A New Measure of Kurtosis Adjusted for Skewness

Studies of kurtosis often concentrate on only symmetric distributions. This paper identifies a process through which the standardized measure of kurtosis based on the fourth moment about the mean can be written in terms of two parts: (i) an irreducible component, about L₄, which can be seen to occur naturally in the analysis of fourth moments; (ii) terms that depend only on moments of lower order, in particular including the effects of asymmetry attached to the third moment about the mean. This separation of the effect of skewness allows definition of an improved measure of kurtosis. This paper calculates and discusses examples of the new measure of kurtosis for a range of standard distributions.     

An Influence Function Approach to Describing the Skewness of a Distribution

Skewness, like kurtosis, is a qualitative property of a distribution. A comparison of several measures of skewness of univariate distributions is carried out. Hampel's influence function is used to clarify the differences and similarities among these measures. A general concept of skewness as a location- and scale-free deformation of the probability mass of a symmetric distribution emerges. Positive skewness can be thought of as resulting from movement of mass at the right of the median from the center to the right tail of the distribution together with movement of mass at the left of the median from the left tail to the center of the distribution.     

Relative effect sizes for measures of risk

Effect sizes are an important component of experimental design, data analysis, and interpretation of statistical results. In some situations, an effect size of clinical or practical importance may be unknown to the researcher. In other situations, the researcher may be interested in comparing observed effect sizes to known standards to quantify clinical importance. In these cases, the notion of relative effect sizes (small, medium, large) can be useful as benchmarks. Although there is generally an extensive literature on relative effect sizes for continuous data, little of this research has focused on relative effect sizes for measures of risk that are common in epidemiological or biomedical studies. The aim of this paper, therefore, is to extend existing relative effect sizes to the relative risk, odds ratio, hazard ratio, rate ratio, and Mantel–Haenszel odds ratio for related samples. In most scenarios with equal group allocation, effect sizes of 1.22, 1.86, and 3.00 can be taken as small, medium, and large, respectively. The odds ratio for a non rare event is a notable exception and modified relative effect sizes are 1.32, 2.38, and 4.70 in that situation.     

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.     

On implementation of a test for Kronecker product covariance structure for multivariate repeated measures data

Under the assumption of multivariate normality the likelihood ratio test is derived to test a hypothesis for Kronecker product structure on a covariance matrix in the context of multivariate repeated measures data. Although the proposed hypothesis testing can be computationally performed by indirect use of Proc Mixed of SAS, the Proc Mixed algorithm often fails to converge. We provide an alternative algorithm. The algorithm is illustrated with two real data sets. A simulation study is also conducted for the purpose of sample size consideration.     

Rank transformations when oovariables are measured with error or mismodelled

when covariables are measured with error which, under minimal assumptions, is more accurate and efficient than the OLS in some situations, especially when the measurement error (ME) is skewly distributed or mismodel led. No prior information about the ME is necessary. Simulations studies is investigated for the performance of the proposed method.     

Persistence and existence of stationary measures for a logistic growth model with predation

We consider a stochastic logistic growth model with a predation term, and a diffusive stochastic part with a power-type coefficient. We provide criteria for the persistence of the population and for the existence and uniqueness of a stationary measure. Furthermore, we perform a detailed study of the densities of the stationary measures resorting to the forward Kolmogorov equation. We compile our results in a stochastic bifurcation diagram, drawing comparisons with the corresponding deterministic model.     

Recursive computation for repeated measures models with patterned variance matrices

A convenient recursive computational method for repeated measures analysis, provided by McGilchrist and Cullis (1990), has been extended by the authors to heterogeneous error structures and also to the repeated measures model with random coefficients. The approach is outlined briefly in this paper. A computing program for the approach has been written and used to obtain results for simulated data having various error structures. A summary of the results is given. The computing program together with some subroutines is available from the authors.     

Comparison of performance measures for multivariate discrete models

For multivariate probit models, Spiess and Tutz suggest three alternative performance measures, which are all based on the decomposition of the variation. The multivariate probit model can be seen as a special case of the discrete copula model. This paper proposes some new measures based on the value of the likelihood function and the prediction-realization table. In addition, it generalizes the measures from Spiess and Tutz for the discrete copula model. Results of a simulation study designed to compare the different measures in various situations are presented.     

Functions of Concordance and Dependence with Related Measures

Some functions that serve as building blocks for construction of a wider range of modes of concordance and dependence are pointed. We probe into interplays of such modes. From the standpoint of their conformity to stochastic dominance ordering of distributions within a Fréchet class, all such derived modes display some parallelism under certain conditions. We finally suggest a novel numeric measure of dependence that covers similar existing measures in literature.     

