Binominal Mixture Lindley Distribution: Properties and Applications
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RESEARCH ARTICLE
Binominal Mixture Lindley Distribution: Properties and Applications M. R. Irshad1
•
D. S. Shibu2 • R. Maya3 • Veena D’cruz1
Accepted: 18 September 2020 Ó The Indian Society for Probability and Statistics (ISPS) 2020
Abstract In this paper, we introduce a generalized mixture distribution, so-called the binomial mixture Lindley distribution (BMLD). The density function of this distribution is obtained by mixing binomial probabilities with gamma distribution. BMLD have various distributions as its special cases and posses various shapes for its hazard rate function including increasing, decreasing, bathtub shape and upside down bathtub shape depending on its parameters. Several mathematical, structural and statistical properties of the new distribution is presented such as moments, moment generating function, hazard rate function, vitality function, mean residual life function, inequality measures, entropy and extropy etc. The parameters of the model are estimated using the method of maximum likelihood and finally real life data sets are considered to illustrate the relevance of the new model by comparing it with some other lifetime models. Keywords Lindley distribution Maximum likelihood estimator Hazard rate function Inequality measures Entropy Extropy Vitality function
& M. R. Irshad [email protected] D. S. Shibu [email protected] R. Maya [email protected] Veena D’cruz [email protected] 1
CUSAT, Kochi-22, Kerala, India
2
University College, Trivandrum, Kerala, India
3
Govt. Women’s College, Trivandrum, Kerala, India
123
Journal of the Indian Society for Probability and Statistics
1 Introduction Numerous probability distributions are introduced in the literature by mixing, extending and modifying well known distributions and hence provide more flexible hazard rate function for modelling lifetime data. These distributions will then be more suitable for fitting appropriate real data than the base models. Knowledge of the appropriate distribution plays an important role in improving the efficiency of any statistical inference related to data sets. Hence the researchers are more keen to develop new distributions by extending classical distributions to increase model flexibility and adaptability in various aspects of modelling data. Lindley (1958) introduced in the literature one of the most discussed lifetime distribution, the Lindley distribution, in the context of the Bayesian statistics as a counter example of the fiducial statistics. Lindley distribution (LD) have the probability density function (pdf), f1 ðxÞ ¼
h2 ð1 þ xÞehx ; x [ 0; h [ 0; 1þh
ð1:1Þ
which is a mixture of exponential ðhÞ and gamma ð2; hÞ distributions. The corresponding cumulative distribution function (cdf) has been obtained as, F1 ðxÞ ¼ 1
h þ 1 þ hx hx e ; x [ 0; h [ 0; 1þh
ð1:2Þ
where h is the scale parameter. Mixture models provide a mathematical based, flexible and meaningful approach for the wide variety of classification requirements. There are numerous fields in which
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