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

Abstract

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.

1. 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.

The exponential distribution is used in modeling real-life data due to its lack of memory property, and it is also analytically tractable. On the other hand, its applicability was limited because it has only a constant hazard rate and decreasing density function. Hence, many researcher have been interested in proposing modified forms of the exponential distribution to increase its flexibility. Some recent extensions of the exponential distribution include the exponentiated exponential [1], beta exponential [2], beta generalized exponential [3], transmuted generalized exponential [4], Harris extended exponential [5], Kumaraswamy transmuted exponential [6], Marshall–Olkin Nadarajah–Haghighi [7], modified exponential [8], alpha power exponential [9,10], odd exponentiated half-logistic exponential [11], Marshall–Olkin logistic exponential [12], generalized odd log-logistic exponential [13], Marshall–Olkin alpha power exponential [14], extended odd Weibull exponential [15], odd inverse Pareto exponential [16], modified Kies exponential [17], Topp–Leone moment exponential [18], heavy-tailed exponential [19], and odd log-logistic Lindley exponential distributions [20].

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 $X=T^{\frac{-1}{\beta }}$ 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 IPLE distribution can provide symmetrical, right-skewed, left-skewed, reversed-J-shaped, and J-shaped densities and increasing, unimodal, decreasing, reversed-J-shaped, and J-shaped hazard rates.
• 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.

Parameter estimation using several classical methods of estimation were studied by several statisticians. For example, the alpha logarithmic transformed Weibull distribution [26], Weibull–Marshall–Olkin–Lindley distribution [27], quasi xgamma-geometric distribution [28], logarithmic transformed Weibull distribution [29], generalized Ramos–Louzada distribution [30], and alpha power exponential distribution [10], among many others.

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.

2. 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:

$$G\left(t\right)=1-\frac{1}{{\left(e^{\lambda t}-1\right)}^{\alpha }+1},t>0,\alpha ,\lambda >0,$$

and:

$$g\left(t\right)=\frac{\alpha \lambda e^{\lambda t}{\left(e^{\lambda t}-1\right)}^{\alpha -1}}{{\left[{\left(e^{\lambda t}-1\right)}^{\alpha }+1\right]}^2},t>0,\alpha ,\lambda >0,$$

where $\alpha$ and $\lambda$ are respectively the shape and scale parameters. For $\alpha =1$, the exponential distribution follows as a special sub-model from the LE model.

Consider the inverse power transformation, $X=T^{\frac{-1}{\beta }}$, where $T\sim$ LE $(\alpha ,\lambda )$, then the resulting IPLE distribution of X can be specified by the CDF:

$$F\left(x\right)=\frac{1}{{\left(e^{\lambda x^{-\beta }}-1\right)}^{\alpha }+1},x>0,\alpha ,\beta ,\lambda >0.$$

(1)

The PDF of the IPLE distribution reduces to:

$$f\left(x\right)=\frac{\alpha \beta \lambda x^{-\beta -1}{e}^{\lambda x^{-\beta }}{\left(e^{\lambda x^{-\beta }}-1\right)}^{\alpha -1}}{{\left[{\left(e^{\lambda x^{-\beta }}-1\right)}^{\alpha }+1\right]}^2},x>0,\alpha ,\beta ,\lambda >0,$$

(2)

where $\beta$ and $\alpha$ are the shape parameters and $\lambda$ is a scale parameter. By setting $\beta =1$, we obtain the inverse logistic exponential distribution.

The survival function (SF) and hazard rate function (HRF) of the IPLE distribution are, respectively, given by:

$$S\left(x\right)=1-F\left(x\right)=1-\frac{1}{{\left(e^{\lambda x^{-\beta }}-1\right)}^{\alpha }+1},$$

(3)

$$h\left(x\right)=\frac{f\left(x\right)}{1-F\left(x\right)}=\frac{\alpha \beta \lambda x^{-\beta -1}{e}^{\lambda x^{-\beta }}}{\left(e^{\lambda x^{-\beta }}-1\right)\left[{\left(e^{\lambda x^{-\beta }}-1\right)}^{\alpha }+1\right]}.$$

(4)

Some possible plots of the PDF and HRF of the IPLE distribution are depicted in Figure 1 and Figure 2, to show the flexibility of both functions.

3. Mathematical Properties

3.1. 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:

$$Q\left(p\right)={\lambda }^{1/\beta }{\left\{log\left[{\left(\frac{1}{p}-1\right)}^{1/\alpha }+1\right]\right\}}^{-1/\beta },0<p<1,$$

(5)

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:

$$\begin{array}{ccc}\hfill SK& =& \frac{Q(3/4)+Q(1/4)-2Q(1/2)}{Q(3/4)-Q(1/4)},\hfill \\ \hfill KU& =& \frac{Q(7/8)+Q(3/8)-Q(5/8)-Q(1/8)}{Q(6/8)-Q(2/8)}.\hfill \end{array}$$

The skewness and kurtosis of the IPLE model can be displayed graphically for some values of $\alpha$ and $\lambda$ in Figure 3.

