Portmanteau tests based on quadratic forms in the autocorrelations

Many white noise and goodness-of-fit tests are (asymptotically) written as quadratic forms in the ordinary autocorrelation estimates. The properties of such tests are studied by investigating the structure of the matrix of the quadratic form. We suggest to choose the matrix of the quadratic form in such a way that the power is maximized according to the information available about the alternative hypothesis. A simulation study sheds some light on the behavior of the test in finite samples. It is generally found more powerful than the most popular portmanteau tests, i.e., the Box and Pierce and the Ljung and Box tests.     

Multivariate portmanteau test for structural VARMA models with uncorrelated but non-independent error terms

We consider portmanteau tests for testing the adequacy of structural vector autoregressive moving-average (VARMA) models under the assumption that the errors are uncorrelated but not necessarily independent. The structural forms are mainly used in econometrics to introduce instantaneous relationships between economic variables. We first study the joint distribution of the quasi-maximum likelihood estimator (QMLE) and the noise empirical autocovariances. We then derive the asymptotic distribution of residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We deduce the asymptotic distribution of the Ljung-Box (or Box-Pierce) portmanteau statistics in this framework. It is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi-squared random variables, which can be quite different from the usual chi-squared approximation used under independent and identically distributed (iid) assumptions on the noise. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte Carlo experiments illustrate the finite sample performance of the modified portmanteau test.     

Autocontours: Dynamic Specification Testing

We propose a new battery of dynamic specification tests for the joint hypothesis of iid-ness and density function based on the fundamental properties of independent random variables with identical distributions. We introduce a device—the autocontour—whose shape is very sensitive to departures from the null in either direction, thus providing superior power. The tests are parametric with asymptotic t and chi-squared limiting distributions and standard convergence rates. They do not require a transformation of the original data or a Kolmogorov style assessment of goodness-of-fit, explicitly account for parameter uncertainty, and have superior finite sample properties. An application to autoregressive conditional duration (ACD) models for trade durations shows that the difficulty with the assumed densities lies on the probability assigned to very small durations. Supplemental materials for this article are available online.     

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.     

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.     

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.     

A Cauchy estimator test for autocorrelation

This article presents a new test for serial correlation in an observed stationary time series. Rather than using the traditional portmanteau tests based on the sample autocorrelation function, we propose a test based on the Cauchy estimator of correlation. A goodness-of-fit statistic for fitted autoregressive moving average models is also derived and the asymptotic distribution of this statistic is quantified. The test can be employed using either this asymptotic distribution or by using Monte-Carlo quantiles. The small sample behaviour is studied via simulation and the Monte-Carlo-based test seems to be more precise. The method is demonstrated on monthly asset returns for Facebook, Incorporated.     

Time series count data regression

The count data model studied in the paper extends the Poisson model by al-lowing for overdispersion and serial correlation. Alternative approaches to esti-mate nuisance parameters, required for the correction of the Poisson maximum likelihood covariance matrix estimator and for a quasi-likelihood estimator, are studied. The estimators are evaluated by finite sample Monte Carlo experi-mentation. It is found that the Poisson maximum likelihood estimator with corrected covariance matrix estimators provide reliable inferences for longer time series. Overdispersion test statistics are wellbehaved, while conventional portmanteau statistics for white noise have too large sizes. Two empirical illustrations are included.     

A LAN based Neyman smooth test for Pareto distributions

The Pareto distribution is found in a large number of real world situations and is also a well-known model for extreme events. In the spirit of Neyman [1937. Smooth tests for goodness of fit. Skand. Aktuarietidskr. 20, 149–199] and Thomas and Pierce [1979. Neyman's smooth goodness-of-fit test when the hypothesis is composite. J. Amer. Statist. Assoc. 74, 441–445], we propose a smooth goodness of fit test for the Pareto distribution family which is motivated by LeCam's theory of local asymptotic normality (LAN). We establish the behavior of the associated test statistic firstly under the null hypothesis that the sample follows a Pareto distribution and secondly under local alternatives using the LAN framework. Finally, simulations are provided in order to study the finite sample behavior of the test statistic.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Shapiro–Wilk test for skew normal distributions based on data transformations

A probability property that connects the skew normal (SN) distribution with the normal distribution is used for proposing a goodness-of-fit test for the composite null hypothesis that a random sample follows an SN distribution with unknown parameters. The random sample is transformed to approximately normal random variables, and then the Shapiro–Wilk test is used for testing normality. The implementation of this test does not require neither parametric bootstrap nor the use of tables for different values of the slant parameter. An additional test for the same problem, based on a property that relates the gamma and SN distributions, is also introduced. The results of a power study conducted by the Monte Carlo simulation show some good properties of the proposed tests in comparison to existing tests for the same problem.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

Homogeneity testing in a Weibull mixture model

The mixed Weibull distribution provides a flexible model to analyze random durations in a possibly heterogeneous population. To test for homogeneity against unobserved heterogeneity in a Weibull mixture model, a dispersion score test and a goodness-of-fit test are investigated. The empirical power of these tests is assessed and compared on a broad range of alternatives. It comes out that the dispersion score test, as it is based on a Weibull-to-exponential transformation, often breaks down. A simple new procedure is introduced for Weibull mixtures in scale, which combines the dispersion score test and the goodness-of-fit test. The new test is compared with several known procedures and shown to have a good overall power. To detect mixtures in shape and scale, a goodness-of-fit test is recommended. This research has been partially sponsored by a grant of the Deutsche Forschungsgemeinschaft. We thank Lars Haferkamp for computational assistance and Wilfried Seidel and a referee for their remarks on alternative test procedures.     

