The Topp–Leone odd log-logistic family of distributions

We introduce a new class of continuous distributions named the Topp–Leone odd log-logistic family, which extends the one-parameter distribution pioneered by Topp and Leone [A family of J-shaped frequency functions. J Amer Statist Assoc. 1955;50:209–219]. We study some of its mathematical properties and describe two special cases. Further, we propose a regression model based on the new Topp–Leone odd log-logistic Weibull distribution. The usefulness and flexibility of the proposed family are illustrated by means of three real data sets.     

An absolutely continuous bivariate Topp–Leone distribution: A useful model on a bounded domain

We introduce an absolutely continuous bivariate generalization of the Topp–Leone distribution, which is a special member of the proportional reversed hazard family using a one-parameter bivariate exchangeable distribution. We show that a copula approach could also be used in defining the bivariate Topp–Leone distribution. The marginal distributions of the new bivariate distribution have also Topp–Leone distributions. We study its distributional and dependence properties. We estimate the parameters by maximum-likelihood procedure, perform a simulation study on the estimators, and apply them to a real data set. Furthermore, we give a way of generating bivariate distributions using the proposed distribution.     

Moments of some J-shaped distributions

This paper concerns a family of univariate distributions suggested by Topp & Leone in 1955. Topp & Leone provided no motivation for this new family and by way of properties they derived only the first four integer-order moments, i.e. E(Xn) for n=1, r 2, r 3, r 4 . In this paper we provide a motivation for the family of distributions and derive explicit algebraic expressions for: (1) hazard rate function; (2) E(Xn) when n ± 1 is any integer; (3) E(Xn) for n=1, r 2, r … r , r 10 , and (4) E[{X-E(X)} n] , n=2, r 3, r 4 . We also give an expression for the characteristic function and discuss issues on estimation and simulation. The main calculations of this paper use properties of the Gauss hypergeometric function.     

Topp–Leone normal distribution with application to increasing failure rate data

In this article, we proposed a new three-parameter probability distribution, called Topp–Leone normal, for modelling increasing failure rate data. The distribution is obtained by using Topp–Leone-X family of distributions with normal as a baseline model. The basic properties including moments, quantile function, stochastic ordering and order statistics are derived here. The estimation of unknown parameters is approached by the method of maximum likelihood, least squares, weighted least squares and maximum product spacings. An extensive simulation study is carried out to compare the long-run performance of the estimators. Applicability of the distribution is illustrated by means of three real data analyses over existing distributions.     

The generalized two-sided power distribution

The generalized standard two-sided power (GTSP) distribution was mentioned only in passing by Kotz and van Dorp Beyond Beta, Other Continuous Families of Distributions with Bounded Support and Applications, World Scientific Press, Singapore, 2004. In this paper, we shall further investigate this three-parameter distribution by presenting some novel properties and use its more general form to contrast the chronology of developments of various authors on the two-parameter TSP distribution since its initial introduction. GTSP distributions allow for J-shaped forms of its pdf, whereas TSP distributions are limited to U-shaped and unimodal forms. Hence, GTSP distributions possess the same three distributional shapes as the classical beta distributions. A novel method and algorithm for the indirect elicitation of the two-power parameters of the GTSP distribution is developed. We present a Project Evaluation Review Technique example that utilizes this algorithm and demonstrates the benefit of separate powers for the two branches of activity GTSP distributions for project completion time uncertainty estimation.     

Modeling heavy-tailed,skewed and peaked uncertainty phenomena with bounded support

A prevalence of heavy-tailed, peaked and skewed uncertainty phenomena have been cited in literature dealing with economic, physics, and engineering data. This fact has invigorated the search for continuous distributions of this nature. In this paper we shall generalize the two-sided framework presented in Kotz and van Dorp (Beyond beta: other continuous families of distributions with bounded support and applications. World Scientific Press, Singapore, 2004) for the construction of families of distributions with bounded support via a mixture technique utilizing two generating densities instead of one. The family of Elevated Two-Sided Power (ETSP) distributions is studied as an instance of this generalized framework. Through a moment ratio diagram comparison, we demonstrate that the ETSP family allows for a remarkable flexibility when modeling heavy-tailed and peaked, but skewed, uncertainty phenomena. We shall demonstrate its applicability via an illustrative example utilizing 2008 US income data.     

