The Extended Log-Logistic Distribution: Inference and Actuarial Applications

Abstract

Actuaries are interested in modeling actuarial data using loss models that can be adopted to describe risk exposure. This paper introduces a new flexible extension of the log-logistic distribution, called the extended log-logistic (Ex-LL) distribution, to model heavy-tailed insurance losses data. The Ex-LL hazard function exhibits an upside-down bathtub shape, an increasing shape, a J shape, a decreasing shape, and a reversed-J shape. We derived five important risk measures based on the Ex-LL distribution. The Ex-LL parameters were estimated using different estimation methods, and their performances were assessed using simulation results. Finally, the performance of the Ex-LL distribution was explored using two types of real data from the engineering and insurance sciences. The analyzed data illustrated that the Ex-LL distribution provided an adequate fit compared to other competing distributions such as the log-logistic, alpha-power log-logistic, transmuted log-logistic, generalized log-logistic, Marshall–Olkin log-logistic, inverse log-logistic, and Weibull generalized log-logistic distributions.

Applications of Bladder Cancer Data Using a Modified Log-Logistic Model

“…The second data set, shown in Table 6 , represents the number of claims from private motor insurance policies in the United Kingdom studied by Alfaer et al [ 25 ]. The TTT graph is plotted in Figure 6(a) and shows a decreasing FR function.…”

“…Muse et al [ 24 ] applied Bayesian and classical approaches for inference about a generalized LL distribution. Alfaer et al [ 25 ] introduced exponentiated Marshal-Olkin extension of the LL model for modeling high tail data in insurance claims.…”

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“…Also, Zeghdoudi and Nedjar [8] and Grine and Zeghdoudi [9] introduced another new distributions which Poisson-Lindley distribution (see Sankaran [10] is a particular case, by compounding Poisson, pseudo Lindley and quasi Lindley distributions which will add some value to the existing literature on modeling lifetime data and biological sciences. To better study this field, it is advisable to consult the following works: Alshanbari et al [11], Alfaer et al [12], Afify et al [13], Chouia.et al [14], Seghier et al [15] and Benatmane et al [16].…”

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“…Tung et al [17] proposed the arcsine-Weibull (ASin-Weibull) distribution for analyzing data sets in the business and financial sectors. For more studies related to the financial data sets (i.e., data modeling in the financial-related area), we refer to studies by Zhao et al [18]; Alfaro et al [19]; Abubakar and Sabri [20]; and Rana et al [21].…”