The Inverse-Power Logistic-Exponential Distribution: Properties, Estimation Methods, and Application to Insurance Data

The present paper proposes a new distribution called the inverse power logistic exponential distribution that extends the inverse Weibull, inverse logistic exponential, inverse Rayleigh, and inverse exponential distributions. The proposed model accommodates symmetrical, right-skewed, left-skewed, reversed-J-shaped, and J-shaped densities and increasing, unimodal, decreasing, reversed-J-shaped, and J-shaped hazard rates. We derive some mathematical properties of the proposed model. The model parameters were estimated using five estimation methods including the maximum likelihood, Anderson–Darling, least-squares, Cramér–von Mises, and weighted least-squares estimation methods. The performance of these estimation methods was assessed by a detailed simulation study. Furthermore, the flexibility of the introduced model was studied using an insurance real dataset, showing that the proposed model can be used to fit the insurance data as compared with twelve competing models.


Introduction
Reliability and survival analysis has several applications, as an important branch of statistics, in different applied fields, such as actuarial science, engineering, demography, biomedical studies, and industrial reliability. Several lifetime distributions have been proposed in the statistical literature to model data in many applied sciences.
One of the important extensions of the exponential distribution is called the logistic exponential distribution, which was proposed by Lan and Leemis [21]. In this paper, we propose a flexible distribution called the inverse power logistic exponential (IPLE) distribution. The IPLE can provide more flexibility and accuracy in fitting actuarial data. The IPLE distribution also generalizes the inverse Weibull, inverse Rayleigh, inverse logistic exponential, and inverse exponential distributions. The proposed model was generated based on the inverse power transformation.
Let X and T be random variables. The inverse transformation, denoted by X = T −1 , or inverse power transformation, denoted by X = T −1 β have been used in generating inverted distributions. For example, the inverse two parameter Lindley distribution by Alkarni [22], the generalized inverse gamma distribution by Mead [23], the reverse Lindley distribution by Sharma et al. [24], and the inverse Lindley distribution using by Barco et al. [25].
The proposed IPLE model is motivated by some properties as follows.
• The IPLE model includes the inverse Weibull, inverse logistic exponential, inverse Rayleigh, and inverse exponential distributions as special sub-models.

•
The probability density function (PDF), as well as the cumulative distribution function (CDF) of the IPLE model have simple closed forms, and hence, it can be adopted in analyzing censored data. • The IPLE model has been used to model a heavy-tailed insurance dataset from actuarial science, and it provides adequate fits compared to other competing distributions.
The main aim of this paper is to study a new extension of the logistic exponential model based on the inverse power transformation and derive some of its distributional properties. We are also interested in exploring the estimation of the IPLE parameters by five classical estimation methods including the maximum likelihood estimators (MLEs), Anderson-Darling estimators (ADEs), least-squares estimators (LSEs), Cramér-von Mises estimators (CVMEs), and weighted least-squares estimators (WLSEs). These estimation methods were compared using an extensive simulation study to assess their performances and to provide a guideline for choosing the best estimation method that gives better estimates for the IPLE parameters. This would be of deep interest to applied statisticians, actuaries, or engineers.
The paper is organized as follows. We define the IPLE distribution and its special sub-models in Section 2. Its mathematical properties are derived in Section 3. Five methods of estimation are discussed in Section 4. The performance of these estimation methods is explored using a simulation study in Section 5. A real dataset with a heavy tail from insurance science is analyzed to show the usefulness and importance of the IPLE distribution in Section 6. We present some conclusions in Section 7.

The IPLE Distribution
Based on the inverse power transformation and the logistic exponential (LE) (Lan and Leemis [21]) distribution, we generate the IPLE distribution. The CDF and PDF of the LE distributions are given by: and: where α and λ are respectively the shape and scale parameters. For α = 1, the exponential distribution follows as a special sub-model from the LE model.
Consider the inverse power transformation, X = T −1 β , where T ∼LE(α, λ), then the resulting IPLE distribution of X can be specified by the CDF: The PDF of the IPLE distribution reduces to: where β and α are the shape parameters and λ is a scale parameter. By setting β = 1, we obtain the inverse logistic exponential distribution. The survival function (SF) and hazard rate function (HRF) of the IPLE distribution are, respectively, given by: Some possible plots of the PDF and HRF of the IPLE distribution are depicted in Figures 1 and 2, to show the flexibility of both functions.

Quantile Function
The quantile function (QF) of the IPLE distribution is derived by the inverse function of the IPLE CDF (1). The QF of the IPLE distribution takes the form: where p follow a uniform distribution (0, 1). Setting p = 0.25, 0.5, and 0.75, in (5), one can obtain the first, second, and third quartiles of the IPLE distribution.
Using the QF, we can determine the Bowley skewness measure and Moors kurtosis measure, respectively, as follows: The skewness and kurtosis of the IPLE model can be displayed graphically for some values of α and λ in Figure 3.

Moments
The rth moments of the IPLE distribution takes the following form: The first four moments of the IPLE distribution follow respectively by setting r = 1, 2, 3, and 4. The moment generating function of the IPLE distribution takes the form: The characteristic function of the IPLE distribution can be obtained by replacing t with it in the last equation.

