Estimation of quantile mixtures via L-moments and trimmed L-moments

Moments or cumulants have been traditionally used to characterize a probability distribution or an observed data set. Recently, L-moments and trimmed L-moments have been noticed as appealing alternatives to the conventional moments. This paper promotes the use of L-moments proposing new parametric families of distributions that can be estimated by the method of L-moments. The theoretical L-moments are defined by the quantile function i.e. the inverse of cumulative distribution function. An approach for constructing parametric families from quantile functions is presented. Because of the analogy to mixtures of densities, this class of parametric families is called quantile mixtures. The method of L-moments is a natural way to estimate the parameters of quantile mixtures. As an example, two parametric families are introduced: the normal-polynomial quantile mixture and the Cauchy-polynomial quantile mixture. The proposed quantile mixtures are applied to model monthly, weekly and daily returns of some major stock indexes.     

Exponentiated Weibull–Poisson distribution: Model,properties and applications

In this paper we propose a new four-parameters distribution with increasing, decreasing, bathtub-shaped and unimodal failure rate, called as the exponentiated Weibull–Poisson (EWP) distribution. The new distribution arises on a latent complementary risk problem base and is obtained by compounding exponentiated Weibull (EW) and Poisson distributions. This distribution contains several lifetime sub-models such as: generalized exponential-Poisson (GEP), complementary Weibull–Poisson (CWP), complementary exponential-Poisson (CEP), exponentiated Rayleigh–Poisson (ERP) and Rayleigh–Poisson (RP) distributions.We obtain several properties of the new distribution such as its probability density function, its reliability and failure rate functions, quantiles and moments. The maximum likelihood estimation procedure via an EM-algorithm is presented in this paper. Sub-models of the EWP distribution are studied in details. In the end, applications to two real data sets are given to show the flexibility and potentiality of the new distribution.     

Latent Classification Models

One of the simplest, and yet most consistently well-performing set of classifiers is the Naïve Bayes models. These models rely on two assumptions: (i) All the attributes used to describe an instance are conditionally independent given the class of that instance, and (ii) all attributes follow a specific parametric family of distributions. In this paper we propose a new set of models for classification in continuous domains, termed latent classification models. The latent classification model can roughly be seen as combining the Naïve Bayes model with a mixture of factor analyzers, thereby relaxing the assumptions of the Naïve Bayes classifier. In the proposed model the continuous attributes are described by a mixture of multivariate Gaussians, where the conditional dependencies among the attributes are encoded using latent variables. We present algorithms for learning both the parameters and the structure of a latent classification model, and we demonstrate empirically that the accuracy of the proposed model is significantly higher than the accuracy of other probabilistic classifiers.Editors: Pedro Larrañaga, Jose A. Lozano, Jose M. Peña and Iñaki Inza     

Variational Bayesian Mixture Model on a Subspace of Exponential Family Distributions

Exponential principal component analysis (e-PCA) has been proposed to reduce the dimension of the parameters of probability distributions using Kullback information as a distance between two distributions. It also provides a framework for dealing with various data types such as binary and integer for which the Gaussian assumption on the data distribution is inappropriate. In this paper, we introduce a latent variable model for the e-PCA. Assuming the discrete distribution on the latent variable leads to mixture models with constraint on their parameters. This provides a framework for clustering on the lower dimensional subspace of exponential family distributions. We derive a learning algorithm for those mixture models based on the variational Bayes (VB) method. Although intractable integration is required to implement the algorithm for a subspace, an approximation technique using Laplace's method allows us to carry out clustering on an arbitrary subspace. Combined with the estimation of the subspace, the resulting algorithm performs simultaneous dimensionality reduction and clustering. Numerical experiments on synthetic and real data demonstrate its effectiveness for extracting the structures of data as a visualization technique and its high generalization ability as a density estimation model.      

