Abstract
Abstract In this article, we develop a new general family of distributions aimed at unifying some well-established lifetime distributions and offering new work perspectives. A special family member based on the so-called modified Weibull distribution is highlighted and studied. It differs from the competition with a very flexible hazard rate function exhibiting increasing, decreasing, constant, upside-down bathtub and bathtub shapes. This panel of shapes remains rare and particularly desirable for modeling purposes. We provide the main mathematical properties of the special distribution, such as a tractable infinite series expansion of the probability density function, moments of several kinds (raw, incomplete, probability weighted…) with discussions on the skewness and kurtosis. The stochastic ordering structure and stress-strength parameter are also considered, as well as the basics of the order statistics. Then, an emphasis is put on the inferential features of the related model. In particular, the estimation of the model parameters is employed by the maximum likelihood method, with a simulation study to confirm the suitability of the approach. Three practical data sets are then analyzed. It is observed that the proposed model gives better fits than other well-known lifetime models derived from the Weibull model.
Highlights
The Weibull distribution has an advantage over the exponential distribution because of its decreasing and increasing hazard rate function depending upon the shape parameter
As a first contribution, we develop a new general family of distributions having the merit of unifying all the extensions of the Weibull distribution mentioned above
When θ = 0 and α > 0, the distribution defined by F (x) with R(x) = λx corresponds to the two-parameter generalized exponential distribution, with R(x) = λxk, we get the exponentiated Weibull distribution, with R(x) = λx + βx2, it corresponds to the generalized linear failure rate distribution and with R(x) = λxkeγx, we obtain the generalized modified Weibull (GMW) distribution
Summary
The Weibull distribution has an advantage over the exponential distribution because of its decreasing and increasing hazard rate function (hrf) depending upon the shape parameter. It has been widely used in many areas, mainly for the analysis of reliability data, time to failure equipment and testing. As a first contribution, we develop a new general family of distributions having the merit of unifying all the extensions of the Weibull distribution mentioned above It covers many more other existing distributions involving a polynomial-exponential function as the primary term
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