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.     

On estimating the box-cox transformation to normality

This paper studies four methods for estimating the Box-Cox parameter used to transform data to normality. Three of these are based on optimizing test statistics for standard normality tests (the Shapiro-Wilk. skewness, and kurtosis tests); the fourth uses the maximum likelihood estimator of the Box-Cox parameter. The four methods are compared and evaluated with a simulation study, where their performances under different skewness and kurtosis conditions are analyzed. The estimator based on optimizing the Shapiro-Wilk statistic generally gives rise to the best transformations, while the maximum likelihood estimator performs almost as well. Estimators based on optimizing skewness and kurtosis do not perform well in general.     

Beta estimation in the market model: skewness and leptokurtosis

Leptokurtosis and skewness characterize the distributions of the returns for many financial instruments traded in security markets. These departures from normality can adversely affect the efficiency of least squares estimates of the β's in the single index or market model. The proposed new partially adaptive estimation techniques accommodate skewed and fat tailed distributions. The empirical investigation, which is the first application of this procedure in regression models, reveals that both skewness and kurtosis can affect β estimates.     

On a new family of shifted logarithmic transformations

Most parametric statistical methods are based on a set of assumptions: normality, linearity and homoscedasticity. Transformation of a metric response is a popular method to meet these assumptions. In particular, transformation of the response of a linear model is a popular method when attempting to satisfy the Gaussian assumptions on the error components in the model. A particular problem with common transformations such as the logarithm or the Box–Cox family is that negative and zero data values cannot be transformed. This paper proposes a new transformation which allows negative and zero data values. The method for estimating the transformation parameter consider an objective criteria based on kurtosis and skewness for achieving normality. Use of the new transformation and the method for estimating the transformation parameter are illustrated with three data sets.     

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.     

Skewness,kurtosis, and black-scholes option mispricing

The Black-Scholes option pricing model assumes that (instantaneous) common stock returns are normally distributed. However, the observed distribution exhibits deviations from normality; in particular skewness and kurtosis. We attribute these deviations to gross data errors. Using options' transactions data, we establish that the sample standard deviation, sample skewness, and sample kurtosis contribute to the Black-Scholes model's observed mispricing of a sample from the Berkeley Options Data Base of 2323 call options written on 88 common stocks paying no dividends during the options'life. Following Huber's statement that the primary case for robust statistics is when the shape of the observed distribution deviates slightly from the assumed distribution (usually the Gaussian), we show that robust volatility estimators eliminate the mispricing with respect to sample skewness and sample kurtosis, and significantly improve the Black-Scholes model's pricing performance with respect to estimated volatility.     

Measures of multivariate skewness and kurtosis for tests of nonnormality

Measures of multivariate skewness and kurtosis are proposed that are based on the skewness and kurtosis of individual components of standardized sample vectors. Asymptotic properties and small sample critical values of tests for nonnormality based on these measures are provided. It is demonstrated that the tests have favorable power properties. Extensions to time series data are pointed out.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.     

Omnibus test of normality using the Q statistic

A new statistical procedure for testing normality is proposed. The Q statistic is derived as the ratio of two linear combinations of the ordered random observations. The coefficients of the linear combinations are utilizing the expected values of the order statistics from the standard normal distribution. This test is omnibus to detect the deviations from normality that result from either skewness or kurtosis. The statistic is independent of the origin and the scale under the null hypothesis of normality, and the null distribution of Q can be very well approximated by the Cornish-Fisher expansion. The powers for various alternative distributions were compared with several other test statistics by simulations.     

Testing high-dimensional normality based on classical skewness and Kurtosis with a possible small sample size

Abstract

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples.     

Tests for multivariate normality based on canonical correlations

We propose new affine invariant tests for multivariate normality, based on independence characterizations of the sample moments of the normal distribution. The test statistics are obtained using canonical correlations between sets of sample moments in a way that resembles the construction of Mardia’s skewness measure and generalizes the Lin–Mudholkar test for univariate normality. The tests are compared to some popular tests based on Mardia’s skewness and kurtosis measures in an extensive simulation power study and are found to offer higher power against many of the alternatives.     