Inequality Curves
The most important inequality curves are called Lorenz and Bonferroni, which have some applications in applied sciences such as economics, reliability, demography, and medicine.
The Lorenz and Bonferroni curves for the IPLE distribution take the forms: respectively, where A = 1 + (e λx −β p − 1) α and x p is the QF of the IPLE distribution.

Moments of Residual Life
The mth moment of residual life is defined by M m = 1 For the IPLE distribution, mth takes the form: For m = 1, we have the mean residual life, M 1 , of the IPLE distribution, and by setting t = 0, the mean of the IPLE distribution follows from M 1 .

Methods of Estimation
In this section, we explore the estimation of the IPLE parameters by different methods of estimation including the maximum likelihood estimators (MLEs), Anderson-Darling estimators (ADEs), least-squares estimators (LSEs), Cramér-von Mises estimators (CVMEs), and weighted least-squares estimators (WLSEs).

Maximum Likelihood Estimators
Let x 1 , x 2 , . . . , x n be a random sample of size n from the IPLE with PDF (2); hence, the log-likelihood function is specified by: By differentiating Equation (6) with respect to α, β, and λ, we can write: and: We note that there were no explicit solutions for the above three equations, and hence, we require employing the nonlinear numerical techniques to obtain the MLEs of the IPLE parameters.

Anderson-Darling Estimation
Consider the order statistics of a random sample of size n from the IPLE distribution denoted by x 1:n , x 2:n , . . . , x 2:n . The ADEs of the IPLE parameters, α, β, and λ, are obtained by minimizing the following equation: with respect to α, β, and λ. Using Equations (1) and (3), the above equation reduces to: The ADEs can also be calculated by solving the following nonlinear equations: where:

Simulation Study
The performance of the five estimation methods in estimating the IPLE parameters based on simulation results is explored in this section.
The MSEs, bias, and MREs are calculated by equations: Tables 1-5 report the simulation results including the AVEs, bias, MSEs, and MREs of the IPLE parameters using the five estimation approaches. It is noted that the estimates of the IPLE parameters obtained using the five estimation methods are quite reliable and very close to the true values, showing small MSEs, biases, and MREs in all studied cases. The five estimators are consistent, where the MSEs, biases, and MREs decrease as the sample size increases, for all studied cases. We can conclude that the MLE, ADE, CVME, WLSE, and LSE methods perform very well in estimating the IPLE parameters. In summary, the MLEs provide the best estimates for the parameters of the IPLE distribution; hence, the MLEs are adopted in the application section to estimate the IPLE parameters and the parameters of the compared models.  Table 2. Simulation values of the AVEs, biases, MSEs, and MREs for (α = 2, β = 1.5, λ = 1).

Applications
This section is devoted to analyzing a real dataset from the insurance field. The dataset represents losses from a private passenger in the United Kingdom (U.K.) automobile insurance policies. It consists of four variables, and we studied the variable number three in particular. It is available in the R c software library. The aim of this section is to show that the proposed model can be provide the best fit to insurance data compared to other competitive extensions of the exponential distribution.
The simulation results show that the maximum likelihood method provides accurate estimates for the IPLE parameters. Hence, the maximum likelihood is adopted here to estimate the parameters of the IPLE model and other competing models. The MLEs and the goodness-of-fit measures are calculated using the Wolfram Mathematica software Version 10. Tables 6 and 7 report the analytical measures, MLEs, and their standard errors. The results in Tables 6 and 7 illustrate that the IPLE distribution provides the best fit to insurance data compared to other competing distributions, and hence, it can be an adequate distribution to analyze heavy-tailed insurance data. The results in these tables were obtained based on the real insurance data using the Wolfram Mathematica software.

Conclusions
This paper proposes a new three parameter distribution, called the inverse power logistic exponential (IPLE) distribution, which can be used to model heavy-tailed data in insurance and other applied areas. The IPLE model generalizes the inverse Weibull, inverse logistic exponential, inverse Rayleigh, and inverse exponential distributions. The hazard rate function of the IPLE distribution can be decreasing, unimodal, increasing, and reversed-J-shaped, and J-shaped. Some of its mathematical properties were derived. The unknown parameters of the IPLE model were estimated by five classical estimators, called the maximum likelihood estimators, Anderson-Darling estimators, least-squares estimators, Cramér-von Mises estimators, and weighted least-squares estimators. The simulation results showed that all estimators perform very well in estimating the parameters of the IPLE distribution. Based on our study, the maximum likelihood method provided accurate estimates for the parameters of the IPLE distribution. The practical importance of the IPLE distribution was illustrated using real insurance data, showing its adequate fits and superiority as compared with other competing existing models. The proposed model may be used effectively in modeling data in several applied areas such as medicine, economics, reliability, life testing, and engineering, among others.
The work of this paper can be extended in some ways. For example, the IPLE parameters can be estimated under different censoring schemes using classical and Bayesian estimation. Exponentiated or transmuted versions of the IPLE model can be established, and a bivariate extension of it may also be studied. The application of the IPLE model may also be explored in other applied areas.