A Novel Lomax Extension with Statistical Properties, Copulas, Different Estimation Methods and Applications

Abstract

A new compound Lomax model is proposed and analyzed. The novel distribution is derived based on compounding the zero truncated Poisson distribution and the exponentiated exponential Lomax distribution. The new density can be “monotonically left skewed,” “monotonically right skewed” and “symmetric” with various useful shapes. The new hazard rate can be “upside down bathtub-increasing,” “bathtub (U-shape),” “monotonically decreasing,” “increasing-constant” and “monotonically increasing.” Relevant statistical properties are derived. We briefly describe different estimation methods, namely the maximum likelihood, Cramér-von-Mises, ordinary least squares, weighted least square, Anderson–Darling, right tail Anderson–Darling and left tail Anderson–Darling. Monte Carlo simulation experiments are performed for comparing the performances of the proposed methods of estimation for both small and large samples. For facilitating the mathematical modeling of the bivariate real data sets, we derive some new corresponding bivariate distributions. Graphical simulation study is performed for assessing the finite sample behavior of the estimators using the maximum likelihood method. Two applications are provided for illustrating the applicability of the new model.

Flexible Extension of the Lomax Distribution for Asymmetric Data under Different Failure Rate Profiles: Characteristics with Applications for Failure Modeling and Service Times for Aircraft Windshields

“…Table 5 provides an over- The Akaike information criterion (AKIC), the Bayesian information criterion (BYIC), the consistent Akaike information criterion (C-AKIC), the Hannan-Quinn information criterion (HNQIC), the Anderson and Darling (ADg), and the Cramér-von Mises (CVMs) are all used in the process of comparing various competing models. However, many other information tests can be used due to Aboraya et al [34], Abdul-Moniem and Abdel-Hameed [35], Ali et al [36], Ali et al [37], Atkinson and Harrison [38], Ansari et al [39], Cordeiro et al [40], Durbey [41], Hamed et al [42], Harris [43], Hassan and Al-Ghamdi [44], Gad et al [45]. In addition to this, the applicable P-value and the Kolmogorov-Smirnova test are both taken into consideration.…”

The Double Burr Type XII Model: Censored and Uncensored Validation Using a New Nikulin-Rao-Robson Goodness-of-Fit Test with Bayesian and Non-Bayesian Estimation Methods

“…The AVs and their mean MSEs are presented in Table 16 from which one can note that the MLEs are convergent. For defining the NRR test, consider that 𝑇 [1] , 𝑇 [2] , ⋯ , 𝑇 The 𝑌 2 statistic follows a chi-square 𝜒2 distribution with (𝜌 − 1) degrees of freedom.…”

“…where both 𝛼 2 > 0 and 𝛽 2 > 0 are shape parameters. In equation (1), when 𝛼 2 = 1 the BrXII model reduces to the Lomax (Lx) or Pareto type II (PaII) model, when 𝛽 2 = 1 the BrXII model reduces to the log-logistic (LL) model. Details and other properties about the BrXII model can be found in Burr (1942Burr ( , 1968Burr ( and 1973, Burr and Cislak (1968), Rodriguez (1977) and Tadikamalla (1980).…”

Bivariate Exponentiated Lomax Distribution Based on Marshall-olkin Method With a Real-life Application

“…Several studies have discussed bivariate distributions based on copulas. These include the works of Elgohari and Yousof [38], Hamed et al [39], Aboraya et al [40], El-Sherpieny et al [41], El-Sherpieny et al [42], El-Sherpieny et al [43], Almetwally et al [44], El-Sherpieny et al [45], Muhammed et al [46], Abulebda et al [47], Hassan et al [48], Zhao et al [49], and Qura et al [50].…”