Two-Piece location-scale distributions based on scale mixtures of normal family

In this work, we study the maximum likelihood (ML) estimation problem for the parameters of the two-piece (TP) distribution based on the scale mixtures of normal (SMN) distributions. This is a family of skewed distributions that also includes the scales mixtures of normal class, and is flexible enough for modeling symmetric and asymmetric data. The ML estimates of the proposed model parameters are obtained via an expectation-maximization (EM)-type algorithm.     

Comparisons of ten estimation methods for the parameters of Marshall–Olkin extended exponential distribution

The aim of this article is to compare via Monte Carlo simulations the finite sample properties of the parameter estimates of the Marshall–Olkin extended exponential distribution obtained by ten estimation methods: maximum likelihood, modified moments, L-moments, maximum product of spacings, ordinary least-squares, weighted least-squares, percentile, Crámer–von-Mises, Anderson–Darling, and Right-tail Anderson–Darling. The bias, root mean-squared error, absolute and maximum absolute difference between the true and estimated distribution functions are used as criterion of comparison. The simulation study reveals that the L-moments and maximum products of spacings methods are highly competitive with the maximum likelihood method in small as well as in large-sized samples.     

On dynamic weighted survival entropy of order α

Recently, Abbasnejad et al. (2010 Abbasnejad, M., Arghami, N.R., Morgenthaler, S., Mohtashami Borzadaran, G.R. (2010). On the dynamic survival entropy. Stat. Probab. Lett. 80:1962–1971.[Crossref], [Web of Science ®] , [Google Scholar]) proposed a measure of uncertainty based on survival function, called the survival entropy of order α. A dynamic form of the survival entropy of order α is also proposed by them. In this paper, we derive the weighted form of these measures. The properties of the new measures are also discussed.     

A general non-central hypergeometric distribution

We construct a general non-central hypergeometric distribution, which models biased sampling without replacement. Our distribution is constructed from the combined order statistics of two samples: one of independent and identically distributed random variables with absolutely continuous distribution F and the other of independent and identically distributed random variables with absolutely continuous distribution G. The distribution depends on F and G only through F○G^{( ? 1)} (F composed with the quantile function of G), and the standard hypergeometric distribution and Wallenius’ non-central hypergeometric distribution arise as special cases. We show in efficient economic markets the quantity traded has a general non-central hypergeometric distribution.     

A new exponentiated class of distributions: Properties and applications

Motivated by series-parallel-series systems, we introduce a new generator of continuous distributions with three extra parameters called the generalized exponentiated class of distributions. We study its mathematical properties and introduce a bivariate extension of the class. We discuss the estimation of its parameters by maximum likelihood and illustrate the potentiality of the class by applications to two real datasets.     

Sequential testing procedures for a class of distributions representing various life-testing models

A class of probability density functions is considered, which covers several life-testing models as specific cases. Sequential probability ratio tests are developed for testing simple and composite hypotheses regarding the parameters of the probabilistic model. Expressions for the operating characteristic and the average sample number functions are derived and their behaviour is studied by means of graph-plotting.     

On the distribution of linear combinations of the components of a Dirichlet random vector

On making use of a result of Imhof, an integral representation of the distribution function of linear combinations of the components of a Dirichlet random vector is obtained. In fact, the distributions of several statistics such as Moran and Geary's indices, the Cliff‐Ord statistic for spatial correlation, the sample coefficient of determination, F‐ratios and the sample autocorrelation coefficient can be similarly determined. Linear combinations of the components of Dirichlet random vectors also turn out to be a key component in a decomposition of quadratic forms in spherically symmetric random vectors. An application involving the sample spectrum associated with series generated by ARMA processes is discussed.     

Bayesian beta nonlinear models with constrained parameters to describe ruminal degradation kinetics

The models used to describe the kinetics of ruminal degradation are usually nonlinear models where the dependent variable is the proportion of degraded food. The method of least squares is the standard approach used to estimate the unknown parameters but this method can lead to unacceptable predictions. To solve this issue, a beta nonlinear model and the Bayesian perspective is proposed in this article. The application of standard methodologies to obtain prior distributions, such as the Jeffreys prior or the reference priors, involves serious difficulties here because this model is a nonlinear non-normal regression model, and the constrained parameters appear in the log-likelihood function through the Gamma function. This paper proposes an objective method to obtain the prior distribution, which can be applied to other models with similar complexity, can be easily implemented in OpenBUGS, and solves the problem of unacceptable predictions. The model is generalized to a larger class of models. The methodology was applied to real data with three models that were compared using the Deviance Information Criterion and the root mean square prediction error. A simulation study was performed to evaluate the coverage of the credible intervals.     

Using Animated Probability Plots to Explore the Suitability of Mixture Models with Two Component Distributions

Risk assessors often use different probability plots as a way to assess the fit of a particular distribution or model by comparing the plotted points to a straight line and to obtain estimates of the parameters in parametric distributions or models. When empirical data do not fall in a sufficiently straight line on a probability plot, and when no other single parametric distribution provides an acceptable (graphical) fit to the data, the risk assessor may consider a mixture model with two component distributions. Animated probability plots are a way to visualize the possible behaviors of mixture models with two component distributions. When no single parametric distribution provides an adequate fit to an empirical dataset, animated probability plots can help an analyst pick some plausible mixture models for the data based on their qualitative fit. After using animations during exploratory data analysis, the analyst must then use other statistical tools, including but not limited to: Maximum Likelihood Estimation (MLE) to find the optimal parameters, Goodness of Fit (GoF) tests, and a variety of diagnostic plots to check the adequacy of the fit. Using a specific example with two LogNormal components, we illustrate the use of animated probability plots as a tool for exploring the suitability of a mixture model with two component distributions. Animations work well with other types of probability plots, and they may be extended to analyze mixture models with three or more component distributions.