Kurtosis modelling by means of the J-transformation

Summary: The H–family of distributions or H–distributions, introduced by Tukey (1960; 1977), are generated by a single transformation of the standard normal distribution and allow for leptokurtosis represented by the parameter h. Alternatively, Haynes et al. (1997) generated leptokurtic distributions by applying the K–transformation to the normal distribution. In this study we propose a third transformation, the so–called J–transformation, and derive some properties of this transformation. Moreover, so-called elongation generating functions (EGFs) are introduced. By means of EGFs we are able to visualize the strength of tail elongation and to construct new transformations. Finally, we compare the three transformations towards their goodness–of–fit in the context of financial return data.     

An Alpha Half Normal Slash Distribution for Analyzing Non Negative Data

In this paper, we study a new class of slash distribution. We define the distribution through means of a stochastic representation as the mixture of an alpha half normal random variable with respect to the power of a uniform random variable. Properties involving moments and moment generating function are derived. The usefulness and flexibility of the proposed distribution is illustrated through a real application by maximum likelihood procedure.     

How Do Nonlinear Unit Root Tests Perform with Non Normal Errors?

This article examines how popular nonlinear unit root tests perform in the presence of non normal errors. Non normal errors normally do not pose a problem in the usual linear unit root tests since the least squares estimator will still be the most efficient under certain ideal conditions regardless of normal or non normal errors. Whether similar results will carry over to nonlinear unit root tests with non normal errors is a question that merits examination. We find that in contrast to the linear tests, the presence of non normal errors in nonlinear unit root tests will lead to a significant loss of power.     

Bimodal symmetric-asymmetric power-normal families

This article proposes new symmetric and asymmetric distributions applying methods analogous as the ones in Kim (2005 Kim, H.J. (2005). On a class of two-piece skew-normal distributions. Statist.: J. Theoret. Appl. Statist. 39:537–553.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Arnold et al. (2009 Arnold, B.C., H.W. Gómez, and H.S. Salinas. (2009). On multiple constraint skewed models. Statist. J. Theoret. Appl. Statist. 43: 279–293.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) to the exponentiated normal distribution studied in Durrans (1992 Durrans, S.R. (1992). Distributions of fractional order statistics in hydrology. Water Resour. Res. 28:1649–1655.[Crossref], [Web of Science ®] , [Google Scholar]), that we call the power-normal (PN) distribution. The proposed bimodal extension, the main focus of the paper, is called the bimodal power-normal model and is denoted by BPN(α) model, where α is the asymmetry parameter. The authors give some properties including moments and maximum likelihood estimation. Two important features of the model proposed is that its normalizing constant has closed and simple form and that the Fisher information matrix is nonsingular, guaranteeing large sample properties of the maximum likelihood estimators. Finally, simulation studies and real applications reveal that the proposed model can perform well in both situations.     

On the Performance of Sequential Regression Multiple Imputation Methods with Non Normal Error Distributions

Sequential regression multiple imputation has emerged as a popular approach for handling incomplete data with complex features. In this approach, imputations for each missing variable are produced based on a regression model using other variables as predictors in a cyclic manner. Normality assumption is frequently imposed for the error distributions in the conditional regression models for continuous variables, despite that it rarely holds in real scenarios. We use a simulation study to investigate the performance of several sequential regression imputation methods when the error distribution is flat or heavy tailed. The methods evaluated include the sequential normal imputation and its several extensions which adjust for non normal error terms. The results show that all methods perform well for estimating the marginal mean and proportion, as well as the regression coefficient when the error distribution is flat or moderately heavy tailed. When the error distribution is strongly heavy tailed, all methods retain their good performances for the mean and the adjusted methods have robust performances for the proportion; but all methods can have poor performances for the regression coefficient because they cannot accommodate the extreme values well. We caution against the mechanical use of sequential regression imputation without model checking and diagnostics.     

A ratio-of-uniforms method for generating exponential power variates

A ratio-of-uniforms method of generating exponential power variates is presented. It is compared to an established generalized rejection method developed by Tadikamalla (1980) and shown to be faster and more easily implemented.     

Simulating Controlled Variate and Rank Correlations Based on the Power Method Transformation

The power method transformation is a popular algorithm used for simulating correlated non normal continuous variates because of its simplicity and ease of execution. Statistical models may consist of continuous and (or) ranked variates. In view of this, the methodology is derived for simulating controlled correlation structures between non normal (a) variates, (b) ranks, and (c) variates with ranks in the context of the power method. The correlation structure between variate-values and their associated rank-order is also derived for the power method. As such, a measure of the potential loss of information is provided when ranks are used in place of variate-values. The results of a Monte Carlo simulation are provided to confirm and demonstrate the methodology.     