Self-organizing mixture networks for probability density estimation

A self-organizing mixture network (SOMN) is derived for learning arbitrary density functions. The network minimizes the Kullback-Leibler information metric by means of stochastic approximation methods. The density functions are modeled as mixtures of parametric distributions. A mixture needs not to be homogenous, i.e., it can have different density profiles. The first layer of the network is similar to Kohonen's self-organizing map (SOM), but with the parameters of the component densities as the learning weights. The winning mechanism is based on maximum posterior probability, and updating of the weights is limited to a small neighborhood around the winner. The second layer accumulates the responses of these local nodes, weighted by the learned mixing parameters. The network possesses a simple structure and computational form, yet yields fast and robust convergence. The network has a generalization ability due to the relative entropy criterion used. Applications to density profile estimation and pattern classification are presented. The SOMN can also provide an insight to the role of neighborhood function used in the SOM.     

Auxiliary mixture sampling with applications to logistic models

A new method of data augmentation for binary and multinomial logit models is described. First, the latent utilities are introduced as auxiliary latent variables, leading to a latent model which is linear in the unknown parameters, but involves errors from the type I extreme value distribution. Second, for each error term the density of this distribution is approximated by a mixture of normal distributions, and the component indicators in these mixtures are introduced as further latent variables. This leads to Markov chain Monte Carlo estimation based on a convenient auxiliary mixture sampler that draws from standard distributions like normal or exponential distributions and, in contrast to more common Metropolis-Hastings approaches, does not require any tuning. It is shown how the auxiliary mixture sampler is implemented for binary or multinomial logit models, and it is demonstrated how to extend the sampler to mixed effect models and time-varying parameter models for binary and categorical data. Finally, an application to Austrian labor market data is discussed.     

Fitting mixtures of exponentials to long-tail distributions to analyze network performance models

Traffic measurements from communication networks have shown that many quantities charecterizing network performance have long-tail probability distributions, i.e., with tails that decay more slowly than exponentially. File lengths, call holding times, scene lengths in MPEG video streams, and intervals between connection requests in Internet traffic all have been found to have long-tail distributions, being well described by distributions such as the Pareto and Weibull. It is known that long-tail distributions can have a dramatic effect upon performance, e.g., long-tail service-time distributions cause long-tail waiting-time distributions in queues, but it is often difficult to describe this effect in detail, because performance models with component long-tail distributions tend to be difficult to analyze. We address this problem by developing an algorithm for approximating a long-tail distribution by a hyperexponential distribution (a finite mixture of exponentials). We first prove that, in prinicple, it is possible to approximate distributions from a large class, including the Pareto and Weibull distributions, arbitrarily closely by hyperexponential distributions. Then we develop a specific fitting alogrithm. Our fitting algorithm is recursive over time scales, starting with the largest time scale. At each stage, an exponential component is fit in the largest remaining time scale and then the fitted exponential component is subtracted from the distribution. Even though a mixture of exponentials has an exponential tail, it can match a long-tail distribution in the regions of primary interest when there are enough exponential components. When a good fit is achieved, the approximating hyperexponential distribution inherits many of the difficulties of the original long-tail distribution; e.g., it is still difficult to obtain reliable estimates from simulation experiments. However, some difficulties are avoided; e.g., it is possible to solve some queueing models that could not be solved before. We give examples showing that the fitting procedure is effective, both for directly matching a long-tail distribution and for predicting the performance in a queueing model with a long-tail service-time distribution.     

Joint modelling of multivariate longitudinal outcomes and a time-to-event: A nonlinear latent class approach

A joint model based on a latent class approach is proposed to explore the association between correlated longitudinal quantitative markers and a time-to-event. A longitudinal latent class model describes latent profiles of evolution of the latent process underlying the correlated markers. The latent process is linked to the markers by nonlinear transformations including parameters to be estimated. A proportional hazard model describes the joint risk of event according to the latent classes and two specifications of the risk function are considered: a parametric function and a semi-parametric function based on splines. Depending on the chosen risk function, estimation is performed by a maximum likelihood or a maximum penalized likelihood approach. A simulation study validates the estimation procedure. As a latent class model relies on the strong assumption that the markers and the time-to-event are independent conditionally on the latent classes, a test of conditional independence is proposed using the residuals conditional on time-to-event. The procedure does not require any posterior classification and can be conducted using standard statistical softwares. The methodology is applied to describe profiles of cognitive decline in the elderly and their associated risk of dementia.     

