International Journal of Statistics and Probability; Vol. 10, No. 4; July 2021 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education A New Lindley-Burr XII Distribution: Model, Properties and Applications Boikanyo Makubate 1 , Broderick Oluyede 1 & Morongwa Gabanakgosi 1 1 Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana Correspondence: Morongwa Gabanakgosi, Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, P. Bag 16, Palapye, Botswana Received: October 27, 2020 Accepted: May 26, 2021 Online Published: June 1, 2021 doi:10.5539/ijsp.v10n4p33 URL: https://doi.org/10.5539/ijsp.v10n4p33 Abstract A new distribution called the Lindley-Burr XII (LBXII) distribution is proposed and studied. Some structural properties of the new distribution including moments, conditional moments, distribution of the order statistics and R´ enyi entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. A simulation study to examine the bias and mean square error of the maximum likelihood estimators is presented and applications to real data sets in order to illustrate the usefulness of the new distribution are given. Keywords: Lindley distribution, Burr XII distribution, generalized distribution, monte Carlo simulations, maximum likelihood estimation 1. Introduction There are several generalizations of the Lindley distribution that are considered to be useful life models, and are suitable for modeling data with different types of hazard rate functions including increasing, decreasing, bathtub and unimodal. (Lindley, 1958) used a mixture of exponential and length-biased exponential distributions to illustrate the difference between fiducial and posterior distributions. This mixture is called the Lindley (L) distribution. There are several gen- eralizations of the Lindley distribution in the literature including the works by (Oluyede, Yang and Omolo, 2015) and (Nadarajah, Bakouch and Tahmasbi, 2011). (Oluyede and Yang, 2015) presented the beta generalized Lindley distri- bution. (Ghitany, Al-Mutairi, Balakrishnan and Al-Ezeni, 2013) presented a two-parameter power Lindley distribution. (Zakerzadeh and Dolati, 2009) studied an extension of the Lindley distribution. These models constitute flexible family of distributions in terms of the varieties of shapes and hazard functions. The cumulative distribution function (cdf) of the Lindley distribution is given by G L ( x; λ) = 1 − λ + 1 + λx λ + 1 e −λx , for x > 0, and λ> 0. (1) The corresponding Lindley probability density function (pdf) is given by g L ( x; λ) = λ 2 (1 + x) 1 + λ e −λx , for x > 0, and λ> 0. (2) Lindley distribution is a combination of exponential and gamma distributions, that is f ( x; λ) = (1 − p) f G ( x; λ) + pf E ( x; λ) with p = 1 1+λ , where f G ( x; λ) ≡ GAM(2,λ), and f E ( x; λ) ≡ EXP(λ). The cdf of the generalized Lindley (GL) distribution (Nadarajah et al., 2011) is given by G GL ( x; α, λ) = [ 1 − 1 + λ + λx 1 + λ exp(−λx) ] α , (3) and the corresponding pdf is given by g GL ( x; α, λ) = αλ 2 1 + λ (1 + x) [ 1 − 1 + λ + λx 1 + λ exp(−λx) ] α−1 exp(−λx), (4) for x > 0,λ> 0,α> 0. This distribution is the exponentiated Lindley distribution. We conisider the following generalization of the Lindley and Burr XII distributions. Let X 1 and X 2 be independent random variables with Lindley and Burr XII distributions, respectively. That is, F X 1 ( x) = 1 − λ+1+λx λ+1 e −λx , and F X 2 ( x) = 1 −(1 + x c ) −k , 33