On selection of a suitable prior for the posterior analysis of censored mixture of Topp Leone distribution

The paper aims to select a suitable prior for the Bayesian analysis of the two-component mixture of the Topp Leone model under doubly censored samples and left censored samples for the first component and right censored samples for the second component. The posterior analysis has been carried out under the assumption of a class of informative and noninformative priors using a couple of loss functions. The comparison among the different Bayes estimators has been made under a simulation study and a real life example. The model comparison criterion has been used to select a suitable prior for the Bayesian analysis. The hazard rate of the Topp Leone mixture model has been compared for a range of parametric values.     

Improved inference for the shape-scale family of distributions under type-II censoring

The presence of a nuisance parameter may often perturb the quality of the likelihood-based inference for a parameter of interest under small to moderate sample sizes. The article proposes a maximal scale invariant transformation for likelihood-based inference for the shape in a shape-scale family to circumvent the effect of the nuisance scale parameter. The transformation can be used under complete or type-II censored samples. Simulation-based performance evaluation of the proposed estimator for the popular Weibull, Gamma and Generalized exponential distribution exhibits markedly improved performance in all types of likelihood-based inference for the shape under complete and type-II censored samples. The simulation study leads to a linear relation between the bias of the classical maximum likelihood estimator (MLE) and the transformation-based MLE for the popular Weibull and Gamma distributions. The linearity is exploited to suggest an almost unbiased estimator of the shape parameter for these distributions. Allied estimation of scale is also discussed.     

Zero-inflated models and estimation in zero-inflated Poisson distribution

In this paper, we briefly overview different zero-inflated probability distributions. We compare the performance of the estimates of Poisson, Generalized Poisson, ZIP, ZIGP and ZINB models through Mean square error (MSE), bias and Standard error (SE) when the samples are generated from ZIP distribution. We propose a new estimator referred to as probability estimator (PE) of inflation parameter of ZIP distribution based on moment estimator (ME) of the mean parameter and compare its performance with ME and maximum likelihood estimator (MLE) through a simulation study. We use the PE along with ME and MLE to fit ZIP distribution to various zero-inflated datasets and observe that the results do not differ significantly. We recommend using PE in place of MLE since it is easy to calculate and the simulation study in this paper demonstrates that the PE performs as good as MLE irrespective of the sample size.     

Bayesian modeling using a class of bimodal skew-elliptical distributions

We consider Bayesian inference using an extension of the family of skew-elliptical distributions studied by Azzalini [1985. A class of distributions which includes the normal ones. Scand. J. Statist. Theory and Applications 12 (2), 171–178]. This new class is referred to as bimodal skew-elliptical (BSE) distributions. The elements of the BSE class can take quite different forms. In particular, they can adopt both uni- and bimodal shapes. The bimodal case behaves similarly to mixtures of two symmetric distributions and we compare inference under the BSE family with the specific case of mixtures of two normal distributions. We study the main properties of the general class and illustrate its applications to two problems involving density estimation and linear regression.     

A new lifetime model with regression models,characterizations and applications

In this article, we introduce a new extension of Burr XII distribution called Topp Leone Generated Burr XII distribution. We derive some of its properties. Useful characterizations are presented. Simulation study is performed to assess the performance of the maximum likelihood estimators. Censored maximum likelihood estimation is presented in the general case of multi-censored data. The new location-scale regression model based on the proposed distribution is introduced. The usefulness of the proposed models is illustrated empirically by means of three real datasets.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

LOSS OF INFORMATION ASSOCIATED WITH THE ORDER STATISTICS AND RELATED ESTIMATORS IN THE DOUBLE EXPONENTIAL DISTRIBUTION CASE

Fisher (1934) derived the loss of information of the maximum likelihood estimator (MLE) of the location parameter in the case of the double exponential distribution. Takeuchi & Akahira (1976) showed that the MLE is not second order asymptotically efficient. This paper extends these results by obtaining the (asymptotic) losses of information of order statistics and related estimators, and by comparing them via their asymptotic distributions up to the second order.     

