The Extended Log-Logistic Distribution: Inference and Actuarial Applications

: 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.


Introduction
Modeling insurance losses data has received significant interest from actuaries and risk managers who often evaluate and study the unlikely outcomes that the value-at-risk may express by chance. The insurance data are usually unimodal [1], right-skewed [2], positive [3], have a heavy-tailed density [4], and have a unimodal hump shape [5].
There is a clear need to develop and propose more flexible distributions by extending the well-known classical distributions or by introducing a new family to model several insurance datasets such as financial returns, unemployment insurance data, insurance losses data, and risk management data, among others.
The log-logistic (LL) distribution is also known as the Fisk distribution in the income distribution literature [6]. Some authors, such as [7,8], have referred to the Fisk distribution as the LL distribution, whereas Arnold [9] referred to it as the Pareto Type III distribution and included an additional location parameter. Further details about the LL model can be found in [10]. Several authors have studied different generalized forms of the LL distribution to improve its capability and flexibility. Some notable examples are the following: Kumaraswamy-LL [11], beta-LL [12], Marshall-Olkin LL [13], McDonald LL [14], Zografos-Balakrishnan LL [15], and odd Lomax LL distributions [16].
This article suggests a new version of the LL distribution called the extended loglogistic (Ex-LL) distribution, which can provide more flexibility in modeling insurance data than other competing models. Hence, the aim of the paper was three-fold. The first was devoted to proposing the Ex-LL model as a new form of the LL distribution via the

The Ex-LL Distribution
The cumulative distribution function (CDF) of the two-parameter LL model is given , β, λ > 0, and its probability density function (PDF) reduces to g(x) = β λ x −β−1 1 + λ We define the CDF of the Ex-LL model based on the extended family [17] which is specified (for x ∈ R) by the CDF and the PDF where G(x; ϕ) is any baseline CDF that depends on ϕ ∈ R. The CDF (1) reduces to the baseline CDF for α = 1.
The corresponding PDF of the Ex-LL distribution takes the form The LL model follows as a special case from (4) with α = 1. The EX-LL HRF reduces to Plots of the PDF and HRF of the Ex-LL distribution are depicted respectively in Figures 1 and 2. These plots show that the Ex-LL density exhibits a symmetrical shape, an asymmetrical shape, a J shape, a unimodal shape, and a reversed-J shape. Furthermore, its HRF exhibits an increasing shape, an upside-down bathtub shape, a decreasing-J shape, and a reversed-J shape.

Mode and Quantile Function
On differentiating the logarithm of (4) with respect to x and equating to zero, the unique mode of the Ex-LL distribution follows as For β ≤ 1, the Ex-LL distribution has no mode. The quantile function (QF) of the Ex-LL distribution follows as Let p ∼Uniform(0, 1), then the QF of the Ex-LL distribution can be adopted to generate its random data by the formula

Moments and Moment Generating Function
The rth ordinary moment of the Ex-LL distribution is given by The first four ordinary moments of the Ex-LL distribution can be obtained directly by setting r = 1, 2, 3, and 4 in the last equation.
The mean (µ 1 ), variance (Var(X)), skewness (γ 1 (X)), and kurtosis (γ 2 (X)) of the Ex-LL distribution are obtained numerically for some choices of its parameters using the Wolfram Mathematica program version 12.0. These results are reported in Table 1. It is noted that the Ex-LL skewness varies in the interval (−0.84559, 45.48282), whereas the LL skewness varies only within the interval (0.35216, 1.81999). Furthermore, the kurtosis of the Ex-LL distribution spreads much larger in the interval (4.10572, 2573.493), whereas the kurtosis of the LL distribution spreads only in the interval (4.50847, 14.76564) for same parameter values.  1 , Var(X), γ 1 (X), and γ 2 (X) of the Ex-LL distribution for several choices of α, β, and λ. The rth incomplete moment of the Ex-LL distribution has the form is the hyper geometric function. The moment generating function of the Ex-LL distribution takes the form

Mean Residual Life, Mean Inactivity Time and Inequality Curves
The mean residual life of the Ex-LL distribution at age t has the form where I 1 (t) is the first incomplete moments.
The mean inactivity time of the Ex-LL distribution takes the form The Lorenz, Bonferroni and Zenga curves (see, Lorenz [18] and Bonferroni [19]) are considered the most important inequality curves and their useful applications are common in insurance, reliability, medicine, and economics.
The Lorenz curve is defined for the Ex-LL distribution as follows: where F(x p ) = p, x p is the QF, and I 1 (t) refers to the first incomplete moments. The Bonferroni and Zenga inequality curves can be determined, through their relationship with the Lorenz curve, by the following formulae ( [20])

