Asymptotic Chi-square Distribution Research Articles

New approaches to model-free dimension reduction for bivariate regression

Dimension reduction with bivariate responses, especially a mix of a continuous and categorical responses, can be of special interest. One immediate application is to regressions with censoring. In this paper, we propose two novel methods to reduce the dimension of the covariates of a bivariate regression via a model-free approach. Both methods enjoy a simple asymptotic chi-squared distribution for testing the dimension of the regression, and also allow us to test the contributions of the covariates easily without pre-specifying a parametric model. The new methods outperform the current one both in simulations and in analysis of a real data. The well-known PBC data are used to illustrate the application of our method to censored regression.

Testing Serial Correlation in Partial Linear Errors-in-Variables Models Based on Empirical Likelihood

In this article, we propose an empirical likelihood-based test to check the existence of serial correlation in partial linear errors-in-variables models. A nonparametric version of Wilk' theorem is derived, which says that our proposed test has an asymptotic chi-square distribution. Simulation results reveal that the finite sample performance of our proposed test is satisfactory in both size and power.

Instrument endogeneity and identification-robust tests: Some analytical results

When some explanatory variables in a regression are correlated with the disturbance term, instrumental variable methods are typically employed to make reliable inferences. Furthermore, to avoid difficulties associated with weak instruments, identification-robust methods are often proposed. However, it is hard to assess whether an instrumental variable is valid in practice because instrument validity is based on the questionable assumption that some of them are exogenous. In this paper, we focus on structural models and analyze the effects of instrument endogeneity on two identification-robust procedures, the Anderson–Rubin (1949, AR) and the Kleibergen (2002, K) tests, with or without weak instruments. Two main setups are considered: (1) the level of “instrument” endogeneity is fixed (does not depend on the sample size) and (2) the instruments are locally exogenous, i.e. the parameter which controls instrument endogeneity approaches zero as the sample size increases. In the first setup, we show that both test procedures are in general consistent against the presence of invalid instruments (hence asymptotically invalid for the hypothesis of interest), whether the instruments are “strong” or “weak”. We also describe cases where test consistency may not hold, but the asymptotic distribution is modified in a way that would lead to size distortions in large samples. These include, in particular, cases where the 2SLS estimator remains consistent, but the AR and K tests are asymptotically invalid. In the second setup, we find (non-degenerate) asymptotic non-central chi-square distributions in all cases, and describe cases where the non-centrality parameter is zero and the asymptotic distribution remains the same as in the case of valid instruments (despite the presence of invalid instruments). Overall, our results underscore the importance of checking for the presence of possibly invalid instruments when applying “identification-robust” tests.

Zero finite-order serial correlation test in a semi-parametric varying-coefficient partially linear errors-in-variables model

Varying-coefficient partially linear models, as extensions of partially linear models and varying-coefficient models, are frequently used in statistical modeling. This paper proposes an empirical log-likelihood ratio for testing serial correlation in a semi-parametric varying-coefficient partially linear errors-in-variables model. The proposed empirical log-likelihood ratio is shown to have an asymptotic chi-square distribution under the null hypothesis of no serial correlation. Some Monte Carlo experiments are conducted to estimate rejection probabilities under the null hypothesis and in the presence of serial correlation. Simulation results show that the proposed test performs satisfactorily in estimated size and power.

Testing serial correlation in semiparametric varying-coefficient partially linear EV models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = Xτβ + Zτα(T) + e, ξ = X + η with the identifying condition E[(e, ητ)τ] = 0, Cov[(e, ητ)τ] = σ2Ip+1. The estimators of interested regression parameters β, and the model error variance σ2, as well as the nonparametric components α(T), are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α(T) achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.

Empirical likelihood for non-degenerate U-statistics

Standard empirical likelihood for U-statistics is too computationally expensive. To overcome this computational difficulty, we reformulate the non-degenerate U-statistics as a sample mean of some “pseudo” observations in this paper, and show that the empirical log-likelihood ratio has an asymptotic chi-squared distribution under the second moment condition. The method is extremely simple to use, and yet provide better coverage accuracy in general than other alternative methods from our simulation studies.