3.2. Moments

The $r\mathrm{th}$ moments of the IPLE distribution takes the following form:

$${\mu }_r^{\prime }=E\left(X^r\right)={\int }_0^{\infty }{x}^{r}f\left(x\right)dx={\lambda }^{\frac{r}{\beta }}{\int }_1^{\infty }\frac{{\left\{log\left[1+{\left(-1+y\right)}^{\frac{1}{\alpha }}\right]\right\}}^{\frac{-r}{\beta }}}{y^2}dy,$$

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:

$$M\left(t\right)=\sum _{k=0}^{\infty }\frac{{(t{\lambda }^{\frac{1}{\beta }})}^k}{k!}{\int }_1^{\infty }\frac{{\left\{log\left[1+{\left(-1+y\right)}^{\frac{1}{\alpha }}\right]\right\}}^{\frac{-k}{\beta }}}{y^2}dy.$$

The characteristic function of the IPLE distribution can be obtained by replacing t with $it$ in the last equation.

3.3. 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:

$$\begin{array}{ccc}\hfill L\left(p\right)& =& \frac{1}{\mu }{\int }_0^{x_p}xf\left(x\right)dx=\frac{{\lambda }^{\frac{1}{\beta }}}{\mu }{\int }_A^{\infty }\frac{{\left\{log\left[1+{\left(-1+y\right)}^{\frac{1}{\alpha }}\right]\right\}}^{\frac{-1}{\beta }}}{y^2}dy,\hfill \\ \hfill B\left(p\right)& =& {\displaystyle \frac{L\left(p\right)}{p}},\hfill \end{array}$$

respectively, where $A=1+{(e^{\lambda x_p^{-\beta }}-1)}^{\alpha }$ and $x_p$ is the QF of the IPLE distribution.

3.4. Moments of Residual Life

The $m\mathrm{th}$ moment of residual life is defined by $M_m=\frac{1}{S\left(t\right)}{\int }_t^{\infty }{(x-t)}^{m}f\left(x\right)dx$

For the IPLE distribution, $m\mathrm{th}$ takes the form:

$$\begin{array}{c}\hfill M_m=\frac{1}{S\left(t\right)(\alpha -1)}\sum _{k=0}^m{(-1)}^{m+k}\left(\genfrac{}{}{0pt}{}{m}{k}\right)t^{m-k}{\lambda }^{\frac{k}{\beta }}{\int }_1^{1+w}\frac{{\left\{log\left[1+{\left(-1+y\right)}^{\frac{1}{\alpha }}\right]\right\}}^{\frac{-k}{\beta }}}{y^2}dy,\end{array}$$

$w={\left(e^{\lambda t^{-\beta }}-1\right)}^{\alpha }$.

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 $M_1$.

4. 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).

4.1. Maximum Likelihood Estimators

Let $x_1,x_2,\dots ,x_n$ be a random sample of size n from the IPLE with PDF (2); hence, the log-likelihood function is specified by:

$$\begin{array}{cc}\hfill \mathcal{l}=& nlog\left(\alpha \right)+nlog\left(\beta \right)+nlog\left(\lambda \right)-(\beta +1)\sum _{i=1}^{n}log\left(x_i\right)+\lambda \sum _{i=1}^n{x}_i^{-\beta }+(\alpha -1)\sum _{i=1}^{n}log\left(e^{\lambda x_i^{-\beta }}-1\right)\hfill \\ & -2\sum _{i=1}^{n}log\left[{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha }+1\right].\hfill \end{array}$$

(6)

By differentiating Equation (6) with respect to $\alpha$, $\beta$, and $\lambda$, we can write:

$$\frac{\partial \mathcal{l}}{\partial \alpha }=\frac{n}{\alpha }+\sum _{i=1}^{n}log\left(e^{\lambda x_i^{-\beta }}-1\right)-2\sum _{i=1}^n\frac{{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha }log\left(e^{\lambda x_i^{-\beta }}-1\right)}{{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha }+1}=0,$$