Some Entropies
The entropy of the random variable X has important applications in many applied fields including statistics for testing hypotheses [21] and engineering, physics, and information theory to describe dynamical systems or nonlinear chaotic [22]. Furthermore, Song [23] developed the log-likelihood-based distribution measure based on the Rényi information. Song's measure is exist and can be defined for all distributions. Song's measure provides meaningful comparisons between distributions as compared with traditional kurtosis measure.
The Rényi P X (k), Tsallis L X (k), and Shannon H X (1) entropies of the Ex-LL distribution can be derived for the Ex-LL distribution by the following formulae.
The Shannon entropy, say H X (1), follows from P X (k) as r −→ 1. Then, H X (1) follows for the Ex-LL distribution as and Υ refers to the Euler Mascheroni constant.

Order Statistics
The PDF of the kth order statistic, X k:n , for the Ex-LL distribution is defined by The CDF of X k:n for the Ex-LL distribution takes the form denotes the hyper geometric function.
The PDF and CDF of the minimum, say (W n ), and the maximum order statistics, say (Z n ), follows by setting k = 1 and k = n, respectively. The limiting distributions (Theorem 2.1.5 [24]) of W n and Z n are given by

Estimation Methods
Here,the Ex-LL parameters are estimated using some estimation approaches.

Maximum Likelihood Estimation
Let x 1 , x 2 , . . . , x n be a random sample from the PDF (4), then the log-likelihood function takes the form The first derivatives with respect to α, β and λ follows by differentiating Equation (8) and equating them to zero. We obtain Solving the previous equations, we obtain the MLEs of the Ex-LL parameters α, β and λ. However, these equations cannot be solved analytically, hence statistical software including Maple, R or Mathematica are adopted to solve them numerically using iterative methods such as Newton-Raphson algorithm.

Least-Squares and Weighted Least-Squares Estimation
Let x 1:n , x 2:n , . . . , x 2:n be the order statistics of the Ex-LL distribution. Hence, the LSEs of the Ex-LL parameters are provided by minimizing: Besides, by solving the following equations we obtain the LSEs of the Ex-LL parameters: The WLSEs of the Ex-LL parameters are calculated by minimizing: Besides, the WLSEs of α, β and λ can be obtained by solving: where ∆ s (x i:n ), s = 1, 2, 3 are given in (9)-(11), respectively.

Anderson-Darling Estimation
The ADEs of the Ex-LL parameters are calculated by minimizing: The ADEs are also obtained by solving: where ∆ s (x i:n ), s = 1, 2, 3 are given in (9)-(11), respectively.

Cramér-von Mises Estimation
The CVMEs of the Ex-LL parameters are calculated by minimizing: These estimators can be also obtained by solving the nonlinear equations: where ∆ s (x i:n ), s = 1, 2, 3 are given in (9)-(11), respectively.

Numerical Simulations for the Estimation Methods
Now, we provided detailed simulation results to explore the performances of the introduced estimation methods in estimating the parameters of the Ex-LL model. We . We generated N = 5000 random samples using Equation (7). The behavior of the different estimates is compared with respect to their: average absolute biases (|BI AS|), Tables 2-9 show the simulation results, average estimates of the parameters (AVEs), |BI AS|, MSEs, and MREs, of the Ex-LL parameters using different approaches. These results showed that estimates are very close to their true values and have small biases, MSEs and MREs. The results illustrated that the biases, MSEs, and MREs decrease as n increases, showing that the introduced estimators are consistent. We can conclude that the introduced estimation methods performed very well in estimating the Ex-LL parameters. Generally, based on our study, the ordering performance of these estimators, in terms of their MSEs, is the MLEs, WLSEs, ADEs, CVMEs, and LSEs. Hence, the maximum likelihood (ML) method is the best estimation approach to estimate the Ex-LL parameters and it will be adopted in the subsequent section to estimate the parameters of the Ex-LL model from real datasets. Table 2. Numerical values of the Ex-LL distribution for the parameters α = 0.25, β = 0.75, λ = 0.5.    Table 3. Numerical values of the Ex-LL distribution for the parameters α = 1.5, β = 0.5, λ = 0.75.  Table 4. Numerical values of the Ex-LL distribution for the parameters α = 2, β = 1.5, λ = 0.25.   Table 6. Numerical values of the Ex-LL distribution for the parameters α = 0.5, β = 2, λ = 3.  Table 7. Numerical values of the Ex-LL distribution for the parameters α = 0.75, β = 0.25, λ = 2.  Table 9. Numerical values of the Ex-LL distribution for the parameters α = 2, β = 0.75, λ = 1.5.