Computational aspects of pricing foreign exchange options with stochastic volatility and stochastic interest rates

This paper presents an analytic result for the price of a European call option on a foreign exchange currency rate. Market volatility is assumed correlated with the exchange rate and interest rates, domestic and foreign, are assumed to be stochastic. Integrals involving interest rates are derived, characteristic functions are produced, and, with evaluation, the nature of the integrals involved in Fourier inversion is examined. By comparison with FX market data, some of the effects of the nature of stochastic interest rates upon option prices are examined.     

New tests for stochastic order with application to case control studies

It is shown that if a binary regression function is increasing then retrospective sampling induces a stochastic ordering of the covariate distributions among the responders, which we call cases, and the non-responders, which we call controls. We also show that if the covariate distributions are stochastically ordered then the regression function must be increasing. This means that testing whether the regression function is monotone is equivalent to testing whether the covariate distributions are stochastically ordered. Capitalizing on these new probabilistic observations we proceed to develop two new non-parametric tests for stochastic order. The new tests are based on either the maximally selected, or integrated, chi-bar statistic of order one. The tests are easy to compute and interpret and their large sampling distributions are easily found. Numerical comparisons show that they compare favorably with existing methods in both small and large samples. We emphasize that the new tests are applicable to any testing problem involving two stochastically ordered distributions.     

Exact inference for multivariate ordered alternatives

In this paper we discuss three types of ordered alternatives ordered location, stochastic ordering and quadrant dependence. We prove that quadrant dependence is the more general among the three. Then we consider a conditional tests for the equality of c distributions against quadrant dependence in a multivariate setup. An exact simultaneous testing procedure based on dependent conditional tests is presented. Two applications to real data are also given.     

New tests for multivariate normality based on Small's and Srivastava's graphical methods

In this paper, we propose a new measure of fit which can be used in the case of quantile–quantile plots. This measure, when applied to Small's and Srivastava's graphical methods provides two new tests for assessing multivariate normality. For different sample sizes and numbers of variables, the critical values of these tests were evaluated via simulations. The power of the new tests and its comparison with some other tests for multivariate normality are presented herein.     

Regression-based orderings and measures of stochastic depence

General axiomatic approach to the so-called global dependence of a random variable xon a random vector Y= Y _t,Y _n) is proposed. natural orderings and measures of global dependence are discussed and examplified by some real and function-valued measures of dependence. orderings and measures to be introduced are referred o as regression-based as they depend only on the distributions of EX|Y X.     

A characterization of and new consistent tests for symmetry

Let X and Y be independent and identically distributed random variables having a continuous distribution function. We study new consistent tests for symmetry around a known median based on the fact that the distribution of X is symmetric around 0 if, and only if, |X| and |max(X,Y)| have the same distribution.     

Maximum Test versus Adaptive Tests for the Two-Sample Location Problem

For the non-parametric two-sample location problem, adaptive tests based on a selector statistic are compared with a maximum and a sum test, respectively. When the class of all continuous distributions is not restricted, the sum test is not a robust test, i.e. it does not have a relatively high power across the different possible distributions. However, according to our simulation results, the adaptive tests as well as the maximum test are robust. For a small sample size, the maximum test is preferable, whereas for a large sample size the comparison between the adaptive tests and the maximum test does not show a clear winner. Consequently, one may argue in favour of the maximum test since it is a useful test for all sample sizes. Furthermore, it does not need a selector and the specification of which test is to be performed for which values of the selector. When the family of possible distributions is restricted, the maximin efficiency robust test may be a further robust alternative. However, for the family of t distributions this test is not as powerful as the corresponding maximum test.     

Bayesian Analysis of the Censored Regression Model with an AEPD Error Term

We present the censored regression model with the error term following the asymmetric exponential power distribution. We propose three Markov chain Monte Carlo (MCMC) algorithms: the first one uses the probability integral transformation; the second one uses a combination of the probability integral transformation and random walk draws; while the third one uses random walk draws. Using simulated data we compare the performance of the three MCMC algorithms. Then we compare the posterior means, or Bayes estimates, with maximum likelihood estimates. We estimate the stock option portion of executive compensation as an example of the empirical application.     

Multiple deletion measures and conditional influence in regression model

The joint effect of the deletion of the ith and jih cases is given by Gray and Ling (1984), they discussed the influence measures for influential subsets in linear regression analysis. The present paper is concerned with multiple sets of deletion measures in the linear regression model. In particular we are interested in the effects of the jointly and conditional influence analysis for the detection of two influential subsets.