Two-Sided Generalized Exponential Distribution

We derive a generalization of the exponential distribution by making log transformation of the standard two-sided power distribution. We show that this new generalization is in fact a mixture of a truncated exponential distribution and truncated generalized exponential distribution introduced by Gupta and Kundu [Generalized exponential distributions. Aust. N. Z. J. Stat. 41(1999):173–188]. The newly defined distribution is more flexible for modeling data than the ordinary exponential distribution. We study its properties, estimate the parameters, and demonstrate it on some well-known real data sets comparing other existing methods.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Zero-modified power series distribution and its Hurdle distribution version

This paper presents a new family of distributions for count data, the so called zero-modified power series (ZMPS), which is an extension of the power series (PS) distribution family, whose support starts at zero. This extension consists in modifying the probability of observing zero of each PS distribution, enabling the new zero-modified distribution to appropriately accommodate data which have any amount of zero observations (for instance, zero-inflated or zero-deflated data). The Hurdle distribution version of the ZMPS distribution is presented. PS distributions included in the proposed ZMPS family are the Poisson, Generalized Poisson, Geometric, Binomial, Negative Binomial and Generalized Negative Binomial distributions. The paper also describes the properties and particularities of the new distribution family for count data. The distribution parameters are estimated via maximum likelihood method and the use of the new family is illustrated in three real data sets. We emphasize that the new distribution family can accommodate sets of count data without any previous knowledge on the characteristic of zero-inflation or zero-deflation present in the data.     

Bimodal Birnbaum–Saunders distribution with applications to non negative measurements

In this article, we introduce a new extension of the Birnbaum–Saunders (BS) distribution as a follow-up to the family of skew-flexible-normal distributions. This extension produces a family of BS distributions including densities that can be unimodal as well as bimodal. This flexibility is important in dealing with positive bimodal data, given the difficulties experienced by the use of mixtures of distributions. Some basic properties of the new distribution are studied including moments. Parameter estimation is approached by the method of moments and also by maximum likelihood, including a derivation of the Fisher information matrix. Three real data illustrations indicate satisfactory performance of the proposed model.     

On a bounded bimodal two-sided distribution fitted to the Old-Faithful geyser data

In this paper, we shall develop a novel family of bimodal univariate distributions (also allowing for unimodal shapes) and demonstrate its use utilizing the well-known and almost classical data set involving durations and waiting times of eruptions of the Old-Faithful geyser in Yellowstone park. Specifically, we shall analyze the Old-Faithful data set with 272 data points provided in Dekking et al. [3]. In the process, we develop a bivariate distribution using a copula technique and compare its fit to a mixture of bivariate normal distributions also fitted to the same bivariate data set. We believe the fit-analysis and comparison is primarily illustrative from an educational perspective for distribution theory modelers, since in the process a variety of statistical techniques are demonstrated. We do not claim one model as preferred over the other.     

Inference in flexible families of distributions with normal kernel

This paper addresses the inference problem for a flexible class of distributions with normal kernel known as skew-bimodal-normal family of distributions. We obtain posterior and predictive distributions assuming different prior specifications. We provide conditions for the existence of the maximum-likelihood estimators (MLE). An EM-type algorithm is built to compute them. As a by product, we obtain important results related to classical and Bayesian inferences for two special subclasses called bimodal-normal and skew-normal (SN) distribution families. We perform a Monte Carlo simulation study to analyse behaviour of the MLE and some Bayesian ones. Considering the frontier data previously studied in the literature, we use the skew-bimodal-normal (SBN) distribution for density estimation. For that data set, we conclude that the SBN model provides as good a fit as the one obtained using the location-scale SN model. Since the former is a more parsimonious model, such a result is shown to be more attractive.     

A Novel Asymmetric Distribution with Power Tails

In this article, we propose a four-parameter asymmetric doubly Pareto-uniform (DPU) distribution with support (?∞, ∞) whose density and cumulative distribution functions are constructed by seamlessly concatenating the left and right Pareto tails with a uniform central part. Properties of the distribution are described and a maximum likelihood estimation (MLE) procedure for its parameters is obtained. Two illustrative examples of the MLE procedure are provided. The first example utilizes an i.i.d. sample of standardized log-differences of bi-monthly 30-year U.S. conventional mortgage interest rates (1971–2004). The second example deals with the height of 100 female Australian athletes.