$$\begin{array}{cc}\hfill \frac{\partial \mathcal{l}}{\partial \beta }=& \frac{n}{\beta }-\sum _{i=1}^{n}log\left(x_i\right)-\lambda \sum _{i=1}^n{x}_i^{-\beta }log\left(x_i\right)-(\alpha -1)\sum _{i=1}^n\frac{\lambda x_i^{-\beta }{e}^{\lambda x_i^{-\beta }}log\left(x_i\right)}{e^{\lambda x_i^{-\beta }}-1}\hfill \\ & +2\sum _{i=1}^n\frac{\alpha \lambda x_i^{-\beta }{e}^{\lambda x_i^{-\beta }}{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha -1}log\left(x_i\right)}{{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha }+1}=0\hfill \end{array}$$

and:

$$\frac{\partial \mathcal{l}}{\partial \lambda }=\frac{n}{\lambda }+\sum _{i=1}^n{x}_i^{-\beta }+(\alpha -1)\sum _{i=1}^n\frac{x_i^{-\beta }{e}^{\lambda x_i^{-\beta }}}{e^{\lambda x_i^{-\beta }}-1}-2\sum _{i=1}^n\frac{\alpha x_i^{-\beta }{e}^{\lambda x_i^{-\beta }}{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha -1}}{{\left(e^{\lambda x_i^{-\beta }}-1\right)}^{\alpha }+1}=0.$$

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.

4.2. Anderson–Darling Estimation

Consider the order statistics of a random sample of size n from the IPLE distribution denoted by $x_{1:n},x_{2:n},\dots ,x_{2:n}$. The ADEs of the IPLE parameters, $\alpha$, $\beta$, and $\lambda$, are obtained by minimizing the following equation:

$$AD=-n-\frac{1}{n}\sum _{i=1}^n(2i-1)[logF\left(x_{i:n}\right)+logS\left(x_{n+1-i:n}\right)],$$

with respect to $\alpha$, $\beta$, and $\lambda$. Using Equations (1) and (3), the above equation reduces to:

$$AD=-n-\frac{1}{n}\sum _{i=1}^n(2i-1)\left(log\left\{1-{\left[{\left(e^{\lambda x_{n+1-i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}\right\}-log\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]\right).$$

The ADEs can also be calculated by solving the following nonlinear equations:

$$\sum _{i=1}^n(2i-1)\left[\frac{{\Delta }_{\kappa }\left(x_{i:n}\right)}{F\left(x_{i:n}\right)}-\frac{{\Delta }_{\kappa }\left(x_{n+1-i:n}\right)}{S\left(x_{n+1-i:n}\right)}\right]=0,\kappa =\alpha ,\beta ,\lambda .$$

where:

$$\begin{array}{ccc}\hfill {\Delta }_{\alpha }\left(x_{i:n}\right)& =& \frac{\partial F\left(x_{i:n}\right)}{\partial \alpha }=\frac{-{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }log\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^2},\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\Delta }_{\beta }\left(x_{i:n}\right)& =& \frac{\partial F\left(x_{i:n}\right)}{\partial \beta }=\frac{\alpha \lambda x_{i:n}^{-\beta }{e}^{\lambda x_{i:n}^{-\beta }}{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha -1}}{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^2},\hfill \end{array}$$

(8)

$$\begin{array}{ccc}\hfill {\Delta }_{\lambda }\left(x_{i:n}\right)& =& \frac{\partial F\left(x_{i:n}\right)}{\partial \lambda }=\frac{-\alpha x_{i:n}^{-\beta }{e}^{\lambda x_{i:n}^{-\beta }}{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha -1}}{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^2}.\hfill \end{array}$$

(9)

4.3. Cramér–von Mises Estimators

The CVMEs of IPLE parameters, $\alpha$, $\beta$, and $\lambda$, are obtained by minimizing the following equation:

$$\begin{array}{cc}\hfill CV=& \frac{1}{12n}+\sum _{i=1}^n{\left[F\left(x_{i:n}\right)-\frac{2i-1}{2n}\right]}^2\hfill \\ \hfill =& \frac{1}{12n}+\sum _{i=1}^n{\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{2i-1}{2n}\right\}}^2,\hfill \end{array}$$

or by solving the following nonlinear equations with respect to $\alpha$, $\beta$, and $\lambda$:

$$\sum _{i=1}^n\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{2i-1}{2n}\right\}{\Delta }_{\kappa }\left(x_{i:n}\right)=0,$$

where ${\Delta }_{\kappa }\left(x_{i:n}\right),\kappa =\alpha ,\beta ,\lambda$ were defined in (7)–(9), respectively.

