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Abstract
In this paper, a new modification of the Lomax distribution is considered named as Lomax exponential distribution (LE). The proposed distribution is quite flexible in modeling the lifetime data with both decreasing and increasing shapes (non-monotonic). We derive the explicit expressions for the incomplete moments, quantile function, the density function for the order statistics etc. The Renyi entropy for the proposed distribution is also obtained. Moreover, the paper discusses the estimates of the parameters by the usual maximum likelihood estimation method along with determining the information matrix. In addition, the potentiality of the proposed distribution is illustrated using two real data sets. To judge the performance of the model, the goodness of fit measures, AIC, CAIC, BIC, and HQIC are used. Form the results it is concluded that the proposed model performs better than the Lomax distribution, Weibull Lomax distribution, and exponential Lomax distribution.
Highlights
In probability theory, it has been a usual practice for the last few years to modify the existing probability distributions so as to improve the flexibility of the existing models
; Consistent Akaike Information Criterion (CAIC) 1⁄4 À 2L þ PflogðnÞ þ 1g; Bayesian Information Criterion (BIC) 1⁄4 PlogðnÞ À 2Lðc^ ; yiÞ; Hannan Quinn information Criterion (HQIC) 1⁄4 À 2Lmax þ 2PlogflogðnÞg: where L is the maximized likelihood function and yi is the given random sample, c^ is the maximum likelihood estimator and p is the number of parameters in the model
The results reveal that increase in the sample size results in a decrease in both the bias and Mean square error (MSE)
Summary
It has been a usual practice for the last few years to modify the existing probability distributions so as to improve the flexibility of the existing models. Let a positive random variable Y has the Lomax distribution with parameters a and b, the cumulative distribution function (Cdf) takes the form h y iÀ a. Cordeiro et al [2] explored the gamma-Lomax distribution and discussed its applications to real data sets. Ibrahim et al [7] produced a new three parameters probability distribution and referred to it as exponentiated Lomax distribution. Let a random variable Y has the Lomax exponential distribution with parameters a and b. Distribution function of the Lomax exponential distribution is given by yexpðyÞ À a. We find the second time partial derivatives of the Eqs from (22) to (25) and are given as d da2 ‘ðy; a; bÞ 1⁄4 I11 1⁄4 À a2
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