Some Comments on Hassairi et al.'s “Implicit Distributions and Estimation”

In social interaction studies, one commonly encounters repeated displays of behaviors along with their duration data. Statistical methods for the analysis of such data use either parametric (e.g., Weibull) or semi-nonparametric (e.g., Cox) proportional hazard models, modified to include random effects (frailty) which account for the correlation of repeated occurrences of behaviors within a unit (dyad). However, dyad-specific random effects by themselves are not able to account for the ordering of event occurrences within dyads. The occurrence of an event (behavior) can make further occurrences of the same behavior to be more or less likely during an interaction. This article develops event-dependent random effects models for analyzing repeated behaviors data using a Bayesian approach. The models are illustrated by a dataset relating to emotion regulation in families with children who have behavioral or emotional problems.     

A generalized skew slash distribution via gamma-normal distribution

In this article, we introduce a generalization of the slash distribution via the gamma-normal distribution. We define the new slash distribution by relation of a gamma-normal random variable with respect to a power of a uniform random variable. The newly defined distribution generalizes the slash distribution and is more flexible in terms of its kurtosis and skewness than the slash distribution. Basic properties of the new distribution are studied. We derive the maximum likelihood estimators of its parameters and apply the distribution to a real dataset.     

A new class of skew-symmetric distributions and related families

In this work, we investigate a new class of skew-symmetric distributions, which includes the distributions with the probability density function (pdf) given by g _α(x)=2f(x) G(α x), introduced by Azzalini [A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]. We call this new class as the symmetric-skew-symmetric family and it has the pdf proportional to f(x) G _β(α x), where G _β(x) is the cumulative distribution function of g _β(x). We give some basic properties for the symmetric-skew-symmetric family and study the particular case obtained from the normal distribution.     

A new class of generalized Lindley distributions with applications

A new four-parameter class of generalized Lindley (GL) distribution called the beta-generalized Lindley (BGL) distribution is proposed. This class of distributions contains the beta-Lindley, GL and Lindley distributions as special cases. Expansion of the density of the BGL distribution is obtained. The properties of these distributions, including hazard function, reverse hazard function, monotonicity property, shapes, moments, reliability, mean deviations, Bonferroni and Lorenz curves are derived. Measures of uncertainty such as Renyi entropy and s-entropy as well as Fisher information are presented. Method of maximum likelihood is used to estimate the parameters of the BGL and related distributions. Finally, real data examples are discussed to illustrate the applicability of this class of models.     

Hazard functions and life distributions in discrete time

The problem of studying lifelength distributions in discrete time is considered for certain forms of hazard functions. A class of life distributions that consists of the geometric, the Waring and the negative hypergeometric distributions is shown to result when the hazard function is inversely proportional to some linear function of time.     

Slashed power-Lindley distribution

In this article, we introduce the slashed power-Lindley distribution. This model can be seen as an extension of the power-Lindley distribution with more flexibility in terms of the kurtosis of distribution. It arises as the ratio of two independent random variables, the one being a power-Lindley distribution and a power of the uniform distribution. We present properties and carry out estimates of the model parameters by the maximum likelihood method. Finally, we conduct a small simulation study to evaluate the performance of maximum likelihood estimators and we analyze a real data set to illustrate the usefulness of the new model.     

Truncated inverted generalized exponential distribution and its properties

The inverted generalized exponential distribution is defined as an alternative model for lifetime data. The existence of moments of this distribution is shown to hold under some restrictions. However, all the moments exist for the truncated inverted generalized exponential distribution and closed-form expressions for them are derived in this article. The distributional properties of this truncated distribution are studied. Maximum likelihood estimation method is discussed for the estimation of the parameters of the distribution both theoretically and empirically. In order to see the modeling performance of the distribution, two real datasets are analyzed.     

New Generalizations of Cauchy Distribution

The generalized skew-normal distribution introduced by Balakrishnan (2002 Balakrishnan , N. ( 2002 ). Discussion on ‘Skew multivariate models related to hidden truncation and/or selective reporting’ by B. C. Arnold and R. J. Beaver . Test 11 : 37 – 39 .[Web of Science ®] , [Google Scholar]) is used to obtain new generalizations of univariate Cauchy distribution with two parameters, denoted by GC _{m, n} (a, b) with m and n non-negative integer numbers and a, b ∈ R. For cases (m, n) = (1, 2), (m, n) = (2, 1), (m, n) = (0, 3) and (m, n) = (3, 0) explicit forms of the density functions are derived and compared to previous generalizations of Cauchy and skew-Cauchy distributions.     

Some gamma distributions

Three new generalizations of the standard gamma distribution introduced by the author are reviewed. Various properties are derived for each distribution, including its hazard rate function and moments. An application is illustrated to drought data.     

