Abstract
Practitioners in all applied domains value simple and adaptable lifetime distributions. They make it possible to create statistical models that are relatively easy to manage. A novel simple lifetime distribution with two parameters is proposed in this article. It is based on a parametric mixture of the exponential and weighted exponential distributions, with a mixture weight depending on a parameter of the involved distribution; no extra parameter is added in this mixture operation. It can also be viewed as a special generalized mixture of two exponential distributions. This decision is based on sound mathematical and physical reasoning; the weight modification allows us to combine some joint properties of the exponential and weighted exponential distribution, which are known as complementary in several modeling aspects. As a result, the proposed distribution may have a decreasing or unimodal probability density function and possess the demanded increasing hazard rate property. Other properties are studied, such as the moments, Bonferroni and Lorenz curves, Rényi entropy, stress-strength reliability, and mean residual life function. Subsequently, a part is devoted to the associated model, which demonstrates how it can be used in a real-world statistical scenario involving data. In this regard, we demonstrate how the estimated model performs well using five different estimation methods and simulated data. The analysis of two data sets demonstrates these excellent results. The new model is compared to the weighted exponential, Weibull, gamma, and generalized exponential models for performance. The obtained comparison results are overwhelmingly in favor of the proposed model according to some standard criteria.
Highlights
2)/(α + 1) > 0 and b = −1/(α + 1) < 0, meaning that the modified WE (MWE) distribution belongs to the family of generalized mixture of two exponential distributions, following the spirit of the distribution proposed by [16], (ii) The cdf is quite simple to manage and the MWE distribution can be studied in an-depth manner on all the theoretical and practical aspects, (iii) Thanks to the parameter α, the related pdf can be decreasing or unimodal, and the related hrf can be constant or increasing as proven later, (iv) In some concrete scenarios, the MWE model can be more efficient in data fitting than the exponential or weighted exponential (WE) models, among other lifetime models
For any positive integer r, the r th raw moment of a random variable X with the MWE distribution is given by r!
We have innovated by proposing a novel simple lifetime distribution with two parameters derived from a special mixture of the exponential and weighted exponential distributions
Summary
Its simple probability density and distribution functions help to derive various mathematical results in closed forms. Several extensions of this distribution were studied in the statistical literature. The WE distribution is suitable for modelling lifetime data when wear-out or ageing is present, providing a real alternative to the exponential distribution for this aim. The success of this weighted version of the exponential distribution has inspired a generation of researchers and practitioners for more in this direction. Further discussions on the WE distribution can be found in [15,16], and the references therein
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