte-Carlo simulation study, the proposed test is shown to perform well in terms of power when compared to its competitors.     

On the optimality of rao's statistic

This paper establishes the asymptotic optimality of Rao's

test within a very wide class of tests that Includes the likeli hood ratio test and Wald's test. An expression for the defici¬ency of the tests in this class relative to Rao's test has also been obtained     

Score test for testing zero-inflated Poisson regression against zero-inflated generalized Poisson alternatives

In several cases, count data often have excessive number of zero outcomes. This zero-inflated phenomenon is a specific cause of overdispersion, and zero-inflated Poisson regression model (ZIP) has been proposed for accommodating zero-inflated data. However, if the data continue to suggest additional overdispersion, zero-inflated negative binomial (ZINB) and zero-inflated generalized Poisson (ZIGP) regression models have been considered as alternatives. This study proposes the score test for testing ZIP regression model against ZIGP alternatives and proves that it is equal to the score test for testing ZIP regression model against ZINB alternatives. The advantage of using the score test over other alternative tests such as likelihood ratio and Wald is that the score test can be used to determine whether a more complex model is appropriate without fitting the more complex model. Applications of the proposed score test on several datasets are also illustrated.     

Multivariate testing and model-checking for generalized order statistics with applications

The exponential family structure of the joint distribution of generalized order statistics is utilized to establish multivariate tests on the model parameters. For simple and composite null hypotheses, the likelihood ratio test (LR test), Wald's test, and Rao's score test are derived and turn out to have simple representations. The asymptotic distribution of the corresponding test statistics under the null hypothesis is stated, and, in case of a simple null hypothesis, asymptotic optimality of the LR test is addressed. Applications of the tests are presented; in particular, we discuss their use in reliability, and to decide whether a Poisson process is homogeneous. Finally, a power study is performed to measure and compare the quality of the tests for both, simple and composite null hypotheses.     

A modified harmonic mean test procedure for variance components

A complete class of tests of variance components is characterized within the class of tests statistics of the form of a ratio of a linear combination of chi-squared random variables to an independent chi-squared random variable. This result is used in the context of general unbalanced mixed models to show that the harmonic mean method results in an inadmissible test of the random treatment effects. The harmonic mean procedure is then modified in such a way that the modified test uniformly dominates the original test. Two competitive tests are the LMP (locally most powerful) and Wald's tests, which have optimal power properties against small and large alternatives, respectively. A Monte Carlo simulation study reveals that the modified test outperforms both the LMP and Wald's tests in badly unbalanced designs and that it is a viable alternative in less unbalanced designs.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

Distribution-free test in Tobit mean regression model

This paper discusses the problem of fitting a parametric model in Tobit mean regression models. The proposed test is based on the supremum of the Khamaladze-type transformation of a partial sum process of calibrated residuals. The asymptotic null distribution of this transformed process is shown to be the same as that of a time-transformed standard Brownian motion. Consistency of this sequence of tests against some fixed alternatives and asymptotic power under some local nonparametric alternatives are also discussed. Simulation studies are conducted to assess the finite sample performance of the proposed test. The power comparison with some existing tests shows some superiority of the proposed test at the chosen alternatives.     

Hypotheses testing for generalized order statistics with simple order restrictions on model parameters under the alternative

In applications of generalized order statistics as, for instance, reliability analysis of engineering systems, prior knowledge about the order of the underlying model parameters is often available and may therefore be incorporated in inferential procedures. Taking this information into account, we establish the likelihood ratio test, Rao's score test, and Wald's test for test problems arising from the question of appropriate model selection for ordered data, where simple order restrictions are put on the parameters under the alternative hypothesis. For simple and composite null hypothesis, explicit representations of the corresponding test statistics are obtained along with some properties and their asymptotic distributions. A simulation study is carried out to compare the order restricted tests in terms of their power. In the set-up considered, the adapted tests significantly improve the power of the associated omnibus versions for small sample sizes, especially when testing a composite null hypothesis.     

Tests for the Proportional Intensity Assumption Based on the Score Process

Proportional intensity models are widely used for describing the relationship between the intensity of a counting process and associated covariates. A basic assumption in this model is the proportionality, that each covariate has a multiplicative effect on the intensity. We present and study tests for this assumption based on a score process which is equivalent to cumulative sums of the Schoenfeld residuals. Tests within principle power against any type of departure from proportionality can be constructed based on this score process. Among the tests studied, in particular an Anderson-Darling type test turns out to be very useful by having good power properties against general alternatives. A simulation study comparing various tests for proportionality indicates that this test seems to be a good choice for an omnibus test for proportionality.     

Score tests when a nuisance parameter is unidentified under the null hypothesis

Rao's score test normally replaces nuisance parameters by their maximum likelihood estimates under the null hypothesis about the parameter of interest. In some models, however, a nuisance parameter is not identified under the null, so that this approach cannot be followed. This paper suggests replacing the nuisance parameter by its maximum likelihood estimate from the unrestricted model and making the appropriate adjustment to the variance of the estimated score. This leads to a rather natural modification of Rao's test, which is examined in detail for a regression-type model. It is compared with the approach, which has featured most frequently in the literature on this problem, where a test statistic appropriate to a known value of the nuisance parameter is treated as a function of that parameter and maximised over its range. It is argued that the modified score test has considerable advantages, including robustness to a crucial assumption required by the rival approach.     

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.     

Testing generalized linear and semiparametric models against smooth alternatives

We propose goodness-of-fit tests for testing generalized linear models and semiparametric regression models against smooth alternatives. The focus is on models having both continous and factorial covariates. As a smooth extension of a parametric or semiparametric model we use generalized varying-coefficient models as proposed by Hastie and Tibshirani. A likelihood ratio statistic is used for testing. Asymptotic expansions allow us to write the estimates as linear smoothers which in turn guarantees simple and fast bootstrapping of the test statistic. The test is shown to have √ n -power, but in contrast with parametric tests it is powerful against smooth alternatives in general.