Invariance of linearity of regression and related characterizations of some classical discrete distributions

Rao (1963) introduced what we call an additive damage model. In this model, original observation is subjected to damage according to a specified probability law by the survival distribution. In this paper, we consider a bivariate observation with second component subjected to damage. Using the invariance of linearity of regression of the first component on the second under the transition of the second component from the original to the damaged state, we obtain the characterizations of the Poisson, binomial and negative binomial distributions within the framework of the additive damage model.     

Characterizations for the original observations with linearity of regression in additive damage models

This paper investigates the characterizations of certain discrete distributions within the framework of multivariate additive damage models. The univariate case for such models appoared in an article by N. Krishnaii (1974) and Rao and Rubin (1964). In this paper the survival distriution in specified and it is shown that linearity of the regression of the undamaged part on the damaged part, or the damaged part on the undamaged part leads to the characterizations of independent binomials, independent negative binomials, independent Poissons, multinomial and negative multinomial for the original p-dimensional observation.     

Characterization of generalized markov-polya and generalized polya-eggenberger distributions

A discrete model is considered where the original observation is subjected to partial destruction according to the Generalized Markov-Polya (GMP) damage model. A characterization of the Generalized Polya-Eggenberger distribution (GPED) is given in the context of the Rao-Rubin condition. More specifically, if the probability that an observation n of a non-negative integer valued r.v.X is reduced to an integer k during a damage, process is given by the GMPD, and if the resulting r.v.Y is such thatrit satisfies the RR-conditlon, then X has a GPED. Secondly, if N = A + B, where B is the missing part and A is the recorded part such that the conditional distribution P(A= x|N=n) is the GMPD, then the r.v.'s A and B are independent if, and only if, N has a GPED. Several other characterizations are also given for these two distributions. The results of Rao-Rubin ‘1964’, Patil-Ratnaparkhi (1977) and Consul (1975) follow as special cases.     

Generalized additive models for location, scale and shape

Summary.  A general class of statistical models for a univariate response variable is presented which we call the generalized additive model for location, scale and shape (GAMLSS). The model assumes independent observations of the response variable y given the parameters, the explanatory variables and the values of the random effects. The distribution for the response variable in the GAMLSS can be selected from a very general family of distributions including highly skew or kurtotic continuous and discrete distributions. The systematic part of the model is expanded to allow modelling not only of the mean (or location) but also of the other parameters of the distribution of y , as parametric and/or additive nonparametric (smooth) functions of explanatory variables and/or random-effects terms. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models. A Newton–Raphson or Fisher scoring algorithm is used to maximize the (penalized) likelihood. The additive terms in the model are fitted by using a backfitting algorithm. Censored data are easily incorporated into the framework. Five data sets from different fields of application are analysed to emphasize the generality of the GAMLSS class of models.     

On Efron's Parameterization

This paper discusses a curved exponential family of distributions which is defined by a differential equation with respect to the expectation parameters in the two–dimensional exponential family. The differential equation considered here is the same as the one given by Efron (1975) for the trinomial distribution. This equation is extended here to a general exponential family, and called Efron's parameterization in the two–dimensional exponential family. The solution of Efron's parameterization is obtained explicitly in an exponential family, although Kumagai & Inagaki (1996) showed that there exists no proper solution of Efron's equation for the trinomial distribution, in line with the counterexample given by Efron (1975 p. 1206). The paper gives some characterizations of Efron's parameterization with special reference to Fisher's circle model. The implications of these characterizations are the two–dimensional normal distribution and a spiral curve in the plane.     

Longitudinal network models and permutation-uniform Markov chains

Consider longitudinal networks whose edges turn on and off according to a discrete-time Markov chain with exponential-family transition probabilities. We characterize when their joint distributions are also exponential families with the same parameter, improving data reduction. Further we show that the permutation-uniform subclass of these chains permit interpretation as an independent, identically distributed sequence on the same state space. We then apply these ideas to temporal exponential random graph models, for which permutation uniformity is well suited, and discuss mean-parameter convergence, dyadic independence, and exchangeability. Our framework facilitates our introducing a new network model; simplifies analysis of some network and autoregressive models from the literature, including by permitting closed-form expressions for maximum likelihood estimates for some models; and facilitates applying standard tools to longitudinal-network Markov chains from either asymptotics or single-observation exponential random graph models.     

