METRON DOI 10.1007/s40300-014-0052-1 The modified Weibull geometric distribution Min Wang · Ibrahim Elbatal Received: 27 January 2014 / Accepted: 8 August 2014 © Sapienza Università di Roma 2014 Abstract A new class of lifetime distributions is introduced by compounding the modified Weibull and geometric distributions, the so-called modified Weibull geometric distribution. It includes as special submodels such as linear failure rate geometric distribution, Weibull geo- metric distribution, exponential geometric distribution, among others. We study its structural properties including probability density function, hazard functions, moments, generating and quantile functions. The distribution is capable of monotonically increasing, decreas- ing, bathtub-shaped, and upside-down bathtub-shaped hazard rates. Estimation by maximum likelihood and inference for a large sample are presented. An expectation-maximization algorithm is used to determine the maximum likelihood estimates of the parameters. Finally, a real data set is analyzed for illustrative purposes. Keywords Modified Weibull distribution · Geometric distribution · Hazard function · Maximum likelihood estimation · EM algorithm 1 Introduction The study of life length of organisms, materials, etc., plays an important role in the bio- logical and engineering sciences. A substantial part of such study is devoted to modeling the lifetime data by a failure distribution. The exponential, Rayleigh, and Weibull distri- butions are used most commonly in reliability and life testing because they have many desirable properties and nice physical interpretations. Unfortunately, the exponential and M. Wang (B ) Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA e-mail: minwang@mtu.edu I. Elbatal Department of Mathematical Statistics, Institute of Statistical Studies and Research, Cairo University, Cairo 12613, Egypt e-mail: i_elbatal@staff.cu.edu.eg 123