On the distribution of a new measure of heaviness of tails

Distributional properties are given for a statistic T*, which has previously been reported to have power properties as a test of normality as attractive as those of the sample kurtosis or perhaps slightly more attractive. Asymptotic results, the mean and variance under normality, the range of variation, and approximation of critical values for testing normality are obtained     

On Sample Skewness and Kurtosis

It is well documented in the literature that the sample skewness and excess kurtosis can be severely biased in finite samples. In this paper, we derive analytical results for their finite-sample biases up to the second order. In general, the bias results depend on the cumulants (up to the sixth order) as well as the dependency structure of the data. Using an AR(1) process for illustration, we show that a feasible bias-correction procedure based on our analytical results works remarkably well for reducing the bias of the sample skewness. Bias-correction works reasonably well also for the sample kurtosis under some moderate degree of dependency. In terms of hypothesis testing, bias-correction offers power improvement when testing for normality, and bias-correction under the null provides also size improvement. However, for testing nonzero skewness and/or excess kurtosis, there exist nonnegligible size distortions in finite samples and bias-correction may not help.     

Omnibus tests of multinormality based on skewness and kurtosis

Measures of univariate skewness and kurtosis have long been used as a test of univariate normality, several omnibus test procedures based on a combination of the measures having been proposed, see Pearson, D’Agestino and Bowman (1977) and Mardia (1979). Mardia (1970) proposed measures of multivariate skewness and kurtosis, and constructed a test of multinormality based on these measures. we obtain the correlation between these measures and propose several omnibus tests using the two measures. The performances of these tests are compared by means of a Monte Carlo study.     

On the corrections to information matrix tests

This paper addresses the issue of designing finite-sample corrections to information matrix tests. We review a Cornish-Fisher correction that has been propowed elsewhere and propose an alternative, Bartlett-type correction. Simulation results for skewness, excess kurtosis, normality and heteroskedasticity tests are given.     

A tool for systematically comparing the power of tests for normality

In this article, we describe a new approach to compare the power of different tests for normality. This approach provides the researcher with a practical tool for evaluating which test at their disposal is the most appropriate for their sampling problem. Using the Johnson systems of distribution, we estimate the power of a test for normality for any mean, variance, skewness, and kurtosis. Using this characterization and an innovative graphical representation, we validate our method by comparing three well-known tests for normality: the Pearson χ² test, the Kolmogorov–Smirnov test, and the D'Agostino–Pearson K ² test. We obtain such comparison for a broad range of skewness, kurtosis, and sample sizes. We demonstrate that the D'Agostino–Pearson test gives greater power than the others against most of the alternative distributions and at most sample sizes. We also find that the Pearson χ² test gives greater power than Kolmogorov–Smirnov against most of the alternative distributions for sample sizes between 18 and 330.     

A monte carlo study on the robustness of the two-sample sequential t test

The robustness of the two-sample sequentla1 t test was studied against departures from normality and equality of variances The effect of skewness and kurtosis of the underlying distribution on the test 1s relatively mild but the effect of heteroscedasticity serious.     

Sensitivity of Bayesian sampling inspection schemes to a non-normal prior distribution

summary In this paper we derive the predictive density function of a future observation under the assumption of Edgeworth-type non-normal prior distribution for the unknown mean of a normal population. Fixed size single sample and sequential sampling inspection plans, in a decisive prediction framework, are examined for their sensitivity to departures from normality of the prior distribution. Numerical illustrations indicate that the decision to market the remaining items of a given lot for a fixed size plan may be sensitive to the presence of skewness or kurtosis in the prior distribution. However, Bayes'decision based on the sequential plan may not change though expected gains may change with variation in the non-normality of the prior distribution.     

Kurtosis of the logistic-exponential survival distribution

ABSTRACT

In this article, the kurtosis of the logistic-exponential distribution is analyzed. All the moments of this survival distribution are finite, but do not possess closed-form expressions. The standardized fourth central moment, known as Pearson’s coefficient of kurtosis and often used to describe the kurtosis of a distribution, can thus also not be expressed in closed form for the logistic-exponential distribution. Alternative kurtosis measures are therefore considered, specifically quantile-based measures and the L-kurtosis ratio. It is shown that these kurtosis measures of the logistic-exponential distribution are invariant to the values of the distribution’s single shape parameter and hence skewness invariant.     

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.