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.     

General frailty model and stochastic orderings

In this paper, we propose a general frailty model and develop its properties including some results for stochastic comparisons. More specifically, our main results lie in seeing how the well known stochastic orderings between distributions of two frailties translate into the orderings between the corresponding survival functions. These results are used to obtain the properties of the classical multiplicative frailty model and the additive frailty model. Several of the results, in the literature, are obtained as special cases.     

A generic randomization device for quantitative response variables

We propose a generic randomization device (RD) for the estimation of mean of quantitative sensitive characteristic that make use of additive and multiplicative scrambled responses with blank card strategy. For this generic RD, we work with 12 choices and derive statistical properties of each model and give many immediate results in corollaries. It is interesting to note that various existing models can now be viewed as the special cases of this RD. Finally, various empirical studies have been done for each of these models to evaluate the efficiency performance.     

Information geodesics for gamma models of communication clustering

Recently, Akyildiz called for further work on non-Poisson models for communication arrivals in distributed networks such as cellular phone systems. The basic ‘random’ model for stochastic events is the Poisson process; for events on a line this resuits in an exponential disiribuuon of intervals between events. Network designers and managers need too monotor and quantify call clustering in order to optimize resaurce usage; the natural reference state from which to measure departures is that arising from a Poisson, process of calls. Here we consider gamma distributions, which contain exponential distributions as a special case. The surface representing gamma models has a natural Riemannian information metric and we obtain some geodesic sprays for this metric. The exponential distributions form a 1-dimensional subspace of the 2-dimensional space of all gamma distributions, so we have an isometric embedding of the random model as a subspace of the gamma models. This geometry may provide an appropriate structure on which to represent clustering as quantifiable departures from randomness and on which to impose dynamic control algorithms to optimize traffic at receiving nodes in distributed communication networks. In practice, we may expect correlation between call arrival times and call duration, reflecting for example peaks of different users of internet services. This would give rise to a twisted product of two surfaces with the twisting controlled by the correlation. Though bivariate gamma models do exist, such as Kibble's, none has tractabie information geometry nor sufficiently general marginal gammas,but a simulation method of approach is suggested.     

Semiparametric Mixtures of Generalized Exponential Families

Abstract.  A semiparametric mixture model is characterized by a non-parametric mixing distribution Q (with respect to a parameter θ ) and a structural parameter β common to all components. Much of the literature on mixture models has focused on fixing β and estimating Q . However, this can lead to inconsistent estimation of both Q and the order of the model m . Creating a framework for consistent estimation remains an open problem and is the focus of this article. We formulate a class of generalized exponential family (GEF) models and establish sufficient conditions for the identifiability of finite mixtures formed from a GEF along with sufficient conditions for a nesting structure. Finite identifiability and nesting structure lead to the central result that semiparametric maximum likelihood estimation of Q and β fails. However, consistent estimation is possible if we restrict the class of mixing distributions and employ an information-theoretic approach. This article provides a foundation for inference in semiparametric mixture models, in which GEFs and their structural properties play an instrumental role.     

A Composite Likelihood Approach to Multivariate Survival Data

This paper is about the statistical analysis of multivariate survival data. We discuss the additive and multiplicative frailty models which have been the most popular models for multivariate survival data. As an alternative to the additive and multiplicative frailty models, we propose basing inference on a composite likelihood function that only requires modelling of the marginal distribution of pairs of failure times. Each marginal distribution of a pair of failure times is here assumed to follow a shared frailty model. The method is illustrated with a real-life example.     