4.4. Least-Squares and Weighted Least-Squares Estimators

The LSEs of the IPLE parameters, $\alpha$, $\beta$, and $\lambda$, by minimizing the following equation:

$$\begin{array}{cc}\hfill LS=& \sum _{i=1}^n{\left[F\left(x_{i:n}\right)-\frac{i}{n+1}\right]}^2\hfill \\ \hfill =& \sum _{i=1}^n{\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{i}{n+1}\right\}}^2.\hfill \end{array}$$

Furthermore, the LSEs of $\alpha$, $\beta$, and $\lambda$ follow also by solving the following nonlinear equations:

$$\sum _{i=1}^n\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{i}{n+1}\right\}{\Delta }_{\kappa }\left(x_{i:n}\right)=0,\kappa =\alpha ,\beta ,\lambda ,$$

where ${\Delta }_{\kappa }\left(x_{i:n}\right)$ are defined in (7)–(9), respectively.

The WLSEs of the parameters $\alpha$, $\beta$, and $\lambda$ can be determined by minimizing the following equation:

$$\begin{array}{cc}\hfill W=& C{\left[F\left(x_{i:n}\right)-\frac{i}{n+1}\right]}^2\hfill \\ \hfill =& C{\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{i}{n+1}\right\}}^2,\hfill \end{array}$$

where $C={\sum }_{i=1}^n{(n+1)}^2(n+2)/i(n-i+1)$.

Furthermore, the WLSEs of the parameters $\alpha$, $\beta$, and $\lambda$ are obtained by solving the following nonlinear equations:

$$C\left\{{\left[{\left(e^{\lambda x_{i:n}^{-\beta }}-1\right)}^{\alpha }+1\right]}^{-1}-\frac{i}{n+1}\right\}{\Delta }_{\kappa }\left(x_{i:n}\right)=0,$$

where ${\Delta }_{\kappa }\left(x_{i:n}\right),\kappa =\alpha ,\beta ,\lambda$ are given in Equations (7)–(9), respectively.

5. Simulation Study

The performance of the five estimation methods in estimating the IPLE parameters based on simulation results is explored in this section. We considered various sample sizes, $n=\{20,50,100,200,500\}$, and different parametric values of $\alpha ,\beta$, and $\lambda$, $\alpha =\{0.25,0.50,0.75,1.0,2.0\}$, $\beta =\{0.50,0.75,1.0,1.5,2.0\}$, and $\lambda =\{0.5,0.75,1.0,1.5,3.0\}$. We generate $n=1000$ random samples from the IPLE distribution using its QF given in Equation (5). We calculate the average values of the estimates (AVEs) along with their associated average mean squared error (MSEs), average absolute biases, and average mean relative estimates (MREs) for the studied sample sizes and different parameter combinations, using the R software, to assess the performance of the proposed five estimation methods.

The MSEs, bias, and MREs are calculated by equations:

$$MSEs=\frac{1}{N}\sum _{i=1}^N{(\widehat{\mathit{\varphi }}-\mathit{\varphi })}^2,Bias=\frac{1}{N}\sum _{i=1}^N|\widehat{\mathit{\varphi }}-\mathit{\varphi }|,MREs=\frac{1}{N}\sum _{i=1}^N|\widehat{\mathit{\varphi }}-\mathit{\varphi }|/\mathit{\varphi },$$

where $\mathit{\varphi }={(\alpha ,\beta ,\lambda )}^{\prime }$.

Table 1. Simulation values of the averages (AVEs), biases, MSEs, and mean relative estimates (MREs) for $(\alpha =0.25,\beta =0.75,\lambda =0.5)$. ADEs, Anderson–Darling estimators; CVMEs, Cramér–von Mises estimators; WLSEs, weighted least-squares estimators.

