Bimatrix variate gamma-beta distributions

Two bimatrix distributions with beta and gamma marginals are introduced. Various properties (including product moments of determinants and traces, entropies, marginal distributions) are derived. Parameter estimation by the method of maximum likelihood is discussed. The performance and efficiencies of the maximum likelihood estimators and the associated confidence intervals are assessed by simulation. The efficiencies are compared versus those for the maximum likelihood estimators and the associated confidence intervals based on matrix variate gamma distributions. A discussion of possible applications of the bimatrix distributions is given.     

Bounds for the pearson type IV tail probabilities

Bounds for the tail probability of Pearson Type IV distribution are obtained, These bounds are simple to calculate.     

基于Pearson Ⅳ分布的VaR与CVaR估计

基于GARCH模型,用Pearson Ⅳ分布拟合标准残差,给出一种更为精确的VaR和CVaR计算方法.重点研究在Norm-GARCH、t-GARCH与GED-GARCH模型下,用原分布和Pearson Ⅳ分布计算VaR的比较,结果表明,用Pearson Ⅳ分布计算VaR都能得到比原分布更小的失败率,且在三种模型之下用Pearson Ⅳ分布计算VaR结果很接近,都能通过检验,所以选择最简单的Norm-GARCH模型就可以;基于此,研究在Norm-GARCH模型下,用正态分布和Pearson Ⅳ分布计算CVaR,并与VaR进行比较,结果表明,用Pearson Ⅳ分布计算VaR和CVaR的失败率都远远小于由正态分布所得到的失败率,特别在VaR估计失效的交易日里,用Pearson Ⅳ分布得到的CVaR均值与实际损失均值非常接近.因此,Pearson Ⅳ分布能很好地刻画金融数据的特征,相对其他分布而言是一个很好的选择.     

On the Bivariate Kummer-Beta Type IV Distribution

Recent studies have shown the X-bar control chart with variable sampling interval detects shifts in the process mean faster than the traditional X-bar chart. These studies are usually based on the assumption that the process data are independently and normally distributed. However, many situations in practice violate these assumptions. In this study, a methodology is developed to economically design a variable sampling interval X-bar control chart that takes into consideration correlated non normal sample data. An example is provided to illustrate the solution procedure. A sensitivity analysis on the input parameters (i.e., the cost and the process parameters) is performed taking into account the non normality and the correlation on the optimal design of the chart.     

A CLOSED-FORM EXPRESSION FOR THE PEARSON TYPE IV DISTRIBUTION FUNCTION

A closed-form expression is presented for the probability integral of the Pearson Type IV distribution, and a corresponding method of evaluation is given. This analysis addresses a long-standing gap in the theory of the Pearson system of distributions. In addition, a simple derivation is given of an expression for the normalizing constant in the Type IV integral.     

A study of the Gamma-Pareto (IV) distribution and its applications

ABSTRACT

Pareto distributions and their close relatives and generalizations provide very flexible families of heavy-tailed distributions that may be used to model income distributions as well as a wide variety of other social and economic distributions. On the other hand, gamma distribution has a wide application in various social and economic spheres such as survival analysis, to model aggregate insurance claims, and the amount of rainfall accumulated in a reservoir etc. Combining the above two heavy-tailed distributions, using the technique by Alzaatreh et al. (2012 Alzaatreh, A., Famoye, F., Lee, C. (2012). Gamma-Pareto distribution and its applications. J. Modern Appl. Stat. Methods. 11:78–94.[Crossref] , [Google Scholar]), we define a new distribution, namely Gamma-Pareto (IV) distribution, hereafter called as GPD^(IV) distribution. Various properties of the GPD^(IV) are investigated such as limiting behavior, moments, mode, and Shannon entropy. Also some characterizations of the GPD^(IV) distribution are mentioned in this paper. Maximum likelihood method is proposed for estimating the model parameters. For illustrative purposes, real data sets are considered as applications of the GPD^(IV) distribution.     

