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.     

Statistical Comparison of the Goodness of Fit Delivered by Five Families of Distributions Used in Distribution Fitting

A basic assumption in distribution fitting is that a single family of distributions may deliver useful representation to the universe of available distributions. To date, little study has been conducted to compare the relative effectiveness of these families. In this article, five families are compared by fitting them to a sample of 20 distributions, using 2 fitting objectives: minimization of the L ₂ norm and four-moment matching. Values of L ₂ norm associated with the fitted families are used as input data to test for significant differences. The Pearson family and the RMM (Response Modeling Methodology) family significantly outperforms all other families.     

Probabilistic Applications of the Schlömilch Transformation

The Schlömilch transformation, long used by mathematicians for integral evaluation, allows probability mass to be redistributed, thus transforming old distributions to new ones. The transformation is used to introduce some new families of distributions on ⁺. Their general properties are studied, i.e., distributional shape and skewness, moments and inverse moments, hazard function, and random number generation. In general, these distributions are suitable for modeling data where the hazard function initially rises steeply. Their usefulness is illustrated by fitting some human weight data. Besides data fitting, one possible use of the new distributions could be in sensitivity or robustness studies, for example as Bayesian prior distributions.     

Approximating the distribution of a renewal process using generalized poisson distributions

The exact distribution of a renewal counting process is not easy to compute and is rarely of closed form. In this article, we approximate the distribution of a renewal process using families of generalized Poisson distributions. We first compute approximations to the first several moments of the renewal process. In some cases, a closed form approximation is obtained. It is found that each family considered has its own strengths and weaknesses. Some new families of generalized Poisson distributions are recommended. Theorems are obtained determining when these variance to mean ratios are less than (or exceed) one without having to find the mean and variance. Some numerical comparisons are also made.     

A fast robust method for fitting gamma distributions

The art of fitting gamma distributions robustly is described. In particular we compare methods of fitting via minimizing a Cramér Von Mises distance, an L ₂ minimum distance estimator, and fitting a B-optimal M-estimator. After a brief prelude on robust estimation explaining the merits in terms of weak continuity and Fréchet differentiability of all the aforesaid estimators from an asymptotic point of view, a comparison is drawn with classical estimation and fitting. In summary, we give a practical example where minimizing a Cramér Von Mises distance is both efficacious in terms of efficiency and robustness as well as being easily implemented. Here gamma distributions arise naturally for “in control” representation indicators from measurements of spectra when using fourier transform infrared (FTIR) spectroscopy. However, estimating the in-control parameters for these distributions is often difficult, due to the occasional occurrence of outliers.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

Comparison of Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM) as General Platforms for Distribution Fitting

Distribution fitting is widely practiced in all branches of engineering and applied science, yet only a few studies have examined the relative capability of various parameter-rich families of distributions to represent a wide spectrum of diversely shaped distributions. In this article, two such families of distributions, Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM), are compared. For a sample of some commonly used distributions, each family is fitted to each distribution, using two methods: fitting by minimization of the L ₂ norm (minimizing density function distance) and nonlinear regression applied to a sample of exact quantile values (minimizing quantile function distance). The resultant goodness-of-fit is assessed by four criteria: the optimized value of the L ₂ norm, and three additional criteria, relating to quantile function matching. Results show that RMM is uniformly better than GLD. An additional study includes Shore's quantile function (QF) and again RMM is the best performer, followed by Shore's QF and then GLD.     

Local dependence functions for extreme value distributions

Kotz & Nadarajah (2002) introduced a measure of local dependence which is a localized version of the Pearson's correlation coefficient. In this paper we provide detailed analyses (both algebraic and numerical) of the form of the measure for the class of bivariate extreme value distributions. We consider, in particular, five families of bivariate extreme value distributions. We also discuss two applications of the new measure. In the first application we introduce an overall measure of correlation and produce evidence to suggest that it is superior than the usual Pearson's correlation coefficient. The second application introduces two new concepts for ordering of bivariate dependence.     

A new variability order based on tail-heaviness

In this paper, we introduce a new variability order that can be interpreted in terms of tail-heaviness which we will call the tail dispersive order. We provide the new definition, its interpretation and properties and the main characterization. We also study the relationship with other classical variability orders. Finally, we study the tail dispersive order in some classical parametric families and provide some applications in insurance and finance. We conclude with a numerical example applied to log returns distributions.     

Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions

Continuing increases in computing power and availability mean that many maximum likelihood estimation (MLE) problems previously thought intractable or too computationally difficult can now be tackled numerically. However, ML parameter estimation for distributions whose only analytical expression is as quantile functions has received little attention. Numerical MLE procedures for parameters of new families of distributions, the g-and-k and the generalized g-and-h distributions, are presented and investigated here. Simulation studies are included, and the appropriateness of using asymptotic methods examined. Because of the generality of these distributions, the investigations are not only into numerical MLE for these distributions, but are also an initial investigation into the performance and problems for numerical MLE applied to quantile-defined distributions in general. Datasets are also fitted using the procedures here. Results indicate that sample sizes significantly larger than 100 should be used to obtain reliable estimates through maximum likelihood.     

Properties of Coherent Systems with Dependent Components

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.     

