Testing time series for interpolability and whiteness

We propose a test to decide if a time series is represented by its linear interpolator better than by its mean value. The same test can be employed to decide if a time series has to be considered white noise. The test is based on a new estimate of the index of linear determinism (Battaglia, 1983, Inverse autocovariances and a measure of linear determinism for a stationary process, J. Time Series Anal. 4, 79-87) and its asymptotic distribution is derived. Comparison with the popular Ljung-Box portmanteau test has been performed based on both asymptotic power and a simulation experiment. The new test     

On tests for normality of experimental error in ridge regression

Considered are tests for normality of the errors in ridge regression. If an intercept is included in the model, it is shown that test statistics based on the empirical distribution function of the ridge residuals have the same limiting distribution as in the one-sample test for normality with estimated mean and variance. The result holds with weak assumptions on the behavior of the independent variables; asymptotic normality of the ridge estimator is not required.     

Some weighted mixed portmanteau tests for diagnostic checking in linear time series models

The popular diagnostic checking methods in linear time series models are portmanteau tests based on either residual autocorrelation functions (acf) or partial autocorrelation functions (pacf). In this paper, we device some new weighted mixed portmanteau tests by appropriately combining individual tests based on both acf and pacf. We derive the asymptotic distribution of such weighted mixed portmanteau statistics and study their size and power. It is found that the weighted mixed tests outperform when higher order ARMA models are fitted and diagnostic checks are performed via testing lack of residual autocorrelations. Simulation results suggest to use the proposed tests as complementary to those classical tests found in literature. An illustrative application is given to demonstrate the usefulness of the mixed test.     

Testing for residual correlation of any order in the autoregressive process

We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of Ljung–Box and Box–Pierce, and appearing to outperform the Breusch–Godfrey procedure on small-sized samples.     

Adjusted empirical likelihood inference for additive hazards regression

ABSTRACT

This article develops an adjusted empirical likelihood (EL) method for the additive hazards model. The adjusted EL ratio is shown to have a central chi-squared limiting distribution under the null hypothesis. We also evaluate its asymptotic distribution as a non central chi-squared distribution under the local alternatives of order n^{? 1/2}, deriving the expression for the asymptotic power function. Simulation studies and a real example are conducted to evaluate the finite sample performance of the proposed method. Compared with the normal approximation-based method, the proposed method tends to have more larger empirical power and smaller confidence regions with comparable coverage probabilities.     

Generalized portmanteau statistics and tests of randomness

This paper considers the problem of testing the randomness of Gaussian and non–Gaussian time series. A general class of parametric portmanteau statistics, which include the Box–Pierce and the Ljung–Box statistics, is introduced. Using the exact first and second moments of the sample autocorrelations when the observations are i.i.d. normal with unknown mean, the exact expected value of any portmanteau statistics is obtained for this case. Two new portmanteau statistics, which exploit the exact moments of the sample autocorrelations, are studied. For the nonparametric case, a rank portmanteau statistic is introduced. The latter has the same distribution for any series of exchangeable random variables and uses the exact moments of the rank autocorrelations. We show that its asymptotic distribution is chi–squate. Simulation results indicate that the new portmanteau statistics are better approximated by the chi–square asymptotic distribution than the Ljung–Box statistics. Several analytical results presented in the paper were derived by usig a symbolic manipulation program.     

Asymptotic properties of kruskal-wallis test and friedman test in the analysis of variance models with random effects

In this paper, asymptotic properties of the Kruskal-Wallis test in the one-way analysis of variance model and that of the Friedman test in the two-way classification model are investigated under alternatives when the treatment effects are random. It is shown that the asymptotic distribution of each statistic is the same as a mixture of central chi-squared variables. Asymptotic comparisons of the tests with respect to their parametric competitors are also performed     

Empirical likelihood for the difference of quantiles under censorship

In this paper, we use a smoothed empirical likelihood method to investigate the difference of quantiles under censorship. An empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be chi-squared. Approximate confidence regions based on this method are constructed. Simulation studies are used to compare the empirical likelihood and the normal approximation method in terms of its coverage accuracy. It is found that the empirical likelihood method provides a much better performance. The research is supported by NSFC (10231030) and RFDP.     

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.     

Likelihood ratio tests for variance components in linear mixed models

Although the asymptotic distributions of the likelihood ratio for testing hypotheses of null variance components in linear mixed models derived by Stram and Lee [1994. Variance components testing in longitudinal mixed effects model. Biometrics 50, 1171–1177] are valid, their proof is based on the work of Self and Liang [1987. Asymptotic properties of maximum likelihood estimators and likelihood tests under nonstandard conditions. J. Amer. Statist. Assoc. 82, 605–610] which requires identically distributed random variables, an assumption not always valid in longitudinal data problems. We use the less restrictive results of Vu and Zhou [1997. Generalization of likelihood ratio tests under nonstandard conditions. Ann. Statist. 25, 897–916] to prove that the proposed mixture of chi-squared distributions is the actual asymptotic distribution of such likelihood ratios used as test statistics for null variance components in models with one or two random effects. We also consider a limited simulation study to evaluate the appropriateness of the asymptotic distribution of such likelihood ratios in moderately sized samples.     

