Abstract
Certain characterizations of the discrete Lindley and discrete Poisson-Lindley distributions, originally introduced by Bakouch, Jazi and Nadarjah (2014) and Sankaran (1970), respectively, are presented. Al-Babtain, Gemeay and Afify (2020) revisited these distributions and provided estimation methods and actuarial measures as well as their applications in medicine. This short note is intended to complete, in some way, Al-Babtain, Gemeay and Afify (2020)’s work. It should be mentioned that the probability mass functions reported in the two papers mentioned above are not correct. In this note, it will be explained why they are not correct.
Highlights
To understand the behavior of the data obtained through a given process we need to be able to describe this behavior via its approximate probability law
Al-Babtain, Gemeay and Afify (2020) revisited two discrete distributions introduced by Sankaran (1970), which is called Discrete Lindley (DL) distribution and by Bakouch, Jazi and Nadarjah (2014), which is called Discrete Poisson-Lindley (DPL) distribution
Al-Babtain, Gemeay and Afify (2020) use three real-life data sets from medical science to show the superiority of the DL and DPL distributions by comparing them with some well-known discrete distriutions such as discrete Pareto and discrete Burr distributions (Krishna and Pundir, 2009), and discrete Burr-Hatke distribution (El-Morshedy, Eliwa and Altun, 2020)
Summary
To understand the behavior of the data obtained through a given process we need to be able to describe this behavior via its approximate probability law. Al-Babtain, Gemeay and Afify (2020) use three real-life data sets from medical science to show the superiority of the DL and DPL distributions by comparing them with some well-known discrete distriutions such as discrete Pareto and discrete Burr distributions (Krishna and Pundir, 2009), and discrete Burr-Hatke distribution (El-Morshedy, Eliwa and Altun, 2020) In this very short note, we present two characterizations of the DL and DPL distributions based on: (i) the conditional expectation of certain function of the random variable and (ii) the reverse hazard function.
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