Abstract
In this paper, we propose new families of generalized Lomax distributions named T-LomaxfYg. Using the methodology of the Transformed-Transformer, known as T-X framework, the T-Lomax families introduced are arising from the quantile functions of exponential, Weibull, log-logistic, logistic, Cauchy and extreme value distributions. Various structural properties of the new families are derived including moments, modes and Shannon entropies. Several new generalized Lomax distributions are studied. The shapes of these T-LomaxfYg distributions are very flexible and can be symmetric, skewed to the right, skewed to the left, or bimodal. The method of maximum likelihood is proposed for estimating the distributions parameters and a simulation study is carried out to assess its performance. Four applications of real data sets are used to demonstrate the flexibility of T-LomaxfYg family of distributions in fitting unimodal and bimodal data sets from di erent disciplines.
Highlights
The Lomax distribution, known as Pareto type-II distribution, is one of the important continuous distributions with a heavy tail defined by one shape and one scale parameters
The three and four parameters T-Lomax{Y} members: Weibull Lindley (W-L){LL}, G-L{LL}, EW-L{E}, N-L{C}, and the W-L{E} are used to fit these unimodal data sets. Their results are compared to the fitting results of the McDonald Lomax (McLomax), the Beta Lomax (BLomax), and the Kumaraswamy Lomax (KwLomax) distributions which are known extensions of the Lomax distribution defined by Lemonte and Cardeiro (2013) with four and five parameters
The Maximum Likelihood Estimation (MLE) of the parameters as well as the goodness-of-fit tests of the T-Lomax{Y} members W-L{LL}, G-L{LL}, EW-L{E} and N-L{C} with the other competing distributions McLomax, BLomax, and KwLomax are shown in Tables 5 and 6
Summary
The Lomax distribution, known as Pareto type-II distribution, is one of the important continuous distributions with a heavy tail defined by one shape and one scale parameters. Corollary 2 Based on Lemma 2, the quantile functions for the (i) T-Lomax{exponential}, (ii) T-Lomax{Weibull}, (iii) T-Lomax{log-logistic}, (iv) T-Lomax{logistic}, (v) T-Lomax{Cauchy}, and (vi) T-Lomax{extreme value} distributions, respectively, are given by (i) QX(p) = λ e(QT (p)/α) − 1 ,. Corollary 3 Based on Theorem 1, the mode(s) of the (i) T-Lomax{exponential}, (ii) T-Lomax{Weibull}, (iii) T-Lomax{loglogistic}, (iv) T-Lomax{logistic}, (v) T-Lomax{Cauchy}, and (vi) T-Lomax{extreme value} distributions, respectively, are the solutions of the equations (i) (x + λ) Ψ fT α log (1 + x/λ) = 1,. Based on Theorem 4, the rth non-central moments for the (i) T-Lomax{exponential}, (ii) T-Lomax{Weibull}, (iii) T-Lomax{log-logistic}, (iv) T-Lomax{logistic}, (v) T-Lomax{Cauchy}, and (vi) T-Lomax{extreme value} distributions, respectively, are given by r (i) E (Xr) = λr an MT (n/α), exists if MT (n/α) < ∞, n=0 r (ii) E (Xr) = λr an MTk (n/γkα), exists if MTk (n/γkα) < ∞, n=0 r∞. Theorem 5 and Corollary 6 can be used to obtain the mean deviations for T-Lomax{exponential}, T-Lomax{Weibull}, T-Lomax{log-logistic}, T-Lomax{logistic}, T-Lomax{Cauchy}, and T-Lomax{extreme value} distributions
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