Empirical likelihood for the two-sample mean problem

We apply empirical likelihood method to constructing confidence regions for the difference of the means of two d-dimensional samples. It is shown that the empirical likelihood ratio test has an asymptotic chi-squared distribution. The Bartlett correction for the univariate case ( d = 1 ) has been investigated by Jing [1995. Two-sample empirical likelihood method. Statist. Probab. Lett. 24, 315–319]. Unfortunately, the Bartlett correction obtained in that article was incorrect. In this article the correct Bartlett correction is found and its effectiveness is shown by a simulation study.

DISTRIBUTION-FREE TESTS OF FRACTIONAL COINTEGRATION

We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series. Our test statistics have an asymptotic chi-square distribution under the null and rely on generalized least squares–type of corrections that control for the short-run correlation of the weak dependent components of the fractionally integrated processes. We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions that are standard in the literature and, in particular, being able to consider simultaneously (asymptotically) stationary or nonstationary processes. Relatively weak conditions on the corresponding short-run and memory parameter estimates are assumed. The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree. Finite-sample properties of the tests are analyzed by means of a Monte Carlo experiment.We thank Helmut Lütkepohl and two referees for helpful comments and suggestions. We also thank participants at the NSF/NBER Time Series Conference at the University of Heidelberg, Germany, at the Unit Root and Cointegration Testing Conference at the University of Algarve, Faro, Portugal, and seminar participants at the Universidad de Navarra and Ente Luigi Einaudi for helpful comments. Javier Hualde's research is supported by the Spanish Ministerio de Educación y Ciencia through Juan de la Cierva and Ramón y Cajal contracts and ref. SEJ2005-07657/ECON. Carlos Velasco's research is supported by the Spanish Ministerio de Educación y Ciencia, ref. SEJ2004-04583/ECON.

A Note on Permutation Tests for Variance Components in Multilevel Generalized Linear Mixed Models

In many applications of generalized linear mixed models to multilevel data, it is of interest to test whether a random effects variance component is zero. It is well known that the usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold. In this note we propose a permutation test, based on randomly permuting the indices associated with a given level of the model, that has the correct Type I error rate under the null. Results from a simulation study suggest that it is more powerful than tests based on mixtures of chi-square distributions. The proposed test is illustrated using data on the familial aggregation of sleep disturbance.

Test di adattamento nel caso di stima dei parametri per distribuzioni di cui siano note frequenze e quantità

We consider a new inferential procedure which can be used in goodness-of-fit problems when a parametric model has to be adapted to an observed distribution with class amounts known. Referring to a random sample of independent observations grouped in T fixed intervals, the method is based on an approximation to the joint likelihood which can be derived considering both the distributions of quantities. We propose a new test statistics S² which is the sum of the classical X² introduced by Pearson and of the statistic V² considered by Dancelli (1993). We show that, when the s parameters are estimated maximizing the joint likelihood, S² has an asymptotic Chi-Square distribution with (2T - s - 1) degrees of freedom. We have performed some numerical evaluation considering the normal model of order 'rho'. Our results indicate that, in this case, the power function of the new test is always greater than that of the usual test based on X² .

Using entropy distance measures for testing odds ratio parameters under logistic regression models based on case–control data

We propose two retrospective test statistics for testing the vector of odds ratio parameters under the logistic regression model based on case–control data by exploiting the density ratio structure under a two-sample semiparametric model, which is equivalent to the assumed logistic regression model. The proposed test statistics are based on Kullback–Leibler entropy distance and are particularly relevant to the case–control sampling plan. These two test statistics have identical asymptotic chi-squared distributions under the null hypothesis and identical asymptotic noncentral chi-squared distributions under local alternatives to the null hypothesis. Moreover, the proposed test statistics require computation of the maximum semiparametric likelihood estimators of the underlying parameters, but are otherwise easily computed. We present some results on simulation and on the analysis of two real data sets.

A Monte Carlo Approach to Unidimensionality Testing in Polytomous Rasch Models

Many statistical tests are designed to test the different assumptions of the Rasch model, but only few are directed at detecting multidimensionality. The Martin-Löf test is an attractive approach, the disadvantage being that its null distribution deviates strongly from the asymptotic chi-square distribution for most realistic sample sizes. A Monte Carlo test approach to p value computation is proposed and is shown to yield a powerful test. Repeated and sequential Monte Carlo tests that can greatly reduce computing time are discussed.

Optimal sufficient dimension reduction in regressions with categorical predictors

Though partial sliced inverse regression (partial SIR: Chiaromonte et al. [2002. Sufficient dimension reduction in regressions with categorical predictors. Ann. Statist. 30, 475–497]) extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors, its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice. When this condition fails, partial SIR may provide misleading results. In this article, we propose a new estimation method via a minimum discrepancy approach without this restriction. Our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi-squared distribution. It also gives us the ability to test predictor effects. An asymptotic chi-squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained.