Bayesian and maximum likelihood estimations of the inverted exponentiated half logistic distribution under progressive Type II censoring

In this paper, the estimation of parameters, reliability and hazard functions of a inverted exponentiated half logistic distribution (IEHLD) from progressive Type II censored data has been considered. The Bayes estimates for progressive Type II censored IEHLD under asymmetric and symmetric loss functions such as squared error, general entropy and linex loss function are provided. The Bayes estimates for progressive Type II censored IEHLD parameters, reliability and hazard functions are also obtained under the balanced loss functions. However, the Bayes estimates cannot be obtained explicitly, Lindley approximation method and importance sampling procedure are considered to obtain the Bayes estimates. Furthermore, the asymptotic normality of the maximum likelihood estimates is used to obtain the approximate confidence intervals. The highest posterior density credible intervals of the parameters based on importance sampling procedure are computed. Simulations are performed to see the performance of the proposed estimates. For illustrative purposes, two data sets have been analyzed.     

A Matrix Variate Closed Skew-Normal Distribution with Applications to Stochastic Frontier Analysis

In this article, we introduce the matrix extension of the closed skew-normal distribution and give two constructions for it: a marginal one and another based on hidden truncation. Important basic properties of the distribution are presented such as its closure under linear transformation and moment generating function. We also give distributional results for quadratic forms involving random matrices distributed according to two particular cases of it. Using an additive construction, we derive a submodel which can be employed to describe the compound error structure of a very general multivariate stochastic frontier model. Finally, we consider the skew-elliptical extension of the proposed distribution.     

Statistical regularity of firm size distribution: the Pareto IV and truncated Yule for Italian SCI manufacturing

In this paper we model the firm size distribution (FSD) of Italian manufacturing firms of SCI, the GDP survey of ISTAT, by a continuous and a discrete distribution: the Pareto IV distribution on total assets and the Yule distribution on Number of Employees. The Pareto IV distribution is characterized by four parameters and shows a better fit than both the Lognormal and Pareto I, which are the distributions more frequently applied to model firm size. The Pareto IV is inconsistent with Gibrat’s Law according to which the different segments of an Industry are characterized by proportionate growth and the distribution of size is Lognormal. A truncation of the Yule distribution has been necessary because the dataset is characterized by firms with at least 20 employees. The truncated Yule distribution shows a good fit for medium–large firms (firms with more than 50 employees). The partition of the dataset in innovative and non-innovative firms – both of which are well described by the Pareto IV – reveals a beneficial effect of scale on innovation. Finally, the good fit of both distributions holds not only for the composite industry, but for the single sectors too. The present work is part of a more general research project: “Industry evolution: innovation, profitability and firm’s growth”, conducted within the Department of Economic and Social Sciences of the Università Cattolica del Sacro Cuore (UCSC), Piacenza, coordinated by Professor Maurizio Baussola in cooperation with ISTAT (Italian Statistical Office, regional office for Lombardy). Part of this research was done when Lisa Crosato was a visiting research student at the LSE Statistics Department, during her Ph.D program in “Quantitative Models for Policy Analysis” at the UCSC of Piacenza.     

PROMETHEE IV through kernel density estimation

ABSTRACT

This paper presents a new version of PROMETHEE IV, which considers the empirical distribution of the criteria through kernel density estimation to evaluate alternatives. The developed method has the ability to treat criteria according to their distribution. The classic PROMETHEE IV can produce divergent integrals, and this could be the cause for its insufficient exploration in literature. The proposed method overcomes this situation since large values have little weight compared to values near the mean.     

Some Practical Aspects of Canonical Variate Analysis

Canonical variate analysis can be viewed as a two-stage principal component analysis. Explicit consideration of the principal components from the first stage, formalized in the content of shrunken estimators, leads to a number of practical advantages. In morphometric studies, the first eigenvector is often a size vector, with the remaining vectors contrast or shape-type vectors, so that a decomposition of the canonical variates into size and shape components can be achieved. In applied studies, often a small number of the principal components effect most of the separation between groups; plots of group means and associated concentration ellipses (ideally these should be circular) for important principal components facilitate graphical inspection. Of considerable practical importance is the potential for improved stability of the estimated canonical vectors. When the between-groups sum of squares for a particular principal component is small, and the corresponding eigenvalue of the within-groups correlation matrix is also small, marked instability of the canonical vectors can be expected. The introduction of shrunken estimators, by adding shrinkage constrants to the eigenvalues, leads to more stable coefficients.     

