A nonparametric allocation scheme for classification based on transvariation probabilities

In this paper, a nonparametric discriminant analysis procedure that is less sensitive than traditional procedures to deviations from the usual assumptions is proposed. The procedure uses the projection pursuit methodology where the projection index is the two-group transvariation probability. Montanari [A. Montanari, Linear discriminant analysis and transvariation, J. Classification 21 (2004), pp. 71–88] proposed and used this projection index to measure group separation but allocated the new observation using distances. Our procedure employs a method of allocation based on group–group transvariation probability to classify the new observation. A simulation study shows that the procedure proposed in this paper provides lower misclassification error rates than classical procedures like linear discriminant analysis and quadratic discriminant analysis and recent procedures like maximum depth and Montanari's transvariation-based classifiers, when the underlying distributions are skewed and/or the prior probabilities are unequal.     

The Kumaraswamy skew-t distribution and its related properties

Skew normal distribution is an alternative distribution to the normal distribution to accommodate asymmetry. Since then extensive studies have been done on applying Azzalini’s skewness mechanism to other well-known distributions, such as skew-t distribution, which is more flexible and can better accommodate long tailed data than the skew normal one. The Kumaraswamy generalized distribution (Kw ? F) is another new class of distribution which is capable of fitting skewed data that can not be fitted well by existing distributions. Such a distribution has been widely studied and various versions of generalization of this distribution family have been introduced. In this article, we introduce a new generalization of the skew-t distribution based on the Kumaraswamy generalized distribution. The new class of distribution, which we call the Kumaraswamy skew-t (KwST) has the ability of fitting skewed, long, and heavy-tailed data and is more flexible than the skew-t distribution as it contains the skew-t distribution as a special case. Related properties of this distribution family such as mathematical properties, moments, and order statistics are discussed. The proposed distribution is applied to a real dataset to illustrate the estimation procedure.     

Some criteria for projection pursuit

The idea of searching for orthogonal projections, from a multidimensional space into a linear subspace, as an aid to detecting non-linear structure has been named exploratory projection pursuit.Most approaches are tied to the idea of searching for interesting projections. Typically, an interesting projection is one where the distribution of the projected data differs from the normal distribution. In this paper we define two projection indices which are aimed specifically at finding projections that best show grouped structure in the plane, if this exists in the multi-dimensional space. These involve a numerical optimization problem which is tackled in two stages, the projection and the pursuit; the first is based on a procedure to generate pseudo-random rotation matrices in the sense of the grand tour by D. Asimov (1985), and the second is a local numerical optimization procedure. One artificial and one real example illustrate the performance of the suggested indices.     

Interval robust design under contaminated and non normal data

Abstract

Robust parameter design (RPD) is an effective tool, which involves experimental design and strategic modeling to determine the optimal operating conditions of a system. The usual assumptions of RPD are that normally distributed experimental data and no contamination due to outliers. And generally the parameter uncertainties in response models are neglected. However, using normal theory modeling methods for a skewed data and ignoring parameter uncertainties can create a chain of degradation in optimization and production phases such that misleading fit, poor estimated optimal operating conditions, and poor quality products. This article presents a new approach based on confidence interval (CI) response modeling for the process mean. The proposed interval robust design makes the system median unbiased for the mean and uses midpoint of the interval as a measure of location performance response. As an alternative robust estimator for the process variance response modeling, using biweight midvariance is proposed which is both resistant and robust of efficiency where normality is not met. The results further show that the proposed interval robust design gives a robust solution to the skewed structure of the data and to contaminated data. The procedure and its advantages are illustrated using two experimental design studies.     

Testing Equality of Order Constrained Inverse Gaussian Means

The inverse Gaussian family of non negative, skewed random variables is analytically simple, and its inference theory is well known to be analogous to the normal theory in numerous ways. Hence, it is widely used for modeling non negative positively skewed data. In this note, we consider the problem of testing homogeneity of order restricted means of several inverse Gaussian populations with a common unknown scale parameter using an approach based on the classical methods, such as Fisher's, for combining independent tests. Unlike the likelihood approach which can only be readily applied to a limited number of restrictions and the settings of equal sample sizes, this approach is applicable to problems involving a broad variety of order restrictions and arbitrary sample size settings, and most importantly, no new null distributions are needed. An empirical power study shows that, in case of the simple order, the test based on Fisher's combination method compares reasonably with the corresponding likelihood ratio procedure.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

