Abstract
Semi-Markov processes are typical tools for modeling multi state systems by allowing several distributions for sojourn times. In this work, we focus on a general class of distributions based on an arbitrary parent continuous distribution function G with Kumaraswamy as the baseline distribution and discuss some of its properties, including the advantageous property of being closed under minima. In addition, an estimate is provided for the so-called stress–strength reliability parameter, which measures the performance of a system in mechanical engineering. In this work, the sojourn times of the multi-state system are considered to follow a distribution with two shape parameters, which belongs to the proposed general class of distributions. Furthermore and for a multi-state system, we provide parameter estimates for the above general class, which are assumed to vary over the states of the system. The theoretical part of the work also includes the asymptotic theory for the proposed estimators with and without censoring as well as expressions for classical reliability characteristics. The performance and effectiveness of the proposed methodology is investigated via simulations, which show remarkable results with the help of statistical (for the parameter estimates) and graphical tools (for the reliability parameter estimate).
Highlights
The Kumaraswamy distribution is a well-known distribution, especially to those familiar with the hydrological literature [1]
Kumaraswamy distribution is appropriate for the modeling of bounded natural and physical phenomena, such as atmospheric temperatures or hydrological measurements [5,6], record data, such as tests, games or sports [7], economic observations [8], or for empirical data with failure rate with an increasing prior [9]
It is appropriate in situations where one considers a distribution with infinite lower and/or upper bounds to fit data, when, the bounds are finite, which makes Kumaraswamy useful in preventive maintenance
Summary
The Kumaraswamy distribution is a well-known distribution, especially to those familiar with the hydrological literature [1]. Due to the closed form of both its distribution as well as its quantile function (inverse cumulative distribution), Kumaraswamy appears advantageous when it comes to the quantile modeling perspective [12,13]. These characteristics make Kumaraswamy useful and applicable in reliability theory. A general class of distributions with Kumaraswamy as a baseline distribution is considered in this work by using a parent continuous distribution function: G (·). We focus on the general class of distributions of the form (1), using a parent continuous distribution function, and discuss some of its properties, including the stress–strength reliability.
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