关于几个条件独立性检验统计量的构造与比较分析

变量间的条件独立性可视为在概率空间上对其因果关系的一种描述,因而可以通过检验变量之间的条件独立性来检验因果关系。文章详细介绍了几个条件独立性检验统计量的构造方法和基本原理,包括线性模型假设下的Fisher-z检验统计量和在非线性模型下或无法确定变量之间的模型时使用的3个非参数的条件独立性检验统计量,并对这几个不同的条件独立性检验统计量的检验效率进行了对比分析。     

Testing nested or non-nested hypotheses

This paper presents a test procedure for nested or non-nested hypotheses. The test statistic is based on the difference between two estimators of the pseudo-true value as defined for instance by Sawa. This statistic is similar to the usual Wald statistic in the case of nested hypotheses and it can be replaced by an asymptotically equivalent one deduced from the score function.     

Testing for multiplicative heteroskedasticity

The problem of testing for multiplicative heteroskedasticity is considered and a large sample test is proposed. The test statistic is based upon ordinary least squares results, so that only estimation under the null hypothesis of homoskedasticity is required. The test is, however, asymptotically equivalent to the likelihood ratio test and so has good asymptotic power properties. The finite sample behaviour of the test statistic is examined using Monte Carlo experiments which indicate that the test works well for quite small samples.     

A generalization of Friedman's rank statistic

Abstract  In this paper a very natural generalization of the two-way analysis of variance rank statistic of F riedman is given. The general distribution-free test procedure based on this statistic for the effect of J treatments in a random block design can be applied in general two-way layouts without interactions and with different numbers of the continuous observations per cell provided the design scheme is connected. The asymptotic distribution under the null hypothesis of the test statistic is derived. A comparison with the method of m rankings of B enard and van E lteren is made. The disadvantage of B enard and van E lteren's test procedure is that the number of observations per block does influence the statistic twice, namely firstly by the number itself, as it should, and see ondly by the level of the ranks which will be different in different blocks if the numbers of observations per block are different. The proposed test statistic is not sensitive to differences in the levels of the ranks caused by the different numbers of observations per block. The test is derived from considerhg the K ruskal -W allis statistics per block.
Finally, the results of simulation experiments are given. The simulation is carried out for three designs and a number of normal location alternatives and gives some information about the power of the suggested test procedure. A comparison is made with B enard and van E lteren's test and with the classical analysis of variance technique. For some simple orthogonal designs the exact null distributions of B enard and van E lteren's test and the proposed test are compared.     

A robust version of the KPSS test based on indicators

This paper proposes a test of the null hypothesis of stationarity that is robust to the presence of fat-tailed errors. The test statistic is a modified version of the so-called KPSS statistic. The modified statistic uses the “sign” of the data minus the sample median, whereas KPSS used deviations from means. This “indicator” KPSS statistic has the same limit distribution as the standard KPSS statistic under the null, without relying on assumptions about moments, but a different limit distribution under unit root alternatives. The indicator test has lower power than standard KPSS when tails are thin, but higher power when tails are fat.     

Building logistic models by means of a non parametric goodness of fit test: a case study

A sequence of logistic models is fitted to data from a Dutch follow-up study on preterm infants (POPS). To examine the adequacy of the model, a recently developed non parametric method to check goodness of fit is applied (le Cessie and Van Houwelingen (1991)). This method uses a test statistic based upon kernel regression methods.
In this paper the problem of choosing a "best" bandwidth, corresponding to the greatest power of the test statistic, is avoided by computing the test statistic for a range of different bandwidths. Testing is then based upon the asymptotic distribution of the maximum of the test statistics.
The testing method is used as a goodness of fit criterion, and the contribution of each individual observation to the test statistic is used as a diagnostic tool to localize deviations of the model, and to determine directions in which the model can be improved.     