Generalized confidence interval estimation for the mean of delta-lognormal distribution: an application to New Zealand trawl survey data

Highly skewed and non-negative data can often be modeled by the delta-lognormal distribution in fisheries research. However, the coverage probabilities of extant interval estimation procedures are less satisfactory in small sample sizes and highly skewed data. We propose a heuristic method of estimating confidence intervals for the mean of the delta-lognormal distribution. This heuristic method is an estimation based on asymptotic generalized pivotal quantity to construct generalized confidence interval for the mean of the delta-lognormal distribution. Simulation results show that the proposed interval estimation procedure yields satisfactory coverage probabilities, expected interval lengths and reasonable relative biases. Finally, the proposed method is employed in red cod densities data for a demonstration.     

A Bayesian estimate of the concordance correlation coefficient with skewed data

Concordance correlation coefficient (CCC) is one of the most popular scaled indices used to evaluate agreement. Most commonly, it is used under the assumption that data is normally distributed. This assumption, however, does not apply to skewed data sets. While methods for the estimation of the CCC of skewed data sets have been introduced and studied, the Bayesian approach and its comparison with the previous methods has been lacking. In this study, we propose a Bayesian method for the estimation of the CCC of skewed data sets and compare it with the best method previously investigated. The proposed method has certain advantages. It tends to outperform the best method studied before when the variation of the data is mainly from the random subject effect instead of error. Furthermore, it allows for greater flexibility in application by enabling incorporation of missing data, confounding covariates, and replications, which was not considered previously. The superiority of this new approach is demonstrated using simulation as well as real‐life biomarker data sets used in an electroencephalography clinical study. The implementation of the Bayesian method is accessible through the Comprehensive R Archive Network. Copyright © 2015 John Wiley & Sons, Ltd.     

A goodness-of-fit test for generalized error distribution

The Generalized Error Distribution is a widespread flexible family of symmetric probability distribution. Thanks to its properties it is becoming more and more popular in many science fields therefore determining if a sample is drawn from a GED is an important issue that usually is pursued with a graphical approach. In this paper we present a new goodness-of-fit test for GED that shows good performances for detecting non GED distribution when the alternative distribution is either skewed or a mixture. A comparison between well known tests and this new procedure is performed through a simulation study. We have developed a function that performs the analysis described in this paper in the R environment. The computational time required to compute this procedure is negligible.     

A new bivariate distribution with weighted exponential marginals and its multivariate generalization

Gupta and Kundu (Statistics 43:621–643, 2009) recently introduced a new class of weighted exponential distribution. It is observed that the proposed weighted exponential distribution is very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the weighted exponential marginals. Different properties of this new bivariate distribution have been investigated. This new family has three unknown parameters, and it is observed that the maximum likelihood estimators of the unknown parameters can be obtained by solving a one-dimensional optimization procedure. We obtain the asymptotic distribution of the maximum likelihood estimators. Small simulation experiments have been performed to see the behavior of the maximum likelihood estimators, and one data analysis has been presented for illustrative purposes. Finally we discuss the multivariate generalization of the proposed model.     

Estimating the point of intersection of two lines under correlation

This paper introduces a new biostatistical approach to the problems of estimating true values and approximating the distribution of true values from unreliable data.We present the basic rationale for the unmixing method and report on a simulation study of its properties in estimating the centiles of a skewed, outlier-prone class of distributions.We also present an application to highly skewed USDA vitamin A intake data, and a pseudo-code version of the unmixing algorithm that we hope will allow other researchers to experiment with it     

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.     

The new unified representation of multivariate skewed distributions

There are so many proposals in construction skewed distributions, and it is worth finding an overall class which covers all of these proposals. We introduce a new unified representation of multivariate skewed distributions. We will show that this new unified multivariate form of skewed distributions includes all of the continuous multivariate skewed distributions in the literature. This new unified representation is based on the multivariate probability integral transformation and can be decomposed into one factor that is original multivariate symmetric probability density function (pdf) f on ? ^k and skewed factor defined by a pdf p on [0, 1] ^k . This decomposition leads us to prove some useful properties of this new unified form. Stochastic representations and basic properties of this new form are also investigated in this article. Our work is motivated by considering the different skewing mechanisms which lead to different skewed distributions and show that all of these common-used distributions can be viewed as a new unified form.     

