A New Lifetime Distribution Slash Lindley – Rayleigh and Power Series
A Dissertation
Submitted to the Council of College
of Administration and Economics /
University of Karbala as Partial fulfillment of the Requirements for Degree of Philosophy
By
Adel Abbood Najm
Supervision
Prof. Dr. Mahdi Wahab Nea’ama
Most of the probability distributions families Lindley-G (LG) where G denotes to the generator distribution and more specifically, the Lindley- Rayleigh (LR) lifetime distribution were suggested for the sake of increasing the flexibility of Lindley distribution by adding another parameter, in spite of that, the right side tail of the LR distribution is not heavy enough to conciliate and fit data that exhibit high kurtosis and outlying values, namely, those which concern with lifetime data such as, the analysis of reliability and survival. In addition to the above-mentioned, there are certain probability distributions fail to fit and hence predict more accurate when they meet a certain data for which its hazard rate functions exhibit a various patterns. A new lifetime distribution has been proposed with two proposals to deal with and overcome these problems and limitations, the first one will be called the Slash Lindley-Rayleigh (“SLR” for short) distribution with three parameters , where it has been founded by using the transformation of variables technique, where most of the statistical and mathematical properties have been derived and studied , besides to find the maximum likelihood estimation MLE for its three parameters which did not appear in the closed form. The researcher work to study the effect of the observations number N (namely, it means a machine or a system breakdowns number)on the reliability function of SLR distribution by assuming that N is a discrete random variable follows one of the power series distributions which is truncated at the point zero (zero truncated distributions) with the parameter which will represent mean of the breakdowns number or the breakdowns probability according to PS distribution (Poisson, Geometric, Logarithmic, Binomial, and Negative Binomial). And for studying and knowing that effect, an extended distribution has been suggested to represent the second proposed distribution which will be called the Slash Lindley-Rayleigh and Power Series (“SLRPS” for short) distribution in four parameters , distribution. distribution has been obtained by using the compounding distributions technique between the SLR distribution and the PS distribution, where some of its statistical characteristics have been derived, furthermore, the maximum likelihood estimates MLE for its parameters have been founded.
Via results of the empirical side which represented by Monte Carlo simulation technique and results of the practical side which are representing by the real data, namely, it represent the weakly averages of lifetimes measured in hours for four machines three of them represent the weaving machines and the other one represents the completion machine with sample size n=75 lifetimes for each machine taken from The General Company Of Textile Industries/The Iraqi Carpet Factory /Iraq- Baghdad, note that the first and second weaving machines are connected according to the series system, and the third weaving machine and the completion machine (the fourth machine) are connected according to the parallel system. And by depending on statistical indices and different comparison criteria such as (AMSE.RAMSE, AIC, BIC), the researcher reach to that the two proposed distributions show a preference and more flexibility in fitting and modeling data with high or excess kurtosis compared with the original distributions (Lindley, Rayleigh, Lindley- Rayleigh) concerned with estimation of the parameters and the reliability function, and these results have been obtained and implemented via writing a special program by using R- language.