Sur un test d'égalité des autocovariances de deux séries chronologiques

We propose a test for the equality of the autocovariance functions of two independent and stationary time series. The test statistic is a quadratic form in the vector of differences of the first J + 1 autocovariances. Its asymptotic distribution is derived under the null hypothesis, and the finite-sample properties of the test, namely the bias and the power, are investigated by Monte Carlo methods. A by-product of this study is a new estimator of the covariance between two sample autocovariances which provides a positive definite covariance matrix. We establish the convergence of this estimator in the L₁ norm.     

Approximated distributions of the weighted sum of correlated chi-squared random variables

We study the asymptotic behavior of the weighted sum of correlated chi-squared random variables. Both chi-squared and normal distributions are proved to approximate the exact distribution. These two approximations are established by matching the first two cumulants. Simulation comparison is made to study the performance of two approximations numerically. We find that the chi-squared approximation performs better than the normal one in the study.     

Empirical likelihood inference for a semiparametric hazards regression model

ABSTRACT

We investigated the empirical likelihood inference approach under a general class of semiparametric hazards regression models with survival data subject to right-censoring. An empirical likelihood ratio for the full 2p regression parameters involved in the model is obtained. We showed that it converged weakly to a random variable which could be written as a weighted sum of 2p independent chi-squared variables with one degree of freedom. Using this, we could construct a confidence region for parameters. We also suggested an adjusted version for the preceding statistic, whose limit followed a standard chi-squared distribution with 2p degrees of freedom.     

A Comparative Study of Model-based Tests of Independence for Ordinal Data Using the Bootstrap

When analyzing categorical data using loglinear models in sparse contingency tables, asymptotic results may fail. In this paper the empirical properties of three commonly used asymptotic tests of independence, based on the uniform association model for ordinal data, are investigated by means of Monte Carlo simulation. Five different bootstrapped tests of independence are presented and compared to the asymptotic tests. The comparisons are made with respect to both size and power properties of the tests. Results indicate that the asymptotic tests have poor size control. The test based on the estimated association parameter is severely conservative and the two chi-squared tests (Pearson, likelihood-ratio) are both liberal. The bootstrap tests that either use a parametric assumption or are based on non-pivotal test statistics do not perform better than the asymptotic tests in all situations. The bootstrap tests that are based on approximately pivotal statistics provide both adjustment of size and enhancement of power. These tests are therefore recommended for use in situations similar to those included in the simulation study.     

Non parametric portmanteau tests for detecting non linearities in high dimensions

ABSTRACT

We propose two non parametric portmanteau test statistics for serial dependence in high dimensions using the correlation integral. One test depends on a cutoff threshold value, while the other test is freed of this dependence. Although these tests may each be viewed as variants of the classical Brock, Dechert, and Scheinkman (BDS) test statistic, they avoid some of the major weaknesses of this test. We establish consistency and asymptotic normality of both portmanteau tests. Using Monte Carlo simulations, we investigate the small sample properties of the tests for a variety of data generating processes with normally and uniformly distributed innovations. We show that asymptotic theory provides accurate inference in finite samples and for relatively high dimensions. This is followed by a power comparison with the BDS test, and with several rank-based extensions of the BDS tests that have recently been proposed in the literature. Two real data examples are provided to illustrate the use of the test procedure.     

Power transformations to induce normality and their applications

Summary.  Random variables which are positive linear combinations of positive independent random variables can have heavily right-skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson–Hilferty cube root transformation for χ ² random variables as a special case. We also provide some important examples, including test statistics of goodness-of-fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box–Ljung–Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box–Ljung–Pierce portmanteau test, unlike the uncorrected versions.     

Test de différence de contrastes et somme pondérée de khi-deux

After recalling the framework of minimum-contrast estimation, its consistency and its asymptotic normality, we highlight the fact that these results do not require any stationarity or ergodicity assumptions. The asymptotic distribution of the underlying contrast difference test is a weighted sum of independent chi-square variables having one degree of freedom each. We illustrate these results in three contexts: (1) a nonhomogeneous Markov chain with likelihood contrast; (2) a Markov field with coding, pseudolikelihood or likelihood contrasts; (3) a not necessarily Gaussian time series with Whittle's contrast. In contexts (2) and (3), we compare experimentally the power of the likelihood-ratio test with those of other contrast-difference tests.     

Statistical Inference in Partially Linear Varying-Coefficient Models with Missing Responses at Random

This article considers statistical inference for partially linear varying-coefficient models when the responses are missing at random. We propose a profile least-squares estimator for the parametric component with complete-case data and show that the resulting estimator is asymptotically normal. To avoid to estimate the asymptotic covariance in establishing confidence region of the parametric component with the normal-approximation method, we define an empirical likelihood based statistic and show that its limiting distribution is chi-squared distribution. Then, the confidence regions of the parametric component with asymptotically correct coverage probabilities can be constructed by the result. To check the validity of the linear constraints on the parametric component, we construct a modified generalized likelihood ratio test statistic and demonstrate that it follows asymptotically chi-squared distribution under the null hypothesis. Then, we extend the generalized likelihood ratio technique to the context of missing data. Finally, some simulations are conducted to illustrate the proposed methods.     

The distribution of the product of independent beta variables

We investigate the classic distribution and approximate distribution of the product of Beta variables which are independent. We show that the product of independent Beta variables is a Beta variable under the some assumptions. We also obtain the approximate distribution of the product of independent Beta variables.     

Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.