A New Weibull–Pareto Distribution: Properties and Applications

Many distributions have been used as lifetime models. In this article, we propose a new three-parameter Weibull–Pareto distribution, which can produce the most important hazard rate shapes, namely, constant, increasing, decreasing, bathtub, and upsidedown bathtub. Various structural properties of the new distribution are derived including explicit expressions for the moments and incomplete moments, Bonferroni and Lorenz curves, mean deviations, mean residual life, mean waiting time, and generating and quantile functions. The Rényi and q entropies are also derived. We obtain the density function of the order statistics and their moments. The model parameters are estimated by maximum likelihood and the observed information matrix is determined. The usefulness of the new model is illustrated by means of two real datasets on Wheaton river flood and bladder cancer. In the two applications, the new model provides better fits than the Kumaraswamy–Pareto, beta-exponentiated Pareto, beta-Pareto, exponentiated Pareto, and Pareto models.     

The Poisson-X family of distributions

ABSTRACT

Recently, Risti? and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.     

A new compound class of log-logistic Weibull–Poisson distribution: model,properties and applications

A new class of distributions called the log-logistic Weibull–Poisson distribution is introduced and its properties are explored. This new distribution represents a more flexible model for lifetime data. Some statistical properties of the proposed distribution including the expansion of the density function, quantile function, hazard and reverse hazard functions, moments, conditional moments, moment generating function, skewness and kurtosis are presented. Mean deviations, Bonferroni and Lorenz curves, Rényi entropy and distribution of the order statistics are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study is conducted to examine the bias, mean square error of the maximum likelihood estimators and width of the confidence interval for each parameter and finally applications of the model to real data sets are presented to illustrate the usefulness of the proposed distribution.     

An extended Maxwell distribution: Properties and applications

In this article, we propose an extension of the Maxwell distribution, so-called the extended Maxwell distribution. This extension is evolved by using the Maxwell-X family of distributions and Weibull distribution. We study its fundamental properties such as hazard rate, moments, generating functions, skewness, kurtosis, stochastic ordering, conditional moments and moment generating function, hazard rate, mean and variance of the (reversed) residual life, reliability curves, entropy, etc. In estimation viewpoint, the maximum likelihood estimation of the unknown parameters of the distribution and asymptotic confidence intervals are discussed. We also obtain expected Fisher’s information matrix as well as discuss the existence and uniqueness of the maximum likelihood estimators. The EMa distribution and other competing distributions are fitted to two real datasets and it is shown that the distribution is a good competitor to the compared distributions.     

The largest SNR distribution

The probability distribution of the maximum of normalized SNRs (signal-to-noise ratios) is studied for wireless systems with multiple branches. Explicit expressions and bounds are derived for the cumulative distribution function, probability density function, hazard rate function, moment generating function, nth moment, variance, skewness, kurtosis, mean deviation, Shannon entropy, order statistics and the asymptotic distribution of the extreme order statistics. Estimation procedures are derived by the methods of moments and maximum likelihood. An application is illustrated with respect to performance assessment of wireless systems.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.     

An extended fatigue life distribution

A five-parameter extended fatigue life model called the McDonald–Birnbaum–Saunders (McBS) distribution is proposed. It extends the Birnbaum–Saunders and beta Birnbaum–Saunders [G.M. Cordeiro and A.J. Lemonte, The β-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Comput. Statist. Data Anal. 55 (2011), pp. 1445–1461] distributions and also the new Kumaraswamy–Birnbaum–Saunders distribution. We obtain the ordinary moments, generating function, mean deviations and quantile function. The method of maximum likelihood is used to estimate the model parameters and its potentiality is illustrated with an application to a real fatigue data set. Further, we propose a new extended regression model based on the logarithm of the McBS distribution. This model can be very useful to the analysis of real data and could give more realistic fits than other special regression models.     

A new lifetime model with a periodic hazard rate and an application

In this paper, we introduce a new three parameter lifetime distribution for modelling lifetime data. Among its interesting properties, it has the feature of possessing a periodic hazard rate function. Some of its mathematical properties are studied, including moments, moment generating function, mean deviations and order statistics. The estimation of its parameter is investigated using the maximum-likelihood estimation. A real-life data set is considered to illustrate the usefulness and the applicability of the proposed distribution.     

The Marshall-Olkin logistic-exponential distribution

We introduce a three-parameter extension of the exponential distribution which contains as sub-models the exponential, logistic-exponential and Marshall-Olkin exponential distributions. The new model is very flexible and its associated density function can be decreasing or unimodal. Further, it can produce all of the four major shapes of the hazard rate, that is, increasing, decreasing, bathtub and upside-down bathtub. Given that closed-form expressions are available for the survival and hazard rate functions, the new distribution is quite tractable. It can be used to analyze various types of observations including censored data. Computable representations of the quantile function, ordinary and incomplete moments, generating function and probability density function of order statistics are obtained. The maximum likelihood method is utilized to estimate the model parameters. A simulation study is carried out to assess the performance of the maximum likelihood estimators. Two actual data sets are used to illustrate the applicability of the proposed model.     

The McDonald Gompertz distribution: Properties and applications

This article introduces a five-parameter lifetime model called the McDonald Gompertz (McG) distribution to extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz, and Kumaraswamy Gompertz distributions among several other models. The hazard function of new distribution can be increasing, decreasing, upside-down bathtub, and bathtub shaped. We obtain several properties of the McG distribution including moments, entropies, quantile, and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. The flexibility and usefulness of the new distribution are illustrated by means of application to two real datasets.     