Method	n	AVEs			Bias			MSEs			MREs
		$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$
MLEs	20	0.29959	0.88811	0.50733	0.13269	0.26169	0.30401	1.80809	0.12442	0.15792	0.53074	0.34892	0.60802
	50	0.25991	0.79873	0.50798	0.05854	0.13806	0.19014	0.00783	0.03766	0.06174	0.23414	0.18409	0.38027
	100	0.25895	0.77429	0.50633	0.04156	0.08943	0.12938	0.00609	0.01801	0.03136	0.16625	0.11924	0.25877
	200	0.25873	0.75914	0.51297	0.03051	0.05831	0.08815	0.00459	0.01000	0.01983	0.12204	0.07775	0.17631
	500	0.25725	0.75171	0.50998	0.02027	0.03069	0.05142	0.00376	0.00413	0.01196	0.08110	0.04093	0.10285
ADEs	20	1.26579	0.79586	0.57588	1.08235	0.28551	0.34821	116.65009	0.13636	0.19635	4.32941	0.38068	0.69643
	50	0.27595	0.76685	0.54284	0.07209	0.17236	0.24400	0.01854	0.04866	0.09423	0.28836	0.22981	0.48799
	100	0.26022	0.76086	0.51886	0.04580	0.11980	0.17337	0.00380	0.02315	0.04820	0.18320	0.15973	0.34674
	200	0.25536	0.75225	0.51502	0.03015	0.08217	0.12481	0.00159	0.01091	0.02536	0.12058	0.10956	0.24962
	500	0.25218	0.74980	0.50695	0.01832	0.04998	0.07614	0.00057	0.00415	0.00959	0.07329	0.06664	0.15229
CVMEs	20	1.78919	0.85649	0.55473	1.61846	0.37561	0.35747	151.11781	0.25508	0.21699	6.47385	0.50082	0.71494
	50	0.30719	0.79458	0.53295	0.11283	0.22474	0.26263	1.77022	0.08639	0.10349	0.45134	0.29965	0.52526
	100	0.26469	0.76813	0.52365	0.05615	0.15042	0.19391	0.00587	0.03701	0.05812	0.22461	0.20056	0.38783
	200	0.25608	0.76069	0.51137	0.03669	0.10336	0.13915	0.00230	0.01761	0.03072	0.14677	0.13782	0.27829
	500	0.25250	0.75338	0.50575	0.02278	0.06464	0.08860	0.00084	0.00671	0.01238	0.09114	0.08618	0.17720
LSEs	20	1.35563	0.81818	0.57272	1.18807	0.35716	0.36193	122.09909	0.22213	0.21403	4.75226	0.47621	0.72386
	50	0.29041	0.77859	0.54445	0.09799	0.21584	0.26076	0.58039	0.07977	0.10480	0.39198	0.28779	0.52152
	100	0.26299	0.76297	0.52593	0.05524	0.14971	0.19431	0.00564	0.03658	0.05866	0.22095	0.19962	0.38863
	200	0.25413	0.75937	0.51083	0.03643	0.10359	0.13843	0.00221	0.01712	0.02968	0.14570	0.13812	0.27686
	500	0.25195	0.75325	0.50489	0.02293	0.06517	0.08886	0.00084	0.00669	0.01242	0.09172	0.08689	0.17771
WLSEs	20	1.01014	0.77884	0.58895	0.83437	0.30811	0.34971	71.84218	0.16052	0.19307	3.33749	0.41082	0.69943
	50	0.26952	0.77672	0.54040	0.07246	0.18758	0.25074	0.01117	0.05932	0.09816	0.28983	0.25010	0.50149
	100	0.25804	0.76201	0.52441	0.04725	0.12609	0.18030	0.00383	0.02634	0.05161	0.18898	0.16812	0.36061
	200	0.25450	0.75544	0.51125	0.03160	0.08658	0.12822	0.00164	0.01218	0.02601	0.12641	0.11544	0.25644
	500	0.25151	0.75334	0.50269	0.01945	0.05377	0.08022	0.00060	0.00460	0.01027	0.07779	0.07169	0.16044

,

Table 2. Simulation values of the AVEs, biases, MSEs, and MREs for $(\alpha =2,\beta =1.5,\lambda =1)$.