Estimating the parameters of the Marshall-Olkin bivariate Weibull distribution by EM algorithm

In this paper we consider the Marshall-Olkin bivariate Weibull distribution. The Marshall-Olkin bivariate Weibull distribution is a singular distribution, whose both the marginals are univariate Weibull distributions. This is a generalization of the Marshall-Olkin bivariate exponential distribution. The cumulative joint distribution of the Marshall-Olkin bivariate Weibull distribution is a mixture of an absolute continuous distribution function and a singular distribution function. This distribution has four unknown parameters and it is observed that the maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. In this paper we discuss about the computation of the maximum likelihood estimators of the unknown parameters using EM algorithm. We perform some simulations to see the performances of the EM algorithm and re-analyze one data set for illustrative purpose.     

A framework for modelling overdispersed count data, including the Poisson-shifted generalized inverse Gaussian distribution

A variety of methods of modelling overdispersed count data are compared. The methods are classified into three main categories. The first category are ad hoc methods (i.e. pseudo-likelihood, (extended) quasi-likelihood, double exponential family distributions). The second category are discretized continuous distributions and the third category are observational level random effects models (i.e. mixture models comprising explicit and non-explicit continuous mixture models and finite mixture models). The main focus of the paper is a family of mixed Poisson distributions defined so that its mean μ is an explicit parameter of the distribution. This allows easier interpretation when μ is modelled using explanatory variables and provides a more orthogonal parameterization to ease model fitting. Specific three parameter distributions considered are the Sichel and Delaporte distributions. A new four parameter distribution, the Poisson-shifted generalized inverse Gaussian distribution is introduced, which includes the Sichel and Delaporte distributions as a special and a limiting case respectively. A general formula for the derivative of the likelihood with respect to μ, applicable to the whole family of mixed Poisson distributions considered, is given. Within the framework introduced here all parameters of the distributions are modelled as parametric and/or nonparametric (smooth) functions of explanatory variables. This provides a very flexible way of modelling count data. Maximum (penalized) likelihood estimation is used to fit the (non)parametric models.     

Bootstrap control charts in monitoring value at risk in insurance

A risk measure is a mapping from the random variables representing the risks to a number. It is estimated using historical data and utilized in making decisions such as allocating capital to each business line or deposit insurance pricing. Once a risk measure is obtained, an efficient monitoring system is required to quickly detect any drifts in the risk measure. This paper investigates the problem of detecting a shift in value at risk as the most widely used risk measure in insurance companies. The probabilistic C control chart and the parametric bootstrap method are employed to establish a risk monitoring scheme in insurance companies. Since the number of claims in a period is a random variable, the proposed method is a variable sample size scheme. Monte Carlo simulations for Weibull, Burr XII, Birnbaum–Saunders and Pareto distributions are carried out to investigate the behavior and performance of the proposed scheme. In addition, a real example from an insurance company is presented to demonstrate the applicability of the proposed method.     

Using algebraic software to compute the moments of order statistics

Here we examine using the algebraic software Mathematica for computing the moments of the order statistics of several frequently used distribution functions, specifically those for the extreme-value, Weibull, double Weibull, logistic and Cauchy distributions. These are illustrated in obtaining the generalised least-squares estimates of the location and scale parameters of the extreme-value distribution.     

Estimation in step-stress accelerated life tests for the exponentiated exponential distribution with type-I censoring

The step-stress accelerated life tests allow the experimenter to increase the stress levels at fixed times during the experiment. The lifetime of a product at any level of stress is assumed to have an exponentiated distribution, whose baseline distribution is a general class of distributions which includes, among others, Weibull, compound Weibull, Pareto, Gompertz, normal and logistic distributions. The scale parameter of the baseline distribution is assumed to be a log-linear function of the stress and a cumulative exposure model holds. Special attention is paid to an exponentiated exponential distribution. Based on type-I censoring, the maximum likelihood estimates of the parameters under consideration are obtained. A Monte Carlo simulation study is carried out to investigate the precision of the maximum likelihood estimates and to obtain the coverage probabilities of the bootstrap confidence intervals for the parameters involved. Finally, an example is presented to illustrate the two discussed methods of bootstrap confidence intervals.     