On the hidden truncated bivariate Pareto (IV) model and associated inferential issues

Income and wealth data are typically modelled by some variant of the classical Pareto distribution. Often, in practice, the observed data are truncated with respect to some unobserved covariate. In this paper, a hidden truncation formulation of this scenario is proposed and analysed. For this purpose, a bivariate Pareto (IV) distribution is assumed for the variable of interest and the unobserved covariate. Some important distributional properties of the resulting model as well as associated inferential methods are studied. An example is used finally to illustrate the results developed here. In this case, it is noted that hidden truncation on the left does not result in any new model, but the hidden truncation on the right does. The properties and fit of such a model pose a challenging problem and that is what is focused here in this work.     

The Generalized Pascal Triangle and the Matrix Variate Jensen-Logistic Distribution

This article defines the so called Generalized Matrix Variate Jensen-Logistic distribution. The relevant applications of this class of distributions in Configuration Shape Theory consist of a more efficient computation, supported by the corresponding inference. This demands the solution of two important problems: (1) the development of analytical and efficient formulae for their k-th derivatives and (2) the use of the derivatives to transform the configuration density into a polynomial density under some special matrix Kummer relation, indexed in this case by the Jensen-Logistic kernel. In this article, we solve these problems by deriving a simple formula for the k-th derivative of the density function, avoiding the usual partition theory framework and using a generalization of Pascal triangles. Then we apply the results by obtaining the associated Jensen-Logistic Kummer relations and the configuration polynomial density in the setting of Statistical Shape Theory.     

More on progressively Type‐II censored order statistics for multivariate observations

In this paper, we consider some results on distribution theory of multivariate progressively Type‐II censored order statistics. We also establish some characterizations of Freund's bivariate exponential distribution based on the lack of memory property.     

Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

Failure Rate Transform of the Weibull Variate and its Properties

The failure rate of a life time distribution has so for been studied in the time domsin. The present paper extends the failure rate transform in the frequency domain. For this purpose, the parent life time distribution has been taken as the Weibull distribution and different properties of the distribution of the failure rate transform have been presented including some characterisations of the Weibull distribution. Some comparative studies have also been made between the parent distribution and the failure rate transform distribution in respect of convex, star shaped and super additive orderings and expectation.     

Tail Behavior of the General Error Distribution

An inequality on the tail behavior of the general error distribution and an asymptotic Mills-type ratio are established. Two applications are provided. The first application considers the asymptotic behavior of the ratio of probability densities and the ratio of the tails of the general error and normal distributions. The second application establishes the asymptotic distribution of the partial maximum of an independent and identically distributed sequence from the general error distribution.     

Selection and Ranking Procedures for Type I Extreme Value Populations and a Related Homogeneity Test

Consider k (≥2) independent Type I extreme value populations with unknown location parameters and common known scale parameter. With samples of same size, we study procedures based on the sample means for (1) selecting the population having the largest location parameter, (2) selecting the population having the smallest location parameter, and (3) testing for equality of all the location parameters. We use Bechhofer's indifference-zone and Gupta's subset selection formulations. Tables of constants for implemention are provided based on approximation for the distribution of the standardized sample mean by a generalized Tukey's lambda distribution. Examples are provided for all procedures.     

Bias Correction in the Type I Generalized Logistic Distribution

Four strategies for bias correction of the maximum likelihood estimator of the parameters in the Type I generalized logistic distribution are studied. First, we consider an analytic bias-corrected estimator, which is obtained by deriving an analytic expression for the bias to order n ^?1; second, a method based on modifying the likelihood equations; third, we consider the jackknife bias-corrected estimator; and fourth, we consider two bootstrap bias-corrected estimators. All bias correction estimators are compared by simulation. Finally, an example with a real data set is also presented.