A study of geometric and statistical curvatures for the skew-normal family of distributions

With special reference to the family of skew-normal distributions, we consider geometric curvature of a probability density function as a means to define and identify rare or catastrophic events—a phenomenon common in studying the financial instruments. Further, we study the statistical curvature properties of this family of distributions and discuss the sample size issue, to assess, to what extent the linear and likelihood-based inference of exponential family of distribution can be applicable for the skew-normal family.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

A Kolmogorov–Smirnov type test for skew normal distributions based on the empirical moment generating function

In this paper tests of hypothesis are constructed for the family of skew normal distributions. The proposed tests utilize the fact that the moment generating function of the skew normal variable satisfies a simple differential equation. The empirical counterpart of this equation, involving the empirical moment generating function, yields simple consistent test statistics. Finite-sample results as well as results from real data are provided for the proposed procedures.     

Inference and diagnostics in skew scale mixtures of normal regression models

Skew scale mixtures of normal distributions are often used for statistical procedures involving asymmetric data and heavy-tailed. The main virtue of the members of this family of distributions is that they are easy to simulate from and they also supply genuine expectation-maximization (EM) algorithms for maximum likelihood estimation. In this paper, we extend the EM algorithm for linear regression models and we develop diagnostics analyses via local influence and generalized leverage, following Zhu and Lee's approach. This is because Cook's well-known approach cannot be used to obtain measures of local influence. The EM-type algorithm has been discussed with an emphasis on the skew Student-t-normal, skew slash, skew-contaminated normal and skew power-exponential distributions. Finally, results obtained for a real data set are reported, illustrating the usefulness of the proposed method.     

Testing for sub-models of the skew <Emphasis Type="Italic">t</Emphasis>-distribution

The skew t-distribution includes both the skew normal and the normal distributions as special cases. Inference for the skew t-model becomes problematic in these cases because the expected information matrix is singular and the parameter corresponding to the degrees of freedom takes a value at the boundary of its parameter space. In particular, the distributions of the likelihood ratio statistics for testing the null hypotheses of skew normality and normality are not asymptotically \(\chi ^2\). The asymptotic distributions of the likelihood ratio statistics are considered by applying the results of Self and Liang (J Am Stat Assoc 82:605–610, 1987) for boundary-parameter inference in terms of reparameterizations designed to remove the singularity of the information matrix. The Self–Liang asymptotic distributions are mixtures, and it is shown that their accuracy can be improved substantially by correcting the mixing probabilities. Furthermore, although the asymptotic distributions are non-standard, versions of Bartlett correction are developed that afford additional accuracy. Bootstrap procedures for estimating the mixing probabilities and the Bartlett adjustment factors are shown to produce excellent approximations, even for small sample sizes.     

A generalized skew two-piece skew-elliptical distribution

We present a new generalized family of skew two-piece skew-elliptical (GSTPSE) models and derive some its statistical properties. It is shown that the new family of distribution may be written as a mixture of generalized skew elliptical distributions. Also, a new representation theorem for a special case of GSTPSE-distribution is given. Next, we will focus on t kernel density and prove that it is a scale mixture of the generalized skew two-piece skew normal distributions. An explicit expression for the central moments as well as a recurrence relations for its cumulative distribution function and density are obtained. Since, this special case is a uni-/bimodal distribution, a sufficient condition for each cases is given. A real data set on heights of Australian females athletes is analysed. Finally, some concluding remarks and open problems are discussed.     

