A Class of Distributions with Quadratic Hazard Quantile Function
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RESEARCH ARTICLE
A Class of Distributions with Quadratic Hazard Quantile Function I. C. Aswin1 • P. G. Sankaran1 • S. M. Sunoj1 Accepted: 18 September 2020 Ó The Indian Society for Probability and Statistics (ISPS) 2020
Abstract The present paper introduces a class of distributions with quadratic hazard quantile function. We study various distributional properties and reliability characteristics of the proposed model. The proposed class of distributions aims to represent the nonmonotone behaviours of hazard quantile function. Characterizations of the model are presented and method of least square is employed to estimate parameters of the class of distributions. Finally, we illustrate the usefulness of the proposed model through real data sets. Keywords Quantile function Quantile density function Quadratic hazard quantile function L-moments
Mathematics Subject Classification 62E10 62N05
1 Introduction Quantile functions are equivalent alternative to distribution functions on modeling and analysis of statistical data. Let X be a continuous random variable with right ¼ 1 Fð:Þ. Then continuous distribution function F(.) and survival function Fð:Þ the quantile function of X is defined as & S. M. Sunoj [email protected] I. C. Aswin [email protected] P. G. Sankaran [email protected] 1
Department of Statistics, Cochin University of Science and Technology, Cochin, Kerala 682 022, India
123
Journal of the Indian Society for Probability and Statistics
QðuÞ ¼ F 1 ðxÞ ¼ inffx : FðxÞ ug;
0 u 1:
ð1:1Þ
If f(.) is the probability density function of X, then f(Q(u)) is called the density d QðuÞ, is quantile function. The derivative of Q(u) which is expressed as qðuÞ ¼ du known as the quantile density function of X. Differentiating FðQðuÞÞ ¼ u we get, f ðQðuÞÞqðuÞ ¼ 1:
ð1:2Þ
There are many advantageous of using the quantile function over the traditional distributional approach. With heavy tailed distributions commonly encountered in the lifetime data analysis, a single long term survivor can have a marked effect on many of the reliability measures based on distribution function that are currently used. Some of these properties include sum of two quantile functions is again a quantile function and product of two positive quantile functions is again a quantile function. For a unified study on this concept, one could refer to Gilchrist (2000), Nair et al. (2013), Sankaran and Kumar (2018) and the references therein. One of the problems in the analysis of lifetime data is the absence of tractable distribution functions. In such situations, the traditional reliability measures such as hazard rate, mean residual life, reversed hazard rate etc. and other tools and results based on distribution function may not be feasible in modeling and analysis of lifetime data. Nair and Sankaran (2009) introduced basic reliability concepts such as hazard rate, mean residual life etc. in terms of quantile function. Accordingly, Nair and Sankaran (2009) defined quantile version of hazard d as rate hðxÞ ¼ dx log
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