A new class of generalized Lindley distributions with applications

A new four-parameter class of generalized Lindley (GL) distribution called the beta-generalized Lindley (BGL) distribution is proposed. This class of distributions contains the beta-Lindley, GL and Lindley distributions as special cases. Expansion of the density of the BGL distribution is obtained. The properties of these distributions, including hazard function, reverse hazard function, monotonicity property, shapes, moments, reliability, mean deviations, Bonferroni and Lorenz curves are derived. Measures of uncertainty such as Renyi entropy and s-entropy as well as Fisher information are presented. Method of maximum likelihood is used to estimate the parameters of the BGL and related distributions. Finally, real data examples are discussed to illustrate the applicability of this class of models.     

The Kumaraswamy modified Weibull distribution: theory and applications

A five-parameter extension of the Weibull distribution capable of modelling a bathtub-shaped hazard rate function is introduced and studied. The beauty and importance of the new distribution lies in its ability to model both monotone and non-monotone failure rates that are quite common in lifetime problems and reliability. The proposed distribution has a number of well-known lifetime distributions as special sub-models, such as the Weibull, extreme value, exponentiated Weibull, generalized Rayleigh and modified Weibull (MW) distributions, among others. We obtain quantile and generating functions, mean deviations, Bonferroni and Lorenz curves and reliability. We provide explicit expressions for the density function of the order statistics and their moments. For the first time, we define the log-Kumaraswamy MW regression model to analyse censored data. The method of maximum likelihood is used for estimating the model parameters and the observed information matrix is determined. Two applications illustrate the potentiality of the proposed distribution.     

On Some Properties of the Beta Normal Distribution

The beta normal distribution is a generalization of both the normal distribution and the normal order statistics. Some of its mathematical properties and a few applications have been studied in the literature. We provide a better foundation for some properties and an analytical study of its bimodality. The hazard rate function and the limiting behavior are examined. We derive explicit expressions for moments, generating function, mean deviations using a power series expansion for the quantile function, and Shannon entropy.     

The logistic-X family of distributions and its applications

ABSTRACT

The logistic distribution has a prominent role in the theory and practice of statistics. We introduce a new family of continuous distributions generated from a logistic random variable called the logistic-X family. Its density function can be symmetrical, left-skewed, right-skewed, and reversed-J shaped, and can have increasing, decreasing, bathtub, and upside-down bathtub hazard rates shaped. Further, it can be expressed as a linear combination of exponentiated densities based on the same baseline distribution. We derive explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Bonferroni and Lorenz curves, Shannon entropy, and order statistics. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. We also investigate the properties of one special model, the logistic-Fréchet distribution, and illustrate its importance by means of two applications to real data sets.     

Classical estimation of hazard rate and mean residual life functions of Pareto distribution

Abstract

In this paper we find the maximum likelihood estimates (MLEs) of hazard rate and mean residual life functions (MRLF) of Pareto distribution, their asymptotic non degenerate distribution, exact distribution and moments. We also discuss the uniformly minimum variance unbiased estimate (UMVUE) of hazard rate function and MRLF. Finally, two numerical examples with simulated data and real data set, are presented to illustrate the proposed estimates.     

A complementary generalized linear failure rate-geometric distribution

In this article, we shall attempt to introduce a new class of lifetime distributions, which enfolds several known distributions such as the generalized linear failure rate distribution and covers both positive as well as negative skewed data. This new four-parameter distribution allows for flexible hazard rate behavior. Indeed, the hazard rate function here can be increasing, decreasing, bathtub-shaped, or upside-down bathtub-shaped. We shall first study some basic distributional properties of the new model such as the cumulative distribution function, the density of the order statistics, their moments, and Rényi entropy. Estimation of the stress-strength parameter as an important reliability property is also studied. The maximum likelihood estimation procedure for complete and censored data and Bayesian method are used for estimating the parameters involved. Finally, application of the new model to three real datasets is illustrated to show the flexibility and potential of the new model compared to rival models.     

The gamma-linear failure rate distribution: theory and applications

For any continuous baseline G distribution, Zografos and Balakrishnan [On families of beta- and generalized gamma-generated distributions and associated inference. Statist Methodol. 2009;6:344–362] introduced the generalized gamma-generated distribution with an extra positive parameter. A new three-parameter continuous model called the gamma-linear failure rate (LFR) distribution, which extends the LFR model, is proposed and studied. Various structural properties of the new distribution are derived, including some explicit expressions for ordinary and incomplete moments, generating function, probability-weighted moments, mean deviations and Rényi and Shannon entropies. We estimate the model parameters by maximum likelihood and obtain the observed information matrix. The new model is modified to cope with possible long-term survivors in lifetime data. We illustrate the usefulness of the proposed model by means of two applications to real data.     

The exponentiated exponential distribution: a survey

The exponentiated exponential distribution, a most attractive generalization of the exponential distribution, introduced by Gupta and Kundu (Aust. N. Z. J. Stat. 41:173–188, 1999) has received widespread attention. It appears, however, that many mathematical properties of this distribution have not been known or have not been known in simpler/general forms. In this paper, we provide a comprehensive survey of the mathematical properties. We derive expressions for the moment generating function, characteristic function, cumulant generating function, the nth moment, the first four moments, variance, skewness, kurtosis, the nth conditional moment, the first four cumulants, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, Bonferroni concentration index, Gini concentration index, Rényi entropy, Shannon entropy, cumulative residual entropy, Song’s measure, moments of order statistics, L moments, asymptotic distribution of the extreme order statistics, reliability, distribution of the sum of exponentiated exponential random variables, distribution of the product of exponentiated exponential random variables and the distribution of the ratio of exponentiated exponential random variables. We also discuss estimation by the method of maximum likelihood, including the case of censoring, and provide simpler expressions for the Fisher information matrix than those given by Gupta and Kundu. It is expected that this paper could serve as a source of reference for the exponentiated exponential distribution and encourage further research.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.