Method	n	AVEs			Bias			MSEs			MREs
		$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$
MLEs	20	1.98692	1.65613	1.05014	0.40172	0.32998	0.13721	0.18342	0.13625	0.03212	0.20086	0.21999	0.13721
	50	1.96919	1.62393	1.03293	0.39783	0.30905	0.09710	0.18012	0.12099	0.01568	0.19892	0.20604	0.09710
	100	1.96333	1.60204	1.02699	0.38290	0.28849	0.07974	0.16996	0.10569	0.01019	0.19145	0.19233	0.07974
	200	1.98399	1.57231	1.01797	0.34806	0.25748	0.06396	0.14809	0.08706	0.00634	0.17403	0.17165	0.06396
	500	2.00053	1.54050	1.00999	0.28207	0.20466	0.04962	0.10730	0.05913	0.00369	0.14103	0.13644	0.04962
ADEs	20	2.05895	1.53835	1.02221	0.41636	0.30581	0.12429	0.19326	0.12045	0.02598	0.20818	0.20388	0.12429
	50	2.03424	1.55152	1.01309	0.41798	0.29525	0.08847	0.19257	0.11049	0.01295	0.20899	0.19683	0.08847
	100	2.03559	1.54074	1.01178	0.39759	0.28508	0.07501	0.18036	0.10188	0.00887	0.19879	0.19005	0.07501
	100	2.02842	1.53773	1.01109	0.37028	0.26689	0.06550	0.16258	0.09023	0.00638	0.18514	0.17792	0.06550
	200	2.03281	1.52109	1.00526	0.31318	0.22445	0.05283	0.12747	0.06767	0.00406	0.15659	0.14963	0.05283
CVMEs	20	1.87826	1.72135	1.07690	0.38363	0.33235	0.07690	0.17189	0.13969	0.02028	0.19181	0.22156	0.07690
	50	1.85124	1.70679	1.05444	0.35344	0.30482	0.05444	0.15031	0.12129	0.01005	0.17672	0.20321	0.05444
	100	1.82886	1.69365	1.04599	0.32938	0.27898	0.04599	0.13554	0.10594	0.00672	0.16469	0.18599	0.04599
	200	1.83366	1.67706	1.04134	0.28956	0.24702	0.04134	0.11196	0.08856	0.00509	0.14478	0.16468	0.04134
	500	1.84133	1.65510	1.03588	0.23572	0.20102	0.03588	0.08330	0.06709	0.00362	0.11786	0.13402	0.03588
LSEs	10	2.00523	1.46603	1.00127	0.09759	0.24010	0.11181	0.00968	0.08205	0.02051	0.04880	0.16006	0.11181
	50	2.00391	1.49271	1.00076	0.09740	0.16174	0.07225	0.00965	0.04083	0.00846	0.04870	0.10783	0.07225
	100	2.00088	1.49953	1.00131	0.09702	0.12334	0.05038	0.00960	0.02360	0.00405	0.04851	0.08222	0.05038
	200	1.99873	1.50270	1.00131	0.09654	0.09461	0.03641	0.00954	0.01399	0.00212	0.04827	0.06307	0.03641
	500	2.00174	1.50034	0.99943	0.09424	0.07327	0.02476	0.00923	0.00804	0.00097	0.04712	0.04885	0.02476
WLSEs	20	2.06749	1.49826	1.00850	0.41887	0.30856	0.12001	0.19513	0.12231	0.02408	0.20943	0.20571	0.12001
	50	2.04871	1.52778	1.01135	0.41703	0.29708	0.08941	0.19303	0.11136	0.01299	0.20851	0.19805	0.08941
	100	2.04046	1.53485	1.01107	0.40579	0.28887	0.07680	0.18504	0.10366	0.00938	0.20289	0.19258	0.07680
	200	2.03033	1.53654	1.01142	0.37599	0.27231	0.06618	0.16631	0.09317	0.00650	0.18800	0.18154	0.06618
	500	2.02224	1.52710	1.00653	0.31262	0.22640	0.05315	0.12676	0.06940	0.00410	0.15631	0.15094	0.05315

,

Table 3. Simulation values of the AVEs, biases, MSEs, and MREs for $(\alpha =0.75,\beta =1,\lambda =1.5)$.