Extension of the Wald statistic to models with dependent observations

A generalization of the Wald statistic for testing composite hypotheses is suggested for dependent data from exponential models which include Lévy processes and diffusion fields. The generalized statistic is proved to be asymptotically chi-squared distributed under regular composite hypotheses. It is simpler and more easily available than the generalized likelihood ratio statistic. Simulations in an example where the latter statistic is available show that the generalized Wald test achieves higher average power than the generalized likelihood ratio test. Received: February 29, 2000     

PERFORMANCE OF MURPHY'S TEST FOR TWO OUTLIERS IN A LINEAR MODEL

Abstract. In JOSHI and LALITHA (1986) a test for two outliers in the same direction in a linear model is discussed. Here the performance of this statistic is studied. For this, the exact non–null density function of the random variables involved in defining the statistic is obtained. Then a measure of performance is defined and it is applied to the case of a random sample from a normal distribution, as in this case the above said statistic reduces to the well known Murphy's test statistic. These values are then compared with the power values obtained by HAWKINS (1978).     

An exact test for a column of the covariance matrix based on a single observation

In this paper, we derive an exact test for a column of the covariance matrix. The test statistic is calculated by using a single observation. The exact distributions of the test statistic are derived under both the null and alternative hypotheses. We also obtain an analytical expression of the power function of the test for the equality of a column of the covariance matrix to a given vector. It is shown that the information contained in a single vector is large enough to ensure a good performance of the test. Moreover, the suggested test can be applied for time-dependent multivariate Gaussian processes.     

Score tests for zero covariances in recursive linear models for grouped or censored data

The score test statistic for testing whether an error covariance is zero is derived for a normal linear recursive model for fully observed, censored or grouped data. The test, which is obtained by regarding non-zero error covariances as arising from correlated random parameter variation, is shown to be closely related to the Information Matrix test. It turns out that the statistic, which is asymptotically N[0,1] under the null, examines the sample covariance of appropriately defined residuals.     

Testing parameter constancy in linear models against stochastic stationary parameters

This paper considers testing parameter constancy in a linear model when the alternative is that a subset of the parameters follows a stationary vector autoregressive process of known finite order. This kind of a linear model is only identified under the alternative, which usually precludes finding a test statistic with an analytic null distribution. In the present situation, however, it is still possible to derive a test statistic with an asymptotic chi-squared distribution under the null hypothesis and this is done in the paper. The small-sample properties of the test statistic are investigated by simulation and found statisfactory. The test retains its power when the alternative to parameter constancy is a random walk parameter process.     

Testing for exogeneity in limited dependent variable models using a simplified likelihood ratio statistic

A simple likelihood-ratio statistic for the weak exogeneity of the continuously observed endogenous variables is presented for the limited information simultaneous equations models in which a single endogenous variable is censored. The statistic is a likelihood ratio test statistic for the exclusion of the reduced form residuals of the continuously observed endogenous variables and is asymptotically locally most powerful. The procedure is illustrated by an application to a model of female labour supply.     

Generalized reduced rank tests using the singular value decomposition

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson [Annals of Mathematical Statistics (1951), 22, 327–351] sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald [Journal of the American Statistical Association (1996), 91, 1301–1309] and Gill and Lewbel [Journal of the American Statistical Association (1992), 87, 766–776] a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith [Econometric Theory (2000), 16, 151–175] usage of numerical optimization for the objective function statistic of Cragg and Donald [Journal of Econometrics (1997), 76, 223–250] and ignoring the non-negativity restriction on the singular values in Ratsimalahelo [2002, Rank test based on matrix perturbation theory. Unpublished working paper, U.F.R. Science Economique, University de Franche-Comté]. In the non-stationary cointegration case, the limiting distribution of the new rank statistic is identical to that of the Johansen trace statistic.     

Test by adaptive LASSO quantile method for real-time detection of a change-point

This article proposes a test statistic based on the adaptive LASSO quantile method to detect in real-time a change in a linear model. The model can have a large number of explanatory variables and the errors don’t satisfy the classical assumptions for a statistical model. For the proposed test statistic, the asymptotic distribution under \(H_0\) is obtained and the divergence under \(H_1\) is shown. It is shown via Monte Carlo simulations, in terms of empirical sizes, of empirical powers and of stopping time detection, that the useful test statistic for applications is better than other test statistics proposed in literature. Two applications on the air pollution and in the health field data are also considered.     