Actuarial Measures
In this section, we discuss the mathematical and computational aspects of five important actuarial measures in portfolio optimization under uncertainty namely VaR, TVaR, TV, TVP, and ES for the Ex-LL distribution.

VaR Measure
The VaR has some other names such as the quantile premium principle or quantile risk measure, and it can be specified with a typical degree of confidence q, where q = 90%, 95% or 99%).
The VaR, say VaR q , of a random variable X is the qth QF of its CDF, i.e., VaR q = Q(q) (Artzner [25]).
The VaR of the Ex-LL model is derived as

TVaR and TV Measures
The TVaR quantifies the expected value of loss given that an event outside a certain probability level has occurred. The TVaR is defined by the formula The TVaR of the Ex-LL distribution has the form is the hyper geometric function.
The TV measure (Landsman [26]) represents the variance of a loss distribution beyond a particular critical value, and it pays attention to the TV beyond VaR. The TV of the Ex-LL distribution is defined as where Using Equation (12)- (14), we obtain the TV of the Ex-LL distribution.

TVP and ES Measures
The TVP plays an important role in insurance field. The TVP of the Ex-LL distribution is defined by where 0 < λ < 1. The TVP of the Ex-LL distribution follows by substituting the expressions (12) and (13) in Equation (15). Artzner [25] introduced another important measure of financial risk called the ES. The ES of the Ex-LL distribution takes the form

Numerical Computations for Actuarial Measures
In this section, we presented numerical simulations for the studied risk measures, VaR, TVaR, TV, TVP, and ES, of the Ex-LL, ELL, and LL distributions. We generated a random sample, n = 100, from the Ex-LL, ELL, and LL distributions, and the parameters were estimated by the ML method. The results were obtained after 1000 repetitions to calculate the five risk measures. The numerical results of these measures were reported in Tables 10 and 11 for the Ex-LL, ELL, and LL distributions. For visual comparisons, we  displayed the results, in Tables 10 and 11, graphically as shown in Figures 3 and 4.
The values and plots in Tables 10 and 11 and Figures 3 and 4 reveal that the introduced Ex-LL model has a heavier tail than the tails of the ELL and LL distributions. Hence, it may be adopted to model heavy-tailed datasets.   Table 10.    Table 11.

Modeling Real Data from the Engineering and Insurance Fields
In this section, we analyzed two real datasets from the engineering and insurance fields to explore the usefulness of the Ex-LL distribution. The first data was studied by [27]. It contains 74 observations and it refers to gauge lengths of 20 mm. This data was analyzed by [28,29].
The second data set refers to losses from private passenger automobile insurance policies in United Kingdom. It consists of 32 observations and 4 variables. We particularly analyzed the variable number 4 which represents number of claims. These data is available on R©software library.
The competing distributions are checked using some goodness-of-fit measures including Anderson-Darling (AD), Cramér-von Mises (CM), and Kolmogorov-Smirnov (KS) with its p-value (KS-p-value).
The parameters of the competing models are estimated via the ML method. The estimates and analytical measures are obtained using the Mathematica program version 12.0. Tables 14 and 15 provide the analytical measures along with the ML estimates and their standard errors (SEs) in parenthesis. The fitted PDF, CDF, SF, and P-P plots of the Ex-LL model for the two data sets are shown in Figures 5 and 6, respectively. The results in Table 14 and 15 indicate that the Ex-LL distribution provides adequate fit than other competing models for the two datasets.

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Conclusions
In this article, we introduce a flexible extension of the log-logistic distribution called the extended log-logistic (Ex-LL) distribution. The EX-LL distribution can be adopted to model heavy-tailed actuarial data. The EX-LL distribution exhibits increasing, reversed-J, decreasing, upside-down bathtub, and J-shapes hazard rate functions. We derive some of its basic mathematical properties. Some risk measures are obtained for the Ex-LL distribution along with their detailed simulation results which illustrate that the tail of the Ex-LL distribution is heavier than the tails of the log-logistic and exponentiated loglogistic distributions. The parameters of the Ex-LL model are estimated using five classical estimation approaches and simulation results are conducted to explore their behavior for different sample sizes. The simulation results show that the maximum likelihood is the best estimation method for the Ex-LL parameters. The practical importance of the Ex-LL distribution is illustrated by two real data sets from the engineering and insurance sciences, showing its superiority fit as compared by nine competing distributions. We hope that the Ex-LL model will attract wider applications in other applied fields such as medicine, economics, reliability, life testing, and survival analyses.