The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation

In this paper, we consider the family of skew generalized t (SGT) distributions originally introduced by Theodossiou [P. Theodossiou, Financial data and the skewed generalized t distribution, Manage. Sci. Part 1 44 (12) ( 1998), pp. 1650–1661] as a skew extension of the generalized t (GT) distribution. The SGT distribution family warrants special attention, because it encompasses distributions having both heavy tails and skewness, and many of the widely used distributions such as Student's t, normal, Hansen's skew t, exponential power, and skew exponential power (SEP) distributions are included as limiting or special cases in the SGT family. We show that the SGT distribution can be obtained as the scale mixture of the SEP and generalized gamma distributions. We investigate several properties of the SGT distribution and consider the maximum likelihood estimation of the location, scale, and skewness parameters under the assumption that the shape parameters are known. We show that if the shape parameters are estimated along with the location, scale, and skewness parameters, the influence function for the maximum likelihood estimators becomes unbounded. We obtain the necessary conditions to ensure the uniqueness of the maximum likelihood estimators for the location, scale, and skewness parameters, with known shape parameters. We provide a simple iterative re-weighting algorithm to compute the maximum likelihood estimates for the location, scale, and skewness parameters and show that this simple algorithm can be identified as an EM-type algorithm. We finally present two applications of the SGT distributions in robust estimation.     

The alpha–beta skew normal distribution: properties and applications

This paper introduces a new class of skew distributions by extending the alpha skew normal distribution proposed by Elal-Olivero [Elal-Olivero, D. Alpha-skew-normal distribution. Proyecciones. 2010;29:224–240]. Statistical properties of the new family are studied in details. In particular, explicit expressions for the moments and the shape parameters including the skewness and the kurtosis coefficients and the moment generating function are derived. The problem of estimating parameters on the basis of a random sample coming from the new class of distribution is considered. To examine the performance of the obtained estimators, a Monte Carlo simulation study is conducted. Flexibility and usefulness of the proposed family of distributions are illustrated by analysing three real data sets.     

Properties and applications of the sarmanov family of bivariate distributions

We discuss properties of the bivariate family of distributions introduced by Sarmanov (1966). It is shown that correlation coefficients of this family of distributions have wider range than those of the Farlie-Gumbel-Morgenstern distributins. Possible applications of this family of bivariate distributions as prior distributins in Bayesian inference are discussed. The density of the bivariate Sarmanov distributions with beta marginals can be expressed as a linear combination of products of independent beta densities. This pseudoconjugate property greatly reduces the complexity of posterior computations when this bivariate beta distribution is used as a prior. Multivariate extensions are derived.     

Calibration model with skew scale mixtures of normal distributions

This work presents a new linear calibration model with replication by assuming that the error of the model follows a skew scale mixture of the normal distributions family, which is a class of asymmetric thick-tailed distributions that includes the skew normal distribution and symmetric distributions. In the literature, most calibration models assume that the errors are normally distributed. However, the normal distribution is not suitable when there are atypical observations and asymmetry. The estimation of the calibration model parameters are done numerically by the EM algorithm. A simulation study is carried out to verify the properties of the maximum likelihood estimators. This new approach is applied to a real dataset from a chemical analysis.     

A family of skew distributions with mode-invariance through transformation of scale

Recently, a new family of skew distributions was proposed using a specific class of transformation of scale, in which the normalizing constant remains unchanged and unimodality is readily assured. In this paper, we introduce the mode invariance in this family, which allows us to easily study certain properties, including monotonicity of skewness, and incorporate various favorable properties. The entropy maximization for a skew distribution is discussed. A numerical study is also conducted.     

Statistical inference for a general class of asymmetric distributions

We consider a general class of asymmetric univariate distributions depending on a real-valued parameter α, which includes the entire family of univariate symmetric distributions as a special case. We discuss the connections between our proposal and other families of skew distributions that have been studied in the statistical literature. A key element in the construction of such families of distributions is that they can be stochastically represented as the product of two independent random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes as well as extensions to the multivariate case. We also study statistical inference for this class based on the method of moments and maximum likelihood. We give special attention to the skew-power exponential distribution, but other cases like the skew-t distribution are also considered. Finally, the statistical methods are illustrated with 3 examples based on real datasets.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.     

