Jarque–Bera Test and its Competitors for Testing Normality – A Power Comparison

For testing normality we investigate the power of several tests, first of all, the well-known test of Jarque & Bera (1980) and furthermore the tests of Kuiper (1960) and Shapiro & Wilk (1965) as well as tests of Kolmogorov–Smirnov and Cramér-von Mises type. The tests on normality are based, first, on independent random variables (model I) and, second, on the residuals in the classical linear regression (model II). We investigate the exact critical values of the Jarque–Bera test and the Kolmogorov–Smirnov and Cramér-von Mises tests, in the latter case for the original and standardized observations where the unknown parameters μ and σ have to be estimated. The power comparison is carried out via Monte Carlo simulation assuming the model of contaminated normal distributions with varying parameters μ and σ and different proportions of contamination. It turns out that for the Jarque–Bera test the approximation of critical values by the chi-square distribution does not work very well. The test is superior in power to its competitors for symmetric distributions with medium up to long tails and for slightly skewed distributions with long tails. The power of the Jarque–Bera test is poor for distributions with short tails, especially if the shape is bimodal – sometimes the test is even biased. In this case a modification of the Cramér-von Mises test or the Shapiro–Wilk test may be recommended.     

A Correlation Test for Normality Based on the Lévy Characterization

A powerful test of fit for normal distributions is proposed. Based on the Lévy characterization, the test statistic is the sample correlation coefficient of normal quantiles and sums of pairs of observations from a random sample. Since the test statistic is location-scale invariant, critical values can be obtained by simulation without estimating any parameters. It is proved that this test is consistent. A power comparison study including some directed tests shows that the proposed test is competitive, it is more powerful than the well-known Jarque–Bera test, and it is comparable to Shapiro–Wilk test against a number of alternatives.     

Behaviour of skewness, kurtosis and normality tests in long memory data

We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ${\sqrt{n}}We establish the limiting distributions for empirical estimators of the coefficient of skewness, kurtosis, and the Jarque–Bera normality test statistic for long memory linear processes. We show that these estimators, contrary to the case of short memory, are neither ?n{\sqrt{n}}-consistent nor asymptotically normal. The normalizations needed to obtain the limiting distributions depend on the long memory parameter d. A direct consequence is that if data are long memory then testing normality with the Jarque–Bera test by using the chi-squared critical values is not valid. Therefore, statistical inference based on skewness, kurtosis, and the Jarque–Bera normality test, needs a rescaling of the corresponding statistics and computing new critical values of their nonstandard limiting distributions.     

Comparisons of various types of normality tests

Normality tests can be classified into tests based on chi-squared, moments, empirical distribution, spacings, regression and correlation and other special tests. This paper studies and compares the power of eight selected normality tests: the Shapiro–Wilk test, the Kolmogorov–Smirnov test, the Lilliefors test, the Cramer–von Mises test, the Anderson–Darling test, the D'Agostino–Pearson test, the Jarque–Bera test and chi-squared test. Power comparisons of these eight tests were obtained via the Monte Carlo simulation of sample data generated from alternative distributions that follow symmetric short-tailed, symmetric long-tailed and asymmetric distributions. Our simulation results show that for symmetric short-tailed distributions, D'Agostino and Shapiro–Wilk tests have better power. For symmetric long-tailed distributions, the power of Jarque–Bera and D'Agostino tests is quite comparable with the Shapiro–Wilk test. As for asymmetric distributions, the Shapiro–Wilk test is the most powerful test followed by the Anderson–Darling test.     

Two-way ANOVA when the distribution of the error terms is skew t

In this article, we assume that the distribution of the error terms is skew t in two-way analysis of variance (ANOVA). Skew t distribution is very flexible for modeling the symmetric and the skew datasets, since it reduces to the well-known normal, skew normal, and Student's t distributions. We obtain the estimators of the model parameters by using the maximum likelihood (ML) and the modified maximum likelihood (MML) methodologies. We also propose new test statistics based on these estimators for testing the equality of the treatment and the block means and also the interaction effect. The efficiencies of the ML and the MML estimators and the power values of the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. Simulation results show that the proposed methodologies are more preferable. We also show that the test statistics based on the ML estimators are more powerful than the test statistics based on the MML estimators as expected. However, power values of the test statistics based on the MML estimators are very close to the corresponding test statistics based on the ML estimators. At the end of the study, a real life example is given to show the implementation of the proposed methodologies.     

Marshall–Olkin distribution: parameter estimation and application to cancer data

In this study, as alternatives to the maximum likelihood (ML) and the frequency estimators, we propose robust estimators for the parameters of Zipf and Marshall–Olkin Zipf distributions. A small simulation study is given to illustrate the performance of the proposed estimators. We apply the proposed estimators to a real data set from cancer research to illustrate the performance of the proposed estimators over the ML, moments and frequency estimators. We observe that the robust estimators have superiority over the frequency estimators based on classical sample mean.     

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.     

Robust tests for equality of two population means under the normal model

A robust procedure is developed for testing the equality of means in the two sample normal model. This is based on the weighted likelihood estimators of Basu et al. (1993). When the normal model is true the tests proposed have the same asymptotic power as the two sample Student's t-statistic in the equal variance case. However, when the normality assumptions are only approximately true the proposed tests can be substantially more powerful than the classical tests. In a Monte Carlo study for the equal variance case under various outlier models the proposed test using Hellinger distance based weighted likelihood estimator compared favorably with the classical test as well as the robust test proposed by Tiku (1980).     

A note on the Jarque–Bera normality test for GARCH innovations

In this paper, we consider the validity of the Jarque–Bera normality test whose construction is based on the residuals, for the innovations of GARCH (generalized autoregressive conditional heteroscedastic) models. It is shown that the asymptotic behavior of the original form of the JB test adopted in this paper is identical to that of the test statistic based on true errors. The simulation study also confirms the validity of the original form since it outperforms other available normality tests.     