Direction dependence in a regression line

In this paper, we derive some simple formulae to express the association between two random variables in the case of a linear relationship, One of these representations, the cube of the correlation coefficient, is given as the ratio of the skewness of the response variable to that of the explanatory variable. This result, along with other expressions of the correlation coefficient presented in this paper, has implications for choosing the response variable in a linear regression modelling.     

Intrinsic Priors for Testing Two Normal Means with Intrinsic Bayes Factors

ABSTRACT

In this article we consider the problem of comparing two normal means with unknown common variance using a Bayesian approach. Conventional Bayes factors with improper non informative priors are not well defined. The intrinsic Bayes factors are used to overcome such a difficulty. We derive intrinsic priors whose Bayes factors are asymptotically equivalent to the corresponding intrinsic Bayes factors. We illustrate our results with numerical examples.     

Economic Design of VSI x-Bar Control Charts for Non Normally Distributed Data under Gamma (λ, 2) Failure Models

Numerous methods have been developed to calculate confidence intervals for the binomial proportion π. Boundedness and discreteness of the sample space imply that none achieves exactly the nominal α/2 left and right non coverage. We consider whether intervals calculated by a particular method tend to be located too close to, or too far out from, the center of symmetry of the support scale, 1/2. Interval location may be characterized by the balance of mesial and distal non coverage in a study evaluating coverage. A complementary approach, applicable to a calculated interval, is derived from the Box–Cox family of scale transformations.     

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.     

Dependence function for continuous bivariate densities

The notion of cross-product ratio for discrete two-way contingency table is extended to the case of continuous bivariate densities. This results in the “local dependence function” that measues the margin-free dependence between bivariate random variables. Properties and examples of the dependence function are discussed. The bivariate normal density plays a special role since it has constant dependence. Continuous bivariate densities can be constructed by specifying the dependence function along with two marginals in analogy to the construction of two-way contingency tables given marginals and patterns of interaction. The dependence function provides a partial ordering on bivariate dependence.     

Generalized skew-normal models: properties and inference

In this article, we introduce a new family of asymmetric distributions, which depends on two parameters namely, α and β, and in the special case where β = 0, the skew-normal (SN) distribution considered by Azzallini [Azzalini, A., 1985, A class of distributions which includes the normal ones. Scandinavian Journal of Statistics, 12, 171–178.] is obtained. Basic properties such as a stochastic representation and the derivation of maximum likelihood and moment estimators are studied. The asymptotic behaviour of both types of estimators is also investigated. Results of a small-scale simulation study is provided illustrating the usefulness of the new model. An application to a real data set is reported showing that it can present better fit than the SN distribution.     

Consistent Estimation of the Regression Coefficient Through Weighted Arithmetic Mean of the Inconsistent Estimators in Replicated Ultrastructural Model

The problem of consistent estimation of the slope parameter in an ultrastructural model with replicated observations is considered in this article. A consistent estimator based on a weighted arithmetic mean of two inconsistent least squares estimators is proposed which is independent of any unknown quantity. The efficiency properties of this estimator are studied.     

Variance components of the linear regression model with a random intercept

Several estimators for the variance components of the above model are derived. Biases and mean square errors of the estimators for small samples are examined. Results on the skewness and kurtosis coefficients and the large sample biases and mean square errors of these estimators are presented in detail.     

Measuring Kurtosis by Right and Left Inequality Orders

The question of how to measure kurtosis in both symmetric and asymmetric distributions is addressed using the kurtosis diagram of Zenga (2006 Zenga , M. ( 2006 ). Kurtosis . In : Kotz , S. , Read , C. B. , Balakrishnan , N. , Vidakovic , B. , eds. Encyclopedia of Statistical Sciences. 2nd ed . New York : John Wiley and Sons . [Google Scholar]). Kurtosis is related to inequality at either side of the median, and we establish a hierarchy of kurtosis orderings in which the kurtosis diagram stands at the weakest level. A sufficient condition for constructing kurtosis measures compatible with such ordering is provided. The merits of the proposed approach in both clarifying and formalizing the idea of kurtosis are evaluated and examples are discussed throughout.