On multiple constraint skewed models

The Azzalini [A. Azzalini, A class of distributions which includes the normal ones, Scandi. J. Statist. 12 (1985), pp. 171–178.] skew normal model can be viewed as one involving normal components subject to a single linear constraint. As a natural extension of this model, we discuss skewed models involving multiple linear and nonlinear constraints and possibly non-normal components. Particular attention is devoted to a distribution called the extended two-piece normal (ETN) distribution. This model is a two-constraint extension of the two-piece normal model introduced by Kim [H.J. Kim, On a class of two-piece skew normal distributions, Statistics 39(6) (2005), pp. 537–553.]. Likelihood inference for the ETN distribution is developed and illustrated using two data sets.     

The three-parameter two-piece normal family of distributions and its fitting

This note introduces a family of skew and symmetric distributions containing the normal family and indexed by three parameters with clear meanings. Another respect in which this family compares favourably with families like the Pearson family, the Bessel-Gram-Charlier family and the Johnson family is ease of maximum likelihood fitting. Fitting by the method of moments is also considered. Asymptotic distributions of maximum likelihood and moment estimators are worked out. A test of symmetry and normality is suggested.     

Doubly adaptive biased coin designs with heterogeneous responses

Doubly adaptive biased coin design (DBCD) is an important family of response-adaptive randomization procedures for clinical trials. It uses sequentially updated estimation to skew the allocation probability to favor the treatment that has performed better thus far. An important assumption for the DBCD is the homogeneity assumption for the patient responses. However, this assumption may be violated in many sequential experiments. Here we prove the robustness of the DBCD against certain time trends in patient responses. Strong consistency and asymptotic normality of the design are obtained under some widely satisfied conditions. Also, we propose a general weighted likelihood method to reduce the bias caused by the heterogeneity in the inference after a trial. Some numerical studies are also presented to illustrate the finite sample properties of DBCD.     

Intrinsic Bayesian inference on a Poisson rate and on the ratio of two Poisson rates

Our purpose is to explore the intrinsic Bayesian inference on the rate of a Poisson distribution and on the ratio of the rates of two independent Poisson distributions, with the natural conjugate family of priors in the first case and the semi-conjugate family of priors defined by Laurent and Legrand (2011) in the second case. Intrinsic Bayesian inference is derived from the Bayesian decision theory framework based on the intrinsic discrepancy loss function. We cover in particular the case of some objective Bayesian procedures suggested by Bernardo when considering reference priors.     

A skew extension of the t-distribution, with applications

Summary. A tractable skew t -distribution on the real line is proposed. This includes as a special case the symmetric t -distribution, and otherwise provides skew extensions thereof. The distribution is potentially useful both for modelling data and in robustness studies. Properties of the new distribution are presented. Likelihood inference for the parameters of this skew t -distribution is developed. Application is made to two data modelling examples.     

Sine-skewed circular distributions

In this paper, we consider skew-symmetric circular distributions generated by perturbation of a symmetric circular distribution. The main focus of the paper, the sine-skewed family of distributions, is a special case of the construction due to Umbach and Jammalamadaka (Stat Probab Lett 79:659–663, 2009). Very general results are provided for the properties of any such distribution, and the sine-skewed Jones–Pewsey distribution is introduced as a particularly flexible model of this type. We study its properties as well as those of three of its special cases. General results are also provided for maximum likelihood estimation of the parameters of any sine-skewed distribution. The developed models and methods of inference are applied in analyses of three circular data sets. Two of them shed new light on previously published analyses.     

Models for Extremal Dependence Derived from Skew‐symmetric Families

Skew‐symmetric families of distributions such as the skew‐normal and skew‐t represent supersets of the normal and t distributions, and they exhibit richer classes of extremal behaviour. By defining a non‐stationary skew‐normal process, which allows the easy handling of positive definite, non‐stationary covariance functions, we derive a new family of max‐stable processes – the extremal skew‐t process. This process is a superset of non‐stationary processes that include the stationary extremal‐t processes. We provide the spectral representation and the resulting angular densities of the extremal skew‐t process and illustrate its practical implementation.