Parametric and nonparametric confidence intervals for estimating the difference of means of two skewed populations

In this paper, we have reviewed and proposed several interval estimators for estimating the difference of means of two skewed populations. Estimators include the ordinary-t, two versions proposed by Welch [17] and Satterthwaite [15], three versions proposed by Zhou and Dinh [18], Johnson [9], Hall [8], empirical likelihood (EL), bootstrap version of EL, median t proposed by Baklizi and Kibria [2] and bootstrap version of median t. A Monte Carlo simulation study has been conducted to compare the performance of the proposed interval estimators. Some real life health related data have been considered to illustrate the application of the paper. Based on our findings, some possible good interval estimators for estimating the mean difference of two populations have been recommended for the researchers.     

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.     

Discriminating among Weibull,log-normal,and log-logistic distributions

In this article, we consider the problem of the model selection/discrimination among three different positively skewed lifetime distributions. All these three distributions, namely; the Weibull, log-normal, and log-logistic, have been used quite effectively to analyze positively skewed lifetime data. In this article, we have used three different methods to discriminate among these three distributions. We have used the maximized likelihood method to choose the correct model and computed the asymptotic probability of correct selection. We have further obtained the Fisher information matrices of these three different distributions and compare them for complete and censored observations. These measures can be used to discriminate among these three distributions. We have also proposed to use the Kolmogorov–Smirnov distance to choose the correct model. Extensive simulations have been performed to compare the performances of the three different methods. It is observed that each method performs better than the other two for some distributions and for certain range of parameters. Further, the loss of information due to censoring are compared for these three distributions. The analysis of a real dataset has been performed for illustrative purposes.     

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.     

Two-Piece location-scale distributions based on scale mixtures of normal family

In this work, we study the maximum likelihood (ML) estimation problem for the parameters of the two-piece (TP) distribution based on the scale mixtures of normal (SMN) distributions. This is a family of skewed distributions that also includes the scales mixtures of normal class, and is flexible enough for modeling symmetric and asymmetric data. The ML estimates of the proposed model parameters are obtained via an expectation-maximization (EM)-type algorithm.     

Loss-based approach to two-piece location-scale distributions with applications to dependent data

Two-piece location-scale models are used for modeling data presenting departures from symmetry. In this paper, we propose an objective Bayesian methodology for the tail parameter of two particular distributions of the above family: the skewed exponential power distribution and the skewed generalised logistic distribution. We apply the proposed objective approach to time series models and linear regression models where the error terms follow the distributions object of study. The performance of the proposed approach is illustrated through simulation experiments and real data analysis. The methodology yields improvements in density forecasts, as shown by the analysis we carry out on the electricity prices in Nordpool markets.

     

Multivariate log-Birnbaum–Saunders regression models

In this paper, we present a multivariate version of the skewed log-Birnbaum–Saunders regression model. This new family of distributions holds good properties such as marginal variables following univariate skewed log-Birnbaum–Saunders distributions, besides presenting the usual log-Birnbaum–Saunders distribution as a particular case. Furthermore, the model parameters are estimated through maximum-likelihood methods, a closed-form expression for the Fisher’s information matrix is presented, and testing hypothesis for model parameters is performed. Two real datasets are analyzed and results are discussed.     

A log-Birnbaum–Saunders regression model with asymmetric errors

This paper introduces a skewed log-Birnbaum–Saunders regression model based on the skewed sinh-normal distribution proposed by Leiva et al. [A skewed sinh-normal distribution and its properties and application to air pollution, Comm. Statist. Theory Methods 39 (2010), pp. 426–443]. Some influence methods, such as the local influence and generalized leverage, are presented. Additionally, we derived the normal curvatures of local influence under some perturbation schemes. An empirical application to a real data set is presented in order to illustrate the usefulness of the proposed model.     

An exact density-based empirical likelihood ratio test for paired data

The Wilcoxon rank-sum test and its variants are historically well-known to be very powerful nonparametric decision rules for testing no location difference between two groups given paired data versus a shift alternative. In this title, we propose a new alternative empirical likelihood (EL) ratio approach for testing the equality of marginal distributions given that sampling is from a continuous bivariate population. We show that in various shift alternative scenarios the proposed exact test is superior to the classic nonparametric procedures, which may break down completely or are frequently inferior to the density-based EL ratio test. This is particularly true in the cases where there is a nonconstant shift under the alternative or the data distributions are skewed. An extensive Monte Carlo study shows that the proposed test has excellent operating characteristics. We apply the density-based EL ratio test to analyze real data from two medical studies.     

Power-normal distribution

Recently, Gupta and Gupta [Analyzing skewed data by power-normal model, Test 17 (2008), pp. 197–210] proposed the power-normal distribution for which normal distribution is a special case. The power-normal distribution is a skewed distribution, whose support is the whole real line. Our main aim of this paper is to consider bivariate power-normal distribution, whose marginals are power-normal distributions. We obtain the proposed bivariate power-normal distribution from Clayton copula, and by making a suitable transformation in both the marginals. Lindley–Singpurwalla distribution also can be used to obtain the same distribution. Different properties of this new distribution have been investigated in detail. Two different estimators are proposed. One data analysis has been performed for illustrative purposes. Finally, we propose some generalizations to multivariate case also along the same line and discuss some of its properties.     

