M. H. Tahir

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.

WEIBULL-LOMAX DISTRIBUTION: PROPERTIES AND APPLICATIONS

We introduce a new model called the Weibull-Lomax distribution which extends the Lomax distribution and has increasing and decreasing shapes for the hazard rate function. 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, probability weighted moments, generating and quantile function. The Rényi and q entropies are also obtained. We provide the density function of the order statistics and their moments. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The potentiality of the new model is illustrated by means of two real life data sets. For these data, the new model outperforms the McDonald-Lomax, Kumaraswamy-Lomax, gamma-Lomax, beta-Lomax, exponentiated Lomax and Lomax models.

Parameter induction in continuous univariate distributions: Well-established G families

The art of parameter(s) induction to the baseline distribution has received a great deal of attention in recent years. The induction of one or more additional shape parameter(s) to the baseline distribution makes the distribution more flexible especially for studying the tail properties. This parameter(s) induction also proved helpful in improving the goodness-of-fit of the proposed generalized family of distributions. There exist many generalized (or generated) G families of continuous univariate distributions since 1985. In this paper, the well-established and widely-accepted G families of distributions like the exponentiated family, Marshall-Olkin extended family, beta-generated family, McDonald-generalized family, Kumaraswamy-generalized family and exponentiated generalized family are discussed. We provide lists of contributed literature on these well-established G families of distributions. Some extended forms of the Marshall-Olkin extended family and Kumaraswamy-generalized family of distributions are proposed.

Circular Strongly Balanced Repeated Measurements Designs

Magda (1980) and Hedayat (1981) first considered the construction of circular strongly balanced repeated measurements designs. Sen and Mukerjee (1987) and Roy (1988) considered the optimality and existence of circular strongly balanced repeated measurements designs based on the method of differences and Hamiltonian decomposition of lexicographic product of two graphs. In this article, we consider the construction of circular strongly balanced repeated measurements designs using the newly proposed method called cyclic shifts, and propose some new designs for p < v.

Circular First- and Second-Order Balanced Repeated Measurements Designs

Sharma (1977) and Aggarwal et al. (2006) considered non circular construction of first- and second-order balanced repeated measurements designs. Sharma et al. (2002) constructed circular first- and second-order balanced repeated measurements designs only for a class with parameters (v, p = 3n, n = v 2) and also showed its universal optimality. In this article, we consider circular construction of first- and second-order balanced repeated measurements designs and strongly balanced repeated measurements designs by using the method of cyclic shifts. Some new circular designs with parameters (v, p, n) for cases p = v, p < v and p > v are given.

Economical generalized neighbor designs for use in Serology

This paper considers the construction of generalized neighbor designs in circular blocks for several cases, which are useful in Serology. The initial blocks for the proposed designs are developed by using the method of cyclic shifts. Generalized neighbor designs are also constructed in linear blocks for v even. Catalogs of circular binary blocks generalized neighbor designs are compiled for all cases.

The Gumbel-Lomax Distribution: Properties and Applications

We introduce a new four-parameter model called the Gumbel-Lomax distribution arising from the GumbelX generator recently proposed by Al-Aqtash (2013). Its density function can be right-skewed and reversed-J shaped, and can have decreasing and upside-down bathtub shaped hazard rate. Various structural properties of the new distribution are obtained including explicit expressions for the quantile function, ordinary and incomplete moments, Lorenz and Bonferroni curves, mean residual lifetime, mean waiting time, probability weighted moments, generating function and Shannon entropy. We also provide the density function for the order statistics. Some characterizations of the new distribution based on the conditional expectations of certain functions of the random variable are also proposed. The model parameters are estimated by the method of maximum likelihood and the observed information matrix is determined. The flexibility of the new model is illustrated by means of two real lifetime data sets.

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.

A New Weibull-G Family of Distributions

Statistical analysis of lifetime data is an important topic in reliability engineering, biomedical and social sciences and others. We introduce a new generator based on the Weibull random variable called the new Weibull-G family. We study some of its mathematical properties. Its density function can be symmetrical, left-skewed, right-skewed, bathtub and reversed-J shaped, and has increasing, decreasing, bathtub, upside-down bathtub, J, reversed-J and S shaped hazard rates. Some special models are presented. We obtain explicit expressions for the ordinary and incomplete moments, quantile and generating functions, Renyi entropy, order statistics and reliability. Three useful characterizations based on truncated moments are also proposed for the new family. The method of maximum likelihood is used to estimate the model parameters. We illustrate the importance of the family by means of two applications to real data sets.

The Poisson-X family of distributions

ABSTRACT Recently, Ristić 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.