Random effects in generalized linear models and the em algoritham

Nelder and Wedderburn (1972) gave a practical fitting procedure that encompassed a more gencral family of data distributions than the Gaussian distribution and provided an easily understood conceptual framework. In extending the framework to more than one error structure the technical difficulties of the fitting procedure have tended to cloud the concepts. Here we show that a simple extension to the fitting procedure is possible and thus pave the way for a fuller examimtion of mixed effects models in generalized linear model distributions. It is clear that we should not, and do not have to, confine ourselves to fitting random effects using the Gaussian distribiition. In addition, in, some quite general mixing distribution problems the application of the EM algorithm to the complete data likelihood leads to iterative schemes that maximize the marginal likelihood of the observed data variable.     

Fitting with Matrix-Exponential Distributions

Abstract

It is well known that general phase-type distributions are considerably overparameterized, that is, their representations often require many more parameters than is necessary to define the distributions. In addition, phase-type distributions, even those defined by a small number of parameters, may have representations of high order. These two problems have serious implications when using phase-type distributions to fit data. To address this issue we consider fitting data with the wider class of matrix-exponential distributions. Representations for matrix-exponential distributions do not need to have a simple probabilistic interpretation, and it is this relaxation which ensures that the problems of overparameterization and high order do not present themselves. However, when using matrix-exponential distributions to fit data, a problem arises because it is unknown, in general, when their representations actually correspond to a distribution. In this paper we develop a characterization for matrix-exponential distributions and use it in a method to fit data using maximum likelihood estimation. The fitting algorithm uses convex semi-infinite programming combined with a nonlinear search.     

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.     

On proportional odds models

Recently, Marshall and Olkin (Biometrika 84(3):641–652 1997) introduced a family of distributions by adding a new parameter to a survival function. In this paper, we give physical interpretation of the family using odds function. It is shown that the family of distributions satisfies the property of proportional odds function. We, then, develop a generalized family and study its properties. Further, we give various definitions of proportional odds model in the bivariate set up. Based on these, we introduce new families of bivariate distributions and study their properties.     

Adaptive independence samplers

Markov chain Monte Carlo (MCMC) is an important computational technique for generating samples from non-standard probability distributions. A major challenge in the design of practical MCMC samplers is to achieve efficient convergence and mixing properties. One way to accelerate convergence and mixing is to adapt the proposal distribution in light of previously sampled points, thus increasing the probability of acceptance. In this paper, we propose two new adaptive MCMC algorithms based on the Independent Metropolis–Hastings algorithm. In the first, we adjust the proposal to minimize an estimate of the cross-entropy between the target and proposal distributions, using the experience of pre-runs. This approach provides a general technique for deriving natural adaptive formulae. The second approach uses multiple parallel chains, and involves updating chains individually, then updating a proposal density by fitting a Bayesian model to the population. An important feature of this approach is that adapting the proposal does not change the limiting distributions of the chains. Consequently, the adaptive phase of the sampler can be continued indefinitely. We include results of numerical experiments indicating that the new algorithms compete well with traditional Metropolis–Hastings algorithms. We also demonstrate the method for a realistic problem arising in Comparative Genomics.     

On Some Families of Estimators of Variance of Post-Stratified Sample Mean Using Two Auxiliary Variables

In this article, we propose some families of estimators for finite population variance of post-stratified sample mean using information on two auxiliary variables. The families of estimators are discussed in their optimum cases. The MSE of these estimators are derived to the first order of approximation. The percent relative efficiency of proposed families of estimators has been demonstrated with the numerical illustrations.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

A new estimate of skewness with mean-squared error smaller than that of the sample skewness

Recently, a new procedure for distribution fitting, based on matching of the first two moments, partial and complete, was introduced (Shore, 1995). When the sampling skewness of the fitted distribution is compared to the sample skewness, and both are regarded as estimates of the skewness of the underlying distribution, the mean-squared-error of the former is appreciably lower than that of the latter. In this paper we present some simulation results to support this claim and demonstrate its magnitude. An alternative two-moment distributional fitting procedure, based on a new family of four-parameter distributions, is also introduced and studied. Since three-moment distribution fitting is very common practice in simulation studies, these results may have important implications for the current state-of-the-art of simulation     

Bayesian bivariate survival analysis using the power variance function copula

Copula models have become increasingly popular for modelling the dependence structure in multivariate survival data. The two-parameter Archimedean family of Power Variance Function (PVF) copulas includes the Clayton, Positive Stable (Gumbel) and Inverse Gaussian copulas as special or limiting cases, thus offers a unified approach to fitting these important copulas. Two-stage frequentist procedures for estimating the marginal distributions and the PVF copula have been suggested by Andersen (Lifetime Data Anal 11:333–350, 2005), Massonnet et al. (J Stat Plann Inference 139(11):3865–3877, 2009) and Prenen et al. (J R Stat Soc Ser B 79(2):483–505, 2017) which first estimate the marginal distributions and conditional on these in a second step to estimate the PVF copula parameters. Here we explore an one-stage Bayesian approach that simultaneously estimates the marginal and the PVF copula parameters. For the marginal distributions, we consider both parametric as well as semiparametric models. We propose a new method to simulate uniform pairs with PVF dependence structure based on conditional sampling for copulas and on numerical approximation to solve a target equation. In a simulation study, small sample properties of the Bayesian estimators are explored. We illustrate the usefulness of the methodology using data on times to appendectomy for adult twins in the Australian NH&MRC Twin registry. Parameters of the marginal distributions and the PVF copula are simultaneously estimated in a parametric as well as a semiparametric approach where the marginal distributions are modelled using Weibull and piecewise exponential distributions, respectively.