Minimum variance unbiased estimation in natural exponential families generated by stable distributions

The class of nature exponential families generated by stable distributions has been introduced in different contexts by several authors. Tweedie (1984) and Jorgensen (1987) studied this class in the context of generalized liner models and exponential dispersion models. Bar-Lev and Enis (1986) introduced this class in the context of the property of reproducibility in natural exponential families and Hougaard (1986) found the distributions in this class to be natural candidates for applications as survival distributions in life tables for heterogeneous populations. In this note, we consider such a class in the context of minimum variance unbiased estimation. For each family in this class, we obtain an explicit expression for the uniformly minimum variance unbiased estimator for the r-th cumlant, the density function, and the reliability function.     

Discriminating between generalized exponential,geometric extreme exponential and Weibull distributions

Generalized exponential, geometric extreme exponential and Weibull distributions are three non-negative skewed distributions that are suitable for analysing lifetime data. We present diagnostic tools based on the likelihood ratio test (LRT) and the minimum Kolmogorov distance (KD) method to discriminate between these models. Probability of correct selection has been calculated for each model and for several combinations of shape parameters and sample sizes using Monte Carlo simulation. Application of LRT and KD discrimination methods to some real data sets has also been studied.     

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.     

Modelling Poisson marked point processes using bivariate mixture transition distributions

A class of bivariate continuous-discrete distributions is proposed to fit Poisson dynamic models in a single unified framework via bivariate mixture transition distributions (BMTDs). Potential advantages of this class over the current models include its ability to capture stretches, bursts and nonlinear patterns characterized by Internet network traffic, high-frequency financial data and many others. It models the inter-arrival times and the number of arrivals (marks) in a single unified model which benefits from the dependence structure of the data. The continuous marginal distributions of this class include as special cases the exponential, gamma, Weibull and Rayleigh distributions (for the inter-arrival times), whereas the discrete marginal distributions are geometric and negative binomial. The conditional distributions are Poisson and Erlang. Maximum-likelihood estimation is discussed and parameter estimates are obtained using an expectation–maximization algorithm, while the standard errors are estimated using the missing information principle. It is shown via real data examples that the proposed BMTD models appear to capture data features better than other competing models.     

Perturbation by multiplicative noise and the Simulation Extrapolation method

While most of the literature on measurement error focuses on additive measurement error, we consider in this paper the multiplicative case. We apply the Simulation Extrapolation method (SIMEX)—a procedure which was originally proposed by Cook and Stefanski (J. Am. Stat. Assoc. 89:1314–1328, 1994) in order to correct the bias due to additive measurement error—to the case where data are perturbed by multiplicative noise and present several approaches to account for multiplicative noise in the SIMEX procedure. Furthermore, we analyze how well these approaches reduce the bias caused by multiplicative perturbation. Using a binary probit model, we produce Monte Carlo evidence on how the reduction of data quality can be minimized. For helpful comments, we would like to thank Helmut Küchenhoff, Winfried Pohlmeier, and Gerd Ronning. Sandra Nolte gratefully acknowledges financial support by the DFG. Elena Biewen and Martin Rosemann gratefully acknowledge the financial support by the Federal Ministry of Education and Research (BMBF). The usual disclaimer applies.     

The odd log-logistic Lindley Poisson model for lifetime data

We define two new lifetime models called the odd log-logistic Lindley (OLL-L) and odd log-logistic Lindley Poisson (OLL-LP) distributions with various hazard rate shapes such as increasing, decreasing, upside-down bathtub, and bathtub. Various structural properties are derived. Certain characterizations of OLL-L distribution are presented. The maximum likelihood estimators of the unknown parameters are obtained. We propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest has a Poisson distribution and the time to event has an OLL-L distribution. The applicability of the new models is illustrated by means real datasets.     

Identifiability of income distributions in the context of damage and generating models

In the context of additive or multiplicative damage models, and under mild conditions, it is shown that the functional forms of suitably chosen regressions on a random variable X or/and its recorded part Y are characteristic of the distribution of X. The paper treats the cases where the recorded value is either an understatement or an overstatement of the true observation.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.