Stochastic analysis of covariance when the error distribution is long-tailed symmetric

In this study, we consider stochastic one-way analysis of covariance model when the distribution of the error terms is long-tailed symmetric. Estimators of the unknown model parameters are obtained by using the maximum likelihood (ML) methodology. Iteratively reweighting algorithm is used to compute the ML estimates of the parameters. We also propose new test statistic based on ML estimators for testing the linear contrasts of the treatment effects. In the simulation study, we compare the efficiencies of the traditional least-squares (LS) estimators of the model parameters with the corresponding ML estimators. We also compare the power of the test statistics based on LS and ML estimators, respectively. A real-life example is given at the end of the study.     

Robust pairwise multiple comparisons under short-tailed symmetric distributions

In one-way ANOVA, most of the pairwise multiple comparison procedures depend on normality assumption of errors. In practice, errors have non-normal distributions so frequently. Therefore, it is very important to develop robust estimators of location and the associated variance under non-normality. In this paper, we consider the estimation of one-way ANOVA model parameters to make pairwise multiple comparisons under short-tailed symmetric (STS) distribution. The classical least squares method is neither efficient nor robust and maximum likelihood estimation technique is problematic in this situation. Modified maximum likelihood (MML) estimation technique gives the opportunity to estimate model parameters in closed forms under non-normal distributions. Hence, the use of MML estimators in the test statistic is proposed for pairwise multiple comparisons under STS distribution. The efficiency and power comparisons of the test statistic based on sample mean, trimmed mean, wave and MML estimators are given and the robustness of the test obtained using these estimators under plausible alternatives and inlier model are examined. It is demonstrated that the test statistic based on MML estimators is efficient and robust and the corresponding test is more powerful and having smallest Type I error.     

A Monte Carlo comparison of Jarque–Bera type tests and Henze–Zirkler test of multivariate normality

In the paper, tests for multivariate normality (MVN) of Jarque-Bera type, based on skewness and kurtosis, have been considered. Tests proposed by Mardia and Srivastava, and the combined tests based on skewness and kurtosis defined by Jarque and Bera have been taken into account. In the Monte Carlo simulations, for each combination of p = 2, 3, 4, 5 number of traits and n = 10(5)50(10)100 sample sizes 10,000 runs have been done to calculate empirical Type I errors of tests under consideration, and empirical power against different alternative distributions. Simulation results have been compared to the Henze–Zirkler’s test. It should be stressed that no test yet proposed is uniformly better than all the others in every combination of conditions examined.     

Robust estimation and testing in one-way ANOVA for Type II censored samples: skew normal error terms

Censoring can be occurred in many statistical analyses in the framework of experimental design. In this study, we estimate the model parameters in one-way ANOVA under Type II censoring. We assume that the distribution of the error terms is Azzalini's skew normal. We use Tiku's modified maximum likelihood (MML) methodology which is a modified version of the well-known maximum likelihood (ML) in the estimation procedure. Unlike ML methodology, MML methodology is non-iterative and gives explicit estimators of the model parameters. We also propose new test statistics based on the proposed estimators. The performances of the proposed estimators and the test statistics based on them are compared with the corresponding normal theory results via Monte Carlo simulation study. A real life data is analysed to show the implementation of the methodology presented in this paper at the end of the study.     

A note on Jarque–Bera normality test for ARMA–GARCH innovations

In this paper, we study the Jarque–Bera (JB) normality test for the innovations of ARMA–GARCH models, whose construction is based on the residuals. The validity of the JB test for ARMA–GARCH innovations should be carefully investigated in advance of actual practice, since the residual-based test may behave differently, depending upon the structure of the time series models and the form of the test statistic (cf. Chen and Kuan, 2003, Hwang and Baek, 2009, Lee and Wei, 1999). In order to demonstrate the validity of the JB test, we prove that the asymptotic distribution of the original form of the JB test is identical to that of the test statistic based on true errors under mild conditions. Simulation results are provided for illustration.     

Estimating Box-Cox power transformation parameter via goodness-of-fit tests

Box–Cox power transformation is a commonly used methodology to transform the distribution of the data into a normal distribution. The methodology relies on a single transformation parameter. In this study, we focus on the estimation of this parameter. For this purpose, we employ seven popular goodness-of-fit tests for normality, namely Shapiro–Wilk, Anderson–Darling, Cramer-von Mises, Pearson Chi-square, Shapiro-Francia, Lilliefors and Jarque–Bera tests, together with a searching algorithm. The searching algorithm is based on finding the argument of the minimum or maximum depending on the test, i.e., maximum for the Shapiro–Wilk and Shapiro–Francia, minimum for the rest. The artificial covariate method of Dag et al. (2014) is also included for comparison purposes. Simulation studies are implemented to compare the performances of the methods. Results show that Shapiro–Wilk and the artificial covariate method are more effective than the others and Pearson Chi-square is the worst performing method. The methods are also applied to two real-life datasets. The R package AID is proposed for implementation of the aforementioned methods.     

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.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

Conditional tests for elliptical symmetry using robust estimators

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

Comparison of Regression Curves with Censored Responses

Abstract.  In this article, we introduce a procedure to test the equality of regression functions when the response variables are censored. The test is based on a comparison of Kaplan–Meier estimators of the distribution of the censored residuals. Kolmogorov–Smirnov- and Cramér–von Mises-type statistics are considered. Some asymptotic results are proved: weak convergence of the process of interest, convergence of the test statistics and behaviour of the process under local alternatives. We also describe a bootstrap procedure in order to approximate the critical values of the test. A simulation study and an application to a real data set conclude the paper.