Absolute continuous bivariate generalized exponential distribution

Generalized exponential distribution has been used quite effectively to model positively skewed lifetime data as an alternative to the well known Weibull or gamma distributions. In this paper we introduce an absolute continuous bivariate generalized exponential distribution by using a simple transformation from a well known bivariate exchangeable distribution. The marginal distributions of the proposed bivariate generalized exponential distributions are generalized exponential distributions. The joint probability density function and the joint cumulative distribution function can be expressed in closed forms. It is observed that the proposed bivariate distribution can be obtained using Clayton copula with generalized exponential distribution as marginals. We derive different properties of this new distribution. It is a five-parameter distribution, and the maximum likelihood estimators of the unknown parameters cannot be obtained in closed forms. We propose some alternative estimators, which can be obtained quite easily, and they can be used as initial guesses to compute the maximum likelihood estimates. One data set has been analyzed for illustrative purposes. Finally we propose some generalization of the proposed model.     

Construction of bivariate and multivariate weighted distributions via conditioning

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this article, we propose an alternative method to construct a new family of bivariate and multivariate weighted distributions. For illustrative purposes, some examples of the proposed method are presented. Several structural properties of the bivariate weighted distributions including marginal distributions together with distributions of the minimum and maximum, evaluation of the reliability parameter, and verification of total positivity of order two are also presented. In addition, we provide some multivariate extensions of the proposed models. A real-life data set is used to show the applicability of these bivariate weighted distributions.     

On a multidimensional general bootstrap for empirical estimator of continuous-time semi-Markov kernels with applications

The present paper introduces a general notion and presents results of bootstrapped empirical estimators of the semi-Markov kernels and of the conditional transition distributions for semi-Markov processes with countable state space, constructed by exchangeably weighting the sample. Our proposal provides a unification of bootstrap methods in the semi-Markov setting including, in particular, Efron's bootstrap. Asymptotic properties of these generalised bootstrapped empirical distributions are obtained, under mild conditions by a martingale approach. We also obtain some new results on the weak convergence of the empirical semi-Markov processes. We apply these general results in several statistical problems such as the construction of confidence bands and the goodness-of-fit tests where the limiting distributions are derived under the null hypothesis. Finally, we introduce the quantile estimators and their bootstrapped versions in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. Our theoretical results and numerical examples by simulations demonstrate the merits of the proposed techniques.     

Parameter and quantile estimation for the three-parameter gamma distribution based on statistics invariant to unknown location

The three-parameter gamma distribution is widely used as a model for distributions of life spans, reaction times, and for other types of skewed data. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter gamma distribution, which avoids the problem of unbounded likelihood, based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared to other prominent methods in terms of bias and mean squared error. Finally, we present two illustrative examples.     

A progressive shift alternative to evaluate nonparametric tests for skewed data

ABSTRACT

When the power of different nonparametric tests is evaluated by simulation, the alternative hypothesis should be carefully designed to ensure validity of the results in the specific research field. In the article, we propose a probit-based progressive shift alternative that is more realistic than the simple shift alternative for skewed non-negative data that occur in many research areas. Our motivation comes from parasitology. The progressive shift alternative is used to compare the power of six location-scale tests and seven commonly used location tests for several skewed theoretical and empirical parasite distributions. It is shown that location-scale tests are more powerful than location tests. Programs for applying the methods studied in the article are freely available for download.     

On families of beta- and generalized gamma-generated distributions and associated inference

A general family of univariate distributions generated by beta random variables, proposed by Jones, has been discussed recently in the literature. This family of distributions possesses great flexibility while fitting symmetric as well as skewed models with varying tail weights. In a similar vein, we define here a family of univariate distributions generated by Stacy’s generalized gamma variables. For these two families of univariate distributions, we discuss maximum entropy characterizations under suitable constraints. Based on these characterizations, an expected ratio of quantile densities is proposed for the discrimination of members of these two broad families of distributions. Several special cases of these results are then highlighted. An alternative to the usual method of moments is also proposed for the estimation of the parameters, and the form of these estimators is particularly amenable to these two families of distributions.     

A new generalization of the gamma distribution with application to negatively skewed survival data

We propose a three-parameter distribution referred to as the reflected- shifted-truncated gamma (RSTG) distribution to model negatively skewed data. Various properties of the proposed distribution are derived. The estimation of the model parameters is approached by maximum likelihood methods and the observed information matrix is derived. Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation for both small and large samples. Using information theoretic criteria, we compare the RSTG distribution to the exponential, generalized F, generalized gamma, Gompertz, log-logistic, lognormal, Rayleigh, and Weibull distributions in three negatively skewed real datasets.     

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 multimodal skewed extension of normal distribution: its properties and applications

A multimodal skewed extension of normal distribution is proposed by applying the general method as in [Huang WJ, Chen YH. Generalized skew-Cauchy distribution. Stat Probab Lett. 2007;77:1137–1147] for the construction of skew-symmetric distributions by using a trigonometric periodic skew function. Some of its distributional properties are investigated. Properties of maximum likelihood estimation of the parameters are studied numerically by simulation. The suitability of the proposed distribution in empirical data modelling is investigated by carrying out comparative fitting of two real-life data sets.     

A Sample Selection Model with Skew‐normal Distribution

Non‐random sampling is a source of bias in empirical research. It is common for the outcomes of interest (e.g. wage distribution) to be skewed in the source population. Sometimes, the outcomes are further subjected to sample selection, which is a type of missing data, resulting in partial observability. Thus, methods based on complete cases for skew data are inadequate for the analysis of such data and a general sample selection model is required. Heckman proposed a full maximum likelihood estimation method under the normality assumption for sample selection problems, and parametric and non‐parametric extensions have been proposed. We generalize Heckman selection model to allow for underlying skew‐normal distributions. Finite‐sample performance of the maximum likelihood estimator of the model is studied via simulation. Applications illustrate the strength of the model in capturing spurious skewness in bounded scores, and in modelling data where logarithm transformation could not mitigate the effect of inherent skewness in the outcome variable.