Consistent estimation and testing in heteroscedastic polynomial errors-in-variables models

The paper concentrates on consistent estimation and testing in functional polynomial measurement errors models with known heterogeneous variances. We rest on the corrected score methodology which allows the derivation of consistent and asymptotically normal estimators for line parameters and also consistent estimators for the asymptotic covariance matrix. Hence, Wald and score type statistics can be proposed for testing the hypothesis of a reduced linear relationship, for example, with asymptotic chi-square distribution which guarantees correct asymptotic significance levels. Results of small scale simulation studies are reported to illustrate the agreement between theoretical and empirical distributions of the test statistics studied. An application to a real data set is also presented.

Bent-Cable Regression Theory and Applications

We use the so-called “bent-cable” model to describe natural phenomena that exhibit a potentially sharp change in slope. The model comprises two linear segments, joined smoothly by a quadratic bend. The class of bent cables includes, as a limiting case, the popular piecewise-linear model (with a sharp kink), otherwise known as the broken stick. Associated with bent-cable regression is the estimation of the bend-width parameter, through which the abruptness of the underlying transition may be assessed. We present worked examples and simulations to demonstrate the regularity and irregularity of bent-cable regression encountered in finite-sample settings. We also extend existing bent-cable asymptotics that previously were limited to the basic model with known linear slopes of 0 and 1. Practical conditions on the design are given to ensure regularity of the full bent-cable estimation problem if the underlying bend segment has nonzero width. Under such conditions, the least-squares estimators are shown to be consistent and to asymptotically follow a multivariate normal distribution. Furthermore, the deviance statistic (or the likelihood ratio statistic, if the random errors are normally distributed) is shown to have an asymptotic chi-squared distribution.

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.

Maximal reliability and power in covariance structure models

In covariance structure modelling, the non-centrality parameter of the asymptotic chi-squared distribution is typically used as an indicator of asymptotic power for hypothesis tests. When a latent linear regression is of interest, the contribution to power by the maximal reliability coefficient, which is associated with used latent variable indicators, is examined and this relationship is further explicated in the case of congeneric measures. It is also shown that item parcelling may reduce power of tests of latent regression parameters. Recommendations on weights for parcelling to avoid power loss are provided, which are found to be those of optimal linear composites with maximal reliability.

Testing the Fit of Gamma Distributions Using the Empirical Moment Generating Function

ABSTRACT This article presents goodness-of-fit tests for two and three-parameter gamma distributions that are based on minimum quadratic forms of standardized logarithmic differences of values of the moment generating function and its empirical counterpart. The test statistics can be computed without reliance to special functions and have asymptotic chi-squared distributions. Monte Carlo simulations are used to compare the proposed test for the two-parameter gamma distribution with goodness-of-fit tests employing empirical distribution function or spacing statistics. Two data sets are used to illustrate the various tests.

A Bootstrap Test for Circular Data

ABSTRACT This article presents a bootstrap test for simultaneous hypotheses on the mean direction and the dispersion of a sample of i.i.d. circular observations with unimodal distribution. This test is a nonparametric extension of the multivariate saddlepoint test for wrapped symmetric α-stable models proposed by Gatto (2000). Because the test statistic has an asymptotic chi-squared distribution with second-order accuracy, the typical sampling stage of most bootstrap procedures is not required. A simulation study illustrates the numerical accuracy and the effectiveness of this new test. In this simulation, samples are generated from circular distributions within the wrapped symmetric α-stable class.

Consistent estimation and testing in comparing analytical bias models

Regression techniques are commonly applied to compare two or more analytical methods at several concentration levels. The paper considers consistent estimation in measurement error models with known different variances typically used in such situations. The approach is based on the corrected score methodology, which allows the derivation of consistent estimators for the model parameters and also for the asymptotic covariance matrix of the parameter estimators. Thus, Wald type statistics are proposed for testing hypothesis related to the bias of the analytical methods with the asymptotic chi-square distribution, which guarantees correct asymptotic significance levels. Results of small-scale simulation studies are reported to illustrate comparisons with other approaches. Applications to real data sets are also considered. Copyright © 2005 John Wiley & Sons, Ltd.