Abstract
Heavy-tailed distributions have wide ap- plications in life-time contexts, especially in reliability and risk modeling. So we consider the estimation problem of reliability, R = P (X > Y ) for small samples, when X and Y are two independent but not identically distributed random variables belonging to the family of heavy-tailed distributions, using a robust estimator, namely the harmonic moment estimator. Extensive sim- ulation studies are carried out to study the performance of this estimator. The relative efficiency of the estima- tor with the well known Hill estimator is studied. We obtain the sampling distribution of the parameters of the distribution as well as that of estimator of R which will help us to study the properties of the estimators. Also we find out the asymptotic confidence intervals of R and its performance is studied with respect to average width and the coverage probability, through simulations. especially in reliability and risk modeling. We consider the estimation problem of reliability R = P (X > Y ) for small and large samples, when X and Y are two inde- pendent but not identically distributed random variables belonging to the family of heavy-tailed distributions.The heavy-tailed distributions like the student's t and Pareto families have been used to model data with high kurto- sis see, (4). The simplest heavy tailed distribution is the Pareto distribution. It is hyperbolic over its entire range. The Pareto distribution is a power law probability dis- tribution that coincides with social, scientific, geophys- ical, actuarial and many other types of observable phe- nomena. The univariate Pareto distribution is a simple model for non-negative data with a power law probabil- ity tail, at least approximately for large values of X. It is a useful model in the analysis of income data, reliabil- ity studies, risk modeling and business failure data (14). Reference (1) gave an extensive historical survey of its uses in the context of income distribution. References (6), (16) and (19) discussed the application of the Pareto disrribution in various fields. Moreover Pareto Type I distribution has a position of importance in the field of life testing because this distribution can be considered as a distribution of failure time of a component as fol- lows.
Highlights
Heavy-tails are characteristics of many phenomena where the probability of a single huge value impacts heavily
Hill estimator [11] was compared by Henry in [10] and it was shown that the harmonic moment estimator outperforms the Hill estimator for estimating the tail index of a distribution in small sample situations, in terms of bias, variance and mean squared error
We have estimated R=P (X > Y ) with respect to two independent Pareto type I distributions with shape parameters α1 and α2
Summary
Heavy-tails are characteristics of many phenomena where the probability of a single huge value impacts heavily. The univariate Pareto distribution is a simple model for non-negative data with a power law probability tail, at least approximately for large values of X. It is a useful model in the analysis of income data, reliability studies, risk modeling and business failure data [14]. In this study we are estimating the desired probability R = P (X > Y ) where X and Y are two independent but not identically distributed random variables following Pareto type I distribution, with parameters α1 and α2. For details see, [10]
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