Stochastics and Quality Control 2018; aop Research Article Bol A. M. Atem, Suleman Nasiru* and Kwara Nantomah ToppśLeone Linear Exponential Distribution https://doi.org/10.1515/eqc-2017-0022 Received September 21, 2017; revised January 3, 2018; accepted January 4, 2018 Abstract: This article studies the properties of the ToppśLeone linear exponential distribution. The param- eters of the new model are estimated using maximum likelihood estimation, and simulation studies are performed to examine the őnite sample properties of the parameters. An application of the model is demon- strated using a real data set. Finally, a bivariate extension of the model is proposed. Keywords: ToppśLeone, Linear Exponential, Bivariate, Quantile, Moment MSC 2010: 62E15, 60E05 1 Introduction The linear exponential distribution which is made up of the exponential and Rayleigh distributions has applications in applied statistics and reliability analysis. The linear exponential distribution, also known as the linear failure rate distribution in literature, is suitable for modeling data sets with either constant or increasing linear failure rates. However, when the data at hand exhibit decreasing, non-linear increasing, or non-monotonic failure rates such as the bathtub and upside down bathtub among others, which are common in reliability studies, then the linear exponential distribution does not provide reasonable parametric őt. This motivated researchers to propose generalizations of the linear exponential distribution in order to improve its goodness-of-őt. Some of the generalizations include: generalized linear failure rate distribution by Sarhan and Kundu [9], Serial Weibull Rayleigh distribution by Nasiru [7], Kumaraswamy linear exponential distribu- tion by Merovci and Elbatal [5], exponentiated generalized linear exponential distribution by Sarhan, Ahmad and Alasbahi [8], the generalized linear exponential distribution by Mahmoud and Alam [4], new generalized linear exponential distribution by Tian, Tian and Zhu [10] and the odd generalized exponential generalized linear exponential distribution by Luguterah and Nasiru [3]. In this study, we employ the ToppśLeone generator developed by Al-Shomrani, Arif, Shawky, Hanif and Shahbaz [1] to propose an extension of the linear exponential distribution called ToppśLeone linear exponential (TLLE) distribution. The cumulative distribution function (CDF) of the ToppśLeone family of distribution is given by F(x)=[G(x)] α [2 − G(x)] α =[1 − ̄ G 2 (x)] α , x ∈ℝ, α > 0, (1.1) and the corresponding probability density function (PDF) of the family is given by f(x)= 2αg(x) ̄ G(x)[1 − ̄ G 2 (x)] α−1 , Bol A. M. Atem, Institute for Basic Sciences, Technology and Innovation, Pan African University, P.O. Box 62000-00200, Nairobi, Kenya, e-mail: bol207@yahoo.com *Corresponding author: Suleman Nasiru, Institute for Basic Sciences, Technology and Innovation, Pan African University, P.O. Box 62000-00200, Nairobi, Kenya; and Faculty of Mathematical Sciences, University for Development Studies, P.O. Box 24, Navrongo, Upper East Region, Ghana, e-mail: sulemanstat@gmail.com. http://orcid.org/0000-0001-6652-4251 Kwara Nantomah, Faculty of Mathematical Sciences, University for Development Studies, P.O. Box 24, Navrongo, Upper East Region, Ghana, e-mail: mykwarasoft@yahoo.com Brought to you by | Tufts University Authenticated Download Date | 2/22/18 2:21 PM