Method	n	AVEs			Bias			MSEs			MREs
		$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$
MLEs	20	5.00407	1.33856	1.69462	4.58584	0.58300	0.61194	915.57192	0.82702	1.79847	6.11446	0.58300	0.40796
	50	0.80009	1.10950	1.56565	0.25042	0.26100	0.28367	2.77923	0.12885	0.17539	0.33390	0.26100	0.18911
	100	0.75820	1.04701	1.52647	0.14670	0.16658	0.18688	0.03663	0.04629	0.05870	0.19561	0.16658	0.12459
	200	0.75105	1.02483	1.51570	0.09973	0.11192	0.12771	0.01602	0.02034	0.02670	0.13298	0.11192	0.08514
	500	0.75042	1.01062	1.50764	0.06399	0.07115	0.07960	0.00655	0.00813	0.01008	0.08532	0.07115	0.05307
ADEs	20	9.38732	1.06719	1.57006	8.88929	0.50246	0.56440	1880.45673	0.47327	1.06650	11.85239	0.50246	0.37627
	50	1.47665	1.00523	1.49807	0.89390	0.27826	0.29899	61.89330	0.12985	0.15880	1.19187	0.27826	0.19933
	100	0.82120	1.00558	1.49631	0.20048	0.18789	0.19384	0.71525	0.05817	0.06173	0.26731	0.18789	0.12922
	200	0.77776	1.00022	1.49962	0.12315	0.13102	0.13796	0.02781	0.02756	0.03125	0.16420	0.13102	0.09198
	500	0.76053	0.99923	1.49858	0.07278	0.08147	0.08400	0.00888	0.01041	0.01126	0.09704	0.08147	0.05600
CVMEs	20	17.17150	1.20607	1.94422	16.72239	0.71332	0.99773	3834.81526	1.07678	20.16972	22.29651	0.71332	0.66515
	50	3.15400	1.03508	1.53326	2.60571	0.38905	0.39081	273.18805	0.26011	0.31904	3.47427	0.38905	0.26054
	100	0.98389	1.01821	1.50993	0.39120	0.26162	0.25295	7.71210	0.11351	0.11368	0.52160	0.26162	0.16863
	200	0.79328	1.00765	1.50508	0.15907	0.17208	0.16963	0.05690	0.04793	0.04686	0.21209	0.17208	0.11309
	500	0.76726	1.00035	1.50054	0.09351	0.10738	0.10484	0.01522	0.01817	0.01748	0.12467	0.10738	0.06989
LSEs	20	15.51388	1.11072	1.75752	15.06375	0.66087	0.84020	2945.08220	0.87576	5.92644	20.08500	0.66087	0.56013
	50	2.90779	1.00897	1.50329	2.36165	0.37711	0.37609	220.69516	0.24414	0.31469	3.14887	0.37711	0.25073
	100	0.95680	0.99714	1.49358	0.35849	0.25029	0.24414	5.02891	0.10431	0.10430	0.47799	0.25029	0.16276
	200	0.78921	1.00561	1.50073	0.15660	0.17016	0.16939	0.05349	0.04723	0.04736	0.20881	0.17016	0.11293
	500	0.76231	1.00428	1.50377	0.09246	0.10792	0.10395	0.01419	0.01836	0.01722	0.12328	0.10792	0.06930
WLSEs	20	10.83923	1.05075	1.61118	10.35417	0.56427	0.66276	1971.92149	0.63157	4.08047	13.80556	0.56427	0.44184
	50	1.40103	1.01544	1.50136	0.83711	0.31227	0.32132	67.46259	0.16992	0.20701	1.11615	0.31227	0.21421
	100	0.81149	1.00615	1.50063	0.19688	0.20227	0.20938	0.09490	0.06721	0.07468	0.26250	0.20227	0.13959
	200	0.77457	1.00254	1.50089	0.12427	0.13633	0.13784	0.02785	0.03006	0.03156	0.16569	0.13633	0.09189
	500	0.75926	1.00120	1.49797	0.07363	0.08283	0.08370	0.00882	0.01088	0.01110	0.09817	0.08283	0.05580

,

Table 4. Simulation values of the AVEs, biases, MSEs, and MREs for $(\alpha =1,\beta =0.5,\lambda =0.75)$.

Method	n	AVEs			Bias			MSEs			MREs
		$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$
MLEs	20	10.66634	0.69815	0.70434	10.15056	0.35036	0.20796	2560.59384	0.29186	0.10072	10.15056	0.70073	0.27728
	50	1.67072	0.55715	0.73867	0.97109	0.15862	0.10938	63.19275	0.04454	0.02101	0.97109	0.31725	0.14584
	100	1.02670	0.53081	0.74374	0.24073	0.09970	0.07498	0.51058	0.01665	0.00921	0.24073	0.19939	0.09997
	200	1.00792	0.51336	0.74574	0.15385	0.06640	0.04930	0.04167	0.00727	0.00395	0.15385	0.13280	0.06573
	500	1.00540	0.50468	0.74840	0.09867	0.04157	0.03114	0.01568	0.00274	0.00153	0.09867	0.08313	0.04152
ADEs	20	17.29534	0.53508	0.72786	16.65270	0.29523	0.16557	3593.12529	0.15502	0.06100	16.65270	0.59046	0.22076
	50	3.39687	0.50182	0.73745	2.65286	0.17299	0.09914	273.28185	0.04915	0.01733	2.65286	0.34597	0.13219
	100	1.35940	0.49382	0.74544	0.54678	0.11664	0.06810	18.76243	0.02211	0.00775	0.54678	0.23328	0.09080
	200	1.05195	0.50079	0.74686	0.19624	0.07866	0.04900	0.08054	0.00984	0.00393	0.19624	0.15732	0.06533
	500	1.02297	0.49892	0.74925	0.12031	0.05114	0.03129	0.02495	0.00412	0.00156	0.12031	0.10228	0.04172
CVMEs	20	28.80055	0.60443	0.72876	28.22486	0.42195	0.18950	6531.10923	0.35574	0.13008	28.22486	0.84390	0.25266
	50	6.61153	0.53382	0.73173	5.93043	0.24163	0.10781	752.14533	0.09964	0.02226	5.93043	0.48326	0.14374
	100	1.92448	0.50863	0.73822	1.16477	0.16055	0.07253	55.85547	0.04206	0.00923	1.16477	0.32110	0.09670
	200	1.10588	0.50426	0.74532	0.28650	0.10743	0.04998	0.26844	0.01880	0.00420	0.28650	0.21486	0.06664
	500	1.03365	0.50083	0.74742	0.15593	0.06740	0.03104	0.04528	0.00716	0.00155	0.15593	0.13480	0.04138
LSEs	20	24.81221	0.56144	0.73570	24.23066	0.38075	0.18041	5010.98830	0.29532	0.19139	24.23066	0.76149	0.24055
	50	6.35294	0.51441	0.73189	5.66583	0.22901	0.10271	695.64053	0.09017	0.02012	5.66583	0.45802	0.13694
	100	1.69405	0.50453	0.74100	0.93870	0.15992	0.07136	32.30933	0.04220	0.00901	0.93870	0.31984	0.09514
	200	1.15725	0.49988	0.74634	0.34226	0.11035	0.04861	1.99838	0.01957	0.00396	0.34226	0.22069	0.06482
	500	1.03342	0.49961	0.74872	0.15567	0.06693	0.03105	0.04564	0.00716	0.00156	0.15567	0.13386	0.04140
WLSEs	20	18.93983	0.53275	0.73201	18.32293	0.33507	0.16053	3681.49435	0.22646	0.07512	18.32293	0.67014	0.21404
	50	3.21300	0.49467	0.73714	2.47225	0.18704	0.09678	240.64885	0.05822	0.01700	2.47225	0.37407	0.12904
	100	1.18486	0.50121	0.74353	0.38732	0.12284	0.07002	2.46525	0.02467	0.00835	0.38732	0.24568	0.09336
	200	1.06726	0.49571	0.74710	0.20810	0.08176	0.05000	0.09630	0.01085	0.00405	0.20810	0.16353	0.06666
	500	1.01595	0.50088	0.74943	0.11884	0.05091	0.03099	0.02346	0.00408	0.00154	0.11884	0.10181	0.04132

