The Weibull-geometric distribution

Abstract

For the first time, we propose the Weibull-geometric (WG) distribution which generalizes the extended exponential-geometric (EG) distribution introduced by Adamidis et al. [K. Adamidis, T. Dimitrakopoulou, and S. Loukas, On a generalization of the exponential-geometric distribution, Statist. Probab. Lett. 73 (2005), pp. 259-269], the exponential-geometric distribution discussed by Adamidis and Loukas [K. Adamidis and S. Loukas, A lifetime distribution with decreasing failure rate, Statist. Probab. Lett. 39 (1998), pp. 35-42] and the Weibull distribution. We derive many of its standard properties. The hazard function of the EG distribution is monotone decreasing, but the hazard function of the WG distribution can take more general forms. Unlike the Weibull distribution, the new distribution is useful for modelling unimodal failure rates. We derive the cumulative distribution and hazard functions, moments, density of order statistics and their moments. We provide expressions for the Rényi and Shannon entropies. The maximum likelihood estimation procedure is discussed and an EM algorithm [A.P. Dempster, N.M. Laird, and D.B. Rubim, Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. R. Stat. Soc. B 39 (1977), pp. 1-38; G.J. McLachlan and T. Krishnan, The EM Algorithm and Extension, Wiley, New York, 1997] is given for estimating the parameters. We obtain the observed information matrix and discuss inference issues. The flexibility and potentiality of the new distribution is illustrated by means of a real data set.

The Exponentiated Generalized Standardized Half-logistic Distribution

“…6, No. 3;2017 Inserting (5) and (6) in equation (21), the PWMs of X can be expressed in a simple form Table 8, we conclude that, for fixed r, the PWMs increase when s increases. The opposite happens when we fix the parameter s and r increases.…”

“…6, No. 3;2017 where a 0 0, c j,0 = a j 0 and the coefficients c j,i (for i ≥ 1) are determined recursively by…”

Trends in epidemiology in the 21st century: time to adopt Bayesian methods

“…Alternative distributions for time-to-event data have been used by studies mentioned in the literature in recent years, allowing the addition of a parameter representing the proportion of individuals which are "immune" to the event of interest 60 . These distributions are extensions of usual distributions including a greater number of unknown parameters 61,62 . Parameter estimation in survival models based on these distributions can be challenging, especially when covariates are involved, since asymptotic properties cannot be assured.…”

Transmuted Complementary Weibull Geometric Distribution

“…. , of size from TCWGD( , , , , ) with probability density function in (5), then the likelihood function can be expressed as follows…”