Abstract
It is well established that classical one-parameter distributions lack the flexibility to model the characteristics of a complex random phenomenon. This fact motivates clever generalizations of these distributions by applying various mathematical schemes. In this paper, we contribute in extending the one-parameter length-biased Maxwell distribution through the famous Marshall–Olkin scheme. We thus introduce a new two-parameter lifetime distribution called the Marshall–Olkin length-biased Maxwell distribution. We emphasize the pliancy of the main functions, strong stochastic order results and versatile moments measures, including the mean, variance, skewness and kurtosis, offering more possibilities compared to the parental length-biased Maxwell distribution. The statistical characteristics of the new model are discussed on the basis of the maximum likelihood estimation method. Applications to simulated and practical data sets are presented. In particular, for five referenced data sets, we show that the proposed model outperforms five other comparable models, also well known for their fitting skills.
Highlights
The Maxwell (M) distribution, called Maxwell–Boltzmann distribution, is a classical one-parameter distribution, finding numerous applications in engineering, physics, chemistry and reliability
We emphasize the pliancy of the main functions, strong stochastic order results and versatile moments measures, including the mean, variance, skewness and kurtosis, offering more possibilities compared to the parental length-biased Maxwell distribution
We introduced a generalization of the length-biased Maxwell distribution known as Marshall–Olkin length-biased Maxwell distribution
Summary
The Maxwell (M) distribution, called Maxwell–Boltzmann distribution, is a classical one-parameter distribution, finding numerous applications in engineering, physics, chemistry and reliability. Like most one-parameter distributions, the M distribution is not suitable to model certain lifetime phenomena This is especially true for those with highly biased right distributed values or any other type of left skewed distributed values. Modi [11] and Saghir [12] proposed to extend the M distribution through the use of the length-biased scheme, introducing the length-biased Maxwell (LBM) distribution with parameter α > 0. The LBM distribution can be viewed as a special power version of the LBE distribution, the LBE distribution being defined by the following cdf: FLBE(x; γ) = 1 − e−x/γ 1 + x , x > 0, γ and FLBM(x; γ) = 0 for x ≤ 0, where γ > 0 denotes the related parameter. We introduce the Marshall–Olkin LBM (MOLBM) distribution, defined with the following cdf: FMOLBM(x; α, β). Basics of the MOLBM Distribution The fundamental functions of the MOLBM distribution are derived and analyzed
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