对单位根检验PT统计量响应面函数的研究

Elliott等人于1996年提出了针对特定备择假设具有最优特性的PT单位根检验统计量。这一检验方法引起了众多计量经济学家的重视,并被收入一些新版本的计量软件包。但在实际运用时由于现有的临界值是在原文献给出的4个样本容量相应临界值的基础上插值而得,因而准确性较低,使得检验结果的可靠性受到了影响。本文利用Monte Carlo模拟的方法给出了在相当广泛的样本容量范围内临界值的近似计算公式,并对响应面函数法与插值法的结果做了一个简单的比较。     

Tests based on φ-divergences for bivariate symmetry

In this paper we introduce a family of test statistics for testing symmetry based on φ-divergence families. These test statistics yield the likelihood ratio test and the Pearson test statistic as special cases. Asymptotic distribution for the new test statistics are derived under both the null and the alternative hypotheses. A simulation study is presented to see that some new test statistics offer an attractive alternative to the classical Pearson test statistic for the problem of symmetry. Received: May 2000     

Testing for unobserved heterogeneity in exponential and Weibull duration models

We examine the use of the likelihood ratio (LR) statistic to test for unobserved heterogeneity in duration models, based on mixtures of exponential or Weibull distributions. We consider both the uncensored and censored duration cases. The asymptotic null distribution of the LR test statistic is not the standard chi-square, as the standard regularity conditions do not hold. Instead, there is a nuisance parameter identified only under the alternative, and a null parameter value on the boundary of the parameter space, as in Cho and White (2007a). We accommodate these and provide methods delivering consistent asymptotic critical values. We conduct a number of Monte Carlo simulations, comparing the level and power of the LR test statistic to an information matrix (IM) test due to Chesher (1984) and Lagrange multiplier (LM) tests of Kiefer (1985) and Sharma (1987). Our simulations show that the LR test statistic generally outperforms the IM and LM tests. We also revisit the work of van den Berg and Ridder (1998) on unemployment durations and of Ghysels et al. (2004) on interarrival times between stock trades, and, as it turns out, affirm their original informal inferences.     

Granger-causality in multiple time series

The concept of Granger-causality is formulated for a finite-dimensional multiple time series. Special attention is given to causality patterns in autoregressive series, and it is shown how these patterns can be tested under quite general assumptions using a χ² statistic. The power of the test is discussed, and it is shown that the χ² statistic results from a Lagrange multiplier test in the Gaussian case. The causality test is tried both on artificial data and some economic time series. Finally we consider the problem of constrained estimation in models with a known causality structure.     

A Frequency-domain Based Test for Non-correlation between Stationary Time Series

A one-sided asymptotically normal test for non-correlation between two stationary time series is proposed based on the spectral coherence function. The test statistic is a properly standardized version of the integrated spectral coherency and has similar asymptotic properties as a previously introduced time domain based test for non-correlation. Unlike its time domain counterpart, the proposed test does not require prewhitening of the time series and, thus, is a truly nonparametric test for non-correlation. In a simulation study, we evaluate the small sample performance of the proposed test in comparison with the time domain test and address the problem of bandwidth selection. Furthermore, we present a modification of the test statistic that allows to test for non-correlation over frequency bands. This version shows higher power of detecting interrelationships restricted to the frequency band of interest. This work has been carried out at the Institute of Applied Mathematics at the University of Heidelberg and partly while the author was visiting the Department of Statistics at the University of Chicago.     

A Simple Artificial Regression Based Test of the Goodness of Fit of Binary Choice Models

A test of the goodness of fit of a binary choice model is derived using the conditional moment testing approach. The test statistic may be calculated as the explained sum of squares or N:R² in a simple artificial regression. The test statistic has reasonably good small sample properties since it is not based on the outer product gradient (OPG) form.