Lifetime distributions motivated by series and parallel structures

According to Ross, any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. Motivated by this, we propose new three-parameter lifetime distributions by compounding geometric, power series, and exponential distributions. The distributions can allow for decreasing, increasing, bathtub-shaped, and upside down bathtub-shaped hazard rates. A mathematical treatment of the new distributions is provided including expressions for their density functions, Shannon and Rényi entropies, mean residual life functions, hazard rate functions, quantiles, and moments. The method of maximum likelihood is used for estimating parameters. Five of the new distributions are studied in detail. Finally, two illustrative data examples and a sensitivity analysis are presented.     

贝叶斯统计推断及其主要进展

贝叶斯统计推断作为现代统计分析方法的重要内容,对于统计学理论的发展具有里程碑的作用。深入总结其研究的主要进展,具有重要的现实意义。在查阅国内外重要学术研究资料的基础上,从贝叶斯统计推断的思想、与古典统计的研究思路比较和贝叶斯统计推断研究的主要进展三个方面作了综述与介绍,力图达到认识贝叶斯统计推断及其研究现状的目的。     

Optimal cutpoints for random observations

For the discretisation of a continuous random variable into different categories the choice of cutpoints is essential. A popular application is the contingent valuation method. In a parametric approach, the choice of cutpoints directly effects the quality of the estimates. Therefore, optimal cutpoints are desirable in order to estimate the parameters most accurately. We consider an arbitrary number of cutpoints and determine optimal cutpoints for the exponential and Gumbel distribution and prove that the c-optimal cutpoints for the location parameter of the logistic distribution have corresponding equal category probabilities. Furthermore, we show that in the limiting case for infinitely many cutpoints there is no loss of information.     

The inverse problem of multivariate and matrix-variate skew normal distributions

In this paper, we prove that the joint distribution of random vectors Z ₁ and Z ₂ and the distribution of Z ₂ are skew normal provided that Z ₁ is skew normally distributed and Z ₂ conditioning on Z ₁ is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.     

On finite mixtures of geometric and negative binomial distributions

Finite mixtures of distributions have been getting increasing use in the applied literature. In the continuous case, linear combinations of exponentials and gammas have been shown to be well suited for modeling purposes. In the discrete case, the focus has primarily been on continuous mixing, usually of Poisson distributions and typically using gammas to describe the random parameter, But many of these applications are forced, especially when a continuous mixing distribution is used. Instead, it is often prefe-rable to try finite mixtures of geometries or negative binomials, since these are the fundamental building blocks of all discrete random variables. To date, a major stumbling block to their use has been the lack of easy routines for estimating the parameters of such models. This problem has now been alleviated by the adaptation to the discrete case of numerical procedures recently developed for exponential, Weibull, and gamma mixtures. The new methods have been applied to four previously studied data sets, and significant improvements reported in goodness-of-fit, with resultant implications for each affected study.     

A Numerical Study of Small Parameter Behavior of Some Families of Distributions

Limiting distributions play an important role in approximating the exact distributions, especially when they have a rather cumbersome analytic form, or simply when they do not have a closed from. The question that naturally arises is how good the approximation is. In this article, we propose a procedure for the numerical assessment of the “goodness” of some easy-to-calculate limiting distributions, originally proposed in Bar-Lev and Enis, in various cases of the underlying distributions, some of which are inherently computationally challenging. The details of the procedure are provided in three examples. The first example deals with the gamma distributions; the second deals with Bessel distributions related to a symmetric random walk, and the third example deals with positive stable distributions. The details of two additional variations of these examples are also discussed. These examples illustrate the ease with which the limiting approximations could be applied in the various cases, well-demonstrating their computational simplicity and attractiveness.     

Asymptotic relative efficiencies of r-estimators of location

A table of asymptotic relative efficiencies of a large variety of R-estimators is given and some of its uses discussed.The family of distributions used includes both those common in the robustness literature and those that correspondent to various rank tests for location.     

On the asymptotic efficiency of certain correlation tests of fit

Tests of fit based on correlation-type statistics are investigated for the exponential, extreme-value, and logistic distributions. The statistics are shown to be asymptotically normal at the rate log^½n. The result is used to show that such tests have 0 asymptotic relative efficiency.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

Asymptotic calculations of functions of expected values and covariances of order statistics

Suppose m and V are respectively the vector of expected values and the covariance matrix of the order statistics of a sample of size n from a continuous distribution F. A method is presented to calculate asymptotic values of functions of m and V ^–1, for distributions F which are sufficiently regular. Values are given for the normal, logistic, and extreme-value distributions; also, for completeness, for the uniform and exponential distributions, although for these other methods must be used.     

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.