The small sample properties of the reset test as applied to systems of equations

The RESET test for functional misspecification is generalised to cover systems of equations, and the properties of 7 versions are studied using Monte Carlo methods. The Rao F -test clearly exhibits the best performance as regards correct size, whilst the commonly used LRT (uncorrected for degrees-of-freedom), and LM and Wald tests (both corrected and uncorrected) behave badly even in single equations. The Rao test exhibits correct size even in ten equation systems, which is better than previous research concerning autocorrelation tests. The power of the test is low, however, when the number of equations grows and the correlation between the omitted variables and the RESET proxies is small.     

On the existence of transformations preserving the structure of order statistics in lower dimensions

In a Type-II right censored sample from the standard uniform distribution, several transformations of respective order statistics are examined, which transform the censored sample into a complete sample in a lower dimension. Such transformations have been considered by Lin et al. (2008), Michael and Schucany (1979) and O’Reilly and Stephens (1988) in the context of goodness-of-fit tests. It is shown that by dropping the assumption of an underlying uniform distribution, these transformed random variables can no longer be considered themselves as order statistics, in general. In the case of the transformation of Michael and Schucany, it is shown that the uniform distribution is the only one possessing this property.     

Bartlett corrections in beta regression models

We consider the issue of performing accurate small-sample testing inference in beta regression models, which are useful for modeling continuous variates that assume values in (0,1), such as rates and proportions. We derive the Bartlett correction to the likelihood ratio test statistic and also consider a bootstrap Bartlett correction. Using Monte Carlo simulations we compare the finite sample performances of the two corrected tests to that of the standard likelihood ratio test and also to its variant that employs Skovgaard's adjustment; the latter is already available in the literature. The numerical evidence favors the corrected tests we propose. We also present an empirical application.     

Power comparison of the Kolmogorov–Smirnov test under ranked set sampling and simple random sampling

Many studies have been used to compare the power of several goodness-of-fit (GOF) tests under simple random sampling (SRS) and ranked set sampling (RSS). In our study, a different design procedure and ranking process in RSS are thoroughly investigated. A simulation study is conducted to compare the power of the Kolmogorov–Smirnov test under SRS and RSS with different sets and cycle sizes for several distributions. Level-2 sampling design and partially rank-ordered sets are used. Also, we benefited from auxiliary variables in the ranking process. Finally, results are presented with tables and figures. Under these conditions we show that the RSS has better performance against the SRS in finite population.     

Goodness-of-fit tests for location–scale families based on random distance

For location–scale families, we consider a random distance between the sample order statistics and the quasi sample order statistics derived from the null distribution as a measure of discrepancy. The conditional qth quantile and expectation of the random discrepancy on the given sample are chosen as test statistics. Simulation results of powers against various alternatives are illustrated under the normal and exponential hypotheses for moderate sample size. The proposed tests, especially the qth quantile tests with a small or large q, are shown to be more powerful than other prominent goodness-of-fit tests in most cases.     

Estimation of the parameters of the Gompertz distribution under the first failure-censored sampling plan

This article presents the goodness-of-fit tests for the Laplace distribution based on its maximum entropy characterization result. The critical values of the test statistics estimated by Monte Carlo simulations are tabulated for various window and sample sizes. The test statistics use an entropy estimator depending on the window size; so, the choice of the optimal window size is an important problem. The window sizes for yielding the maximum power of the tests are given for selected sample sizes. Power studies are performed to compare the proposed tests with goodness-of-fit tests based on the empirical distribution function. Simulation results report that entropy-based tests have consistently higher power than EDF tests against almost all alternatives considered.     

Multivariate goodness-of-fit tests based on statistically equivalent blocks

Various nonparametric procedures are known for the goodness-of-fit test in the univariate case. The distribution-free nature of these procedures does not extend to the multivariate case. In this paper, we consider an application of the theory of statistically equivalent blocks(SEB)to obtain distribution-free procedures for the multivariate case. The sample values are transformed to random variables which are distributed as sample spacings from a uniform distribution on [0, 1], under the null hypothesis. Various test statistics are known, based on the spacings, which are used for testing uniformity in the univariate case. Any of these statistics can be used in the multivariate situation, based on the spacings generated from the SEB. This paper gives an expository development of the theory of SEB and a review of tests for goodness-of-fit, based on sample spacings. To show an application of the SEB, we consider a test of bivariate normality.