Uncertainty propagation in puff-based dispersion models using polynomial chaos

Atmospheric dispersion is a complex nonlinear physical process with numerous uncertainties in model parameters, inputs, source parameters, initial and boundary conditions. Accurate propagation of these uncertainties through the dispersion models is crucial for a reliable prediction of the probability distribution of the states and assessment of risk. A simple three-dimensional Gaussian puff-based dispersion model is used as a test case to study the effect of uncertainties in the model parameters and initial conditions on the output concentration. A polynomial chaos based approach is used to numerically investigate the evolution of the model output uncertainties due to initial condition and parametric uncertainties. The polynomial chaos solution is found to be an accurate approximation to ground truth, established by Monte Carlo simulation, while offering an efficient computational approach for large nonlinear systems with a relatively small number of uncertainties.     

Probability of failure models in finite element analysis of brittle materials

A model of failure for brittle materials is presented, based on the weakest link hypothesis and Weibull distribution function. Methods of computing the probability of failure of a structural component, and numerical implementation in a finite element program, are shown. Computer subroutines are included.     

Fuzzy divergence, probability measure of fuzzy events and image thresholding

A new measure called divergence between two fuzzy sets is introduced along with a few properties. Its application to clustering problems is indicated and applied to an object extraction problem. A tailored version of the probability measure of a fuzzy event is also used for image segmentation. Both parametric and non-parametric probability distributions are considered in this regard.     

A genetic algorithm-based system for wireless mesh networks: analysis of system data considering different routing protocols and architectures

Wireless mesh networks (WMNs) are attracting a lot of attention from wireless network researchers. Node placement problems have been investigated for a long time in the optimization field due to numerous applications in location science. In our previous work, we evaluated WMN-GA system which is based on genetic algorithms (GAs) to find an optimal location assignment for mesh routers. In this paper, we evaluate the performance of four different distributions of mesh clients for two WMN architectures considering throughput, delay and energy metrics. For simulations, we used ns-3, optimized link state routing (OLSR) and hybrid wireless mesh protocols (HWMP). We compare the performance for Normal, Uniform, Exponential and Weibull distributions of mesh clients by sending multiple constant bit rate flows in the network. The simulation results show that for HWM protocol the throughput of Uniform distribution is higher than other distributions. However, for OLSR protocol, the throughput of Exponential distribution is better than other distributions. For both protocols, the delay and remaining energy are better for Weibull distribution.     

A note on mixed exponential approximations for Gi/G/1 waiting times

A method is offered for the effective estimation of the stationary waiting-time distribution of the GI/G/1 queue by a (possibly nonconvex) mixed exponential CDF. The approach relies on obtaining a generalized exponential mixture as an approximation for the distribution of the service times. This is done by the adaptation of a nonlinear optimization algorithm previously developed for the maximum-likelihood estimation of parameters from mixed Weibull distributions. The approach is particularly well-suited for obtaining the delay distribution beginning from raw interarrivai and service-time data.     

The log-exponentiated Weibull regression model for interval-censored data

In interval-censored survival data, the event of interest is not observed exactly but is only known to occur within some time interval. Such data appear very frequently. In this paper, we are concerned only with parametric forms, and so a location-scale regression model based on the exponentiated Weibull distribution is proposed for modeling interval-censored data. We show that the proposed log-exponentiated Weibull regression model for interval-censored data represents a parametric family of models that include other regression models that are broadly used in lifetime data analysis. Assuming the use of interval-censored data, we employ a frequentist analysis, a jackknife estimator, a parametric bootstrap and a Bayesian analysis for the parameters of the proposed model. We derive the appropriate matrices for assessing local influences on the parameter estimates under different perturbation schemes and present some ways to assess global influences. Furthermore, for different parameter settings, sample sizes and censoring percentages, various simulations are performed; in addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to a modified deviance residual in log-exponentiated Weibull regression models for interval-censored data.