Abstract
Motivated by the fact that the so-called Weibull generated exponentiated exponential distribution (WGEED) accommodates for non-monotone as well as monotone hazard rate functions (HRFs), we give some properties of this WGEED and explore its use in life testing by obtaining Bayes prediction intervals of future observables. The survival function (SF) of the WGEED is constructed by composing a Weibull cumulative distribution function (CDF) with –ln [exponentiated exponential] CDF. Some properties of the WGEED are given and prediction intervals of future observables, using the one- and two-sample schemes, are obtained. A comparison between the WGEE and the Weibull distributions, based on Kolmogorov-Smirnov goodness of fit test, shows that the former fits better than the latter. Real life data shows the possibility of using the WGEED in analyzing lifetime data. Numerical examples of one- and two- sample Bayes interval prediction are given to illustrate the procedure and a simulation study is made to compute the coverage probabilities and the average lengths of intervals.
Highlights
New families of distributions that accommodate for non-monotone as well as monotone hazard rate functions (HRFs) can be constructed by the composition of a given cumulative distribution function (CDF), say H(.), with another CDF G(. ) or a function of G(. )
We have considered in this paper the Weibull generated exponentiated exponential distribution (WGEED) a new model which arises as a composition of Weibull and exponentiated exponential distribution
The CDF (7) of the Weibull generated exponentiated exponential (WGEE) is obtained in closed form that simplifies its use
Summary
New families of distributions that accommodate for non-monotone as well as monotone hazard rate functions (HRFs) can be constructed by the composition of a given cumulative distribution function (CDF), say H(.), with another CDF G(. ) or a function of G(. ). For an arbitrary G, CDF (4) is known as the beta-G distribution. ∫ H ( x) = x γ d yd −1 e−γ y d y and G( x) = (1− e−β x )α , x > 0 , (α, β > 0 ). Ristić and Balakrishnan [10] called F ( x) = 1 − F (x) , (with γ = 1), gamma generated exponentiated exponential distribution. Instead of using the gamma CDF for H(.), we use the Weibull (η,γ ) CDF, given by. The composition of H(.) with – ln G(.), where H and G are given by (6) and (5), respectively, leads to the SF. The CDF, given by (7), shall be called Weibull generated exponentiated exponential (WGEE) distribution with parameters (δ ,γ , β ).
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