Permuting incomplete paired data: a novel exact and asymptotic correct randomization test

Various statistical tests have been developed for testing the equality of means in matched pairs with missing values. However, most existing methods are commonly based on certain distributional assumptions such as normality, 0-symmetry or homoscedasticity of the data. The aim of this paper is to develop a statistical test that is robust against deviations from such assumptions and also leads to valid inference in case of heteroscedasticity or skewed distributions. This is achieved by applying a clever randomization approach to handle missing data. The resulting test procedure is not only shown to be asymptotically correct but is also finitely exact if the distribution of the data is invariant with respect to the considered randomization group. Its small sample performance is further studied in an extensive simulation study and compared to existing methods. Finally, an illustrative data example is analysed.     

Exploring multivariate data using directions of high density

The most common techniques for graphically presenting a multivariate dataset involve projection onto a one or two-dimensional subspace. Interpretation of such plots is not always straightforward because projections are smoothing operations in that structure can be obscured by projection but never enhanced. In this paper an alternative procedure for finding interesting features is proposed that is based on locating the modes of an induced hyperspherical density function, and a simple algorithm for this purpose is developed. Emphasis is placed on identifying the non-linear effects, such as clustering, so to this end the data are firstly sphered to remove all of the location, scale and correlational structure. A set of simulated bivariate data and artistic qualities of painters data are used as examples.     

Bayesian inference for the generalized exponential distribution

The two-parameter generalized exponential (GE) distribution was introduced by Gupta and Kundu [Gupta, R.D. and Kundu, D., 1999, Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41(2), 173–188.]. It was observed that the GE can be used in situations where a skewed distribution for a nonnegative random variable is needed. In this article, the Bayesian estimation and prediction for the GE distribution, using informative priors, have been considered. Importance sampling is used to estimate the parameters, as well as the reliability function, and the Gibbs and Metropolis samplers data sets are used to predict the behavior of further observations from the distribution. Two data sets are used to illustrate the Bayesian procedure.     

Analyzing survival data with highly negatively skewed distribution: The Gompertz-sinh family

In this article, we explore a new two-parameter family of distribution, which is derived by suitably replacing the exponential term in the Gompertz distribution with a hyperbolic sine term. The resulting new family of distribution is referred to as the Gompertz-sinh distribution, and it possesses a thicker and longer lower tail than the Gompertz family, which is often used to model highly negatively skewed data. Moreover, we introduce a useful generalization of this model by adding a second shape parameter to accommodate a variety of density shapes as well as nondecreasing hazard shapes. The flexibility and better fitness of the new family, as well as its generalization, is demonstrated by providing well-known examples that involve complete, group, and censored data.     

Estimating the number of factors to include in a high-dimensional multivariate bilinear model

We present two new statistics for estimating the number of factors underlying in a multivariate system. One of the two new methods, the original NUMFACT, has been used in high profile environmental studies. The two new methods are first explained from a geometrical viewpoint. We then present an algebraic development and asymptotic cutoff points. Next we present a simulation study that shows that for skewed data the new methods are typically superior to traditional methods and for normally distributed data the new methods are competitive to the best of the traditional methods. We finally show how the methods compare by using two environmental data sets.     

A unified view on skewed distributions arising from selections

Parametric families of multivariate nonnormal distributions have received considerable attention in the past few decades. The authors propose a new definition of a selection distribution that encompasses many existing families of multivariate skewed distributions. Their work is motivated by examples that involve various forms of selection mechanisms and lead to skewed distributions. They give the main properties of selection distributions and show how various families of multivariate skewed distributions, such as the skew‐normal and skew‐elliptical distributions, arise as special cases. The authors further introduce several methods of constructing selection distributions based on linear and nonlinear selection mechanisms.     

The multivariate skew-slash distribution

The slash distribution is often used as a challenging distribution for a statistical procedure. In this article, we define a skewed version of the slash distribution in the multivariate setting and derive several of its properties. The multivariate skew-slash distribution is shown to be easy to simulate from and can therefore be used in simulation studies. We provide various examples for illustration.