and

Table 5. Simulation values of the AVEs, biases, MSEs, and MREs for $(\alpha =0.5,\beta =2,\lambda =3)$.

Method	n	AVEs			Bias			MSEs			MREs
		$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$	$\mathbf{\alpha }$	$\mathbf{\beta }$	$\mathbf{\lambda }$
MLE	20	1.88556	2.59014	3.99837	1.58797	0.96638	1.72681	392.11695	2.04166	37.32390	3.17594	0.48319	0.57560
	50	0.50674	2.18512	3.24298	0.12822	0.45987	0.73043	0.02901	0.38797	1.07395	0.25645	0.22994	0.24348
	100	0.50481	2.08042	3.10209	0.08770	0.29555	0.47206	0.01292	0.14872	0.38945	0.17541	0.14778	0.15735
	200	0.50165	2.03826	3.05662	0.06187	0.20242	0.32396	0.00612	0.06561	0.17325	0.12375	0.10121	0.10799
	500	0.50005	2.01613	3.02382	0.03799	0.12291	0.20007	0.00226	0.02407	0.06423	0.07597	0.06146	0.06669
AD	20	3.96553	2.11542	3.34479	3.61726	0.85448	1.44437	535.50972	1.33161	6.85513	7.23452	0.42724	0.48146
	50	0.60271	2.03972	3.06957	0.20560	0.46733	0.73634	2.72839	0.36640	1.03220	0.41120	0.23366	0.24545
	100	0.52541	2.01725	3.02193	0.10073	0.32557	0.49841	0.01861	0.17145	0.41353	0.20145	0.16279	0.16614
	200	0.50979	2.01459	3.01119	0.06788	0.22917	0.34138	0.00783	0.08375	0.19032	0.13575	0.11458	0.11379
	500	0.50379	2.00337	3.00178	0.04195	0.14183	0.21564	0.00281	0.03185	0.07349	0.08391	0.07092	0.07188
CV	20	7.57233	2.31764	7.46770	7.24915	1.19529	5.75209	1160.36446	2.98504	33,000.03944	14.49830	0.59764	1.91736
	50	1.00678	2.09441	3.26840	0.63006	0.63320	1.07383	56.85043	0.71505	3.56632	1.26012	0.31660	0.35794
	100	0.53923	2.04312	3.08599	0.13104	0.41775	0.64473	0.04556	0.28696	0.76081	0.26208	0.20887	0.21491
	200	0.51562	2.02613	3.04866	0.08297	0.28403	0.43812	0.01229	0.13022	0.33084	0.16594	0.14201	0.14604
	500	0.50685	2.00668	3.01755	0.05017	0.17464	0.26905	0.00421	0.04904	0.11532	0.10035	0.08732	0.08968
LS	20	6.74249	2.15412	4.02082	6.42494	1.10222	2.39731	970.48582	2.38090	111.44446	12.84988	0.55111	0.79910
	50	0.87003	2.06527	3.17638	0.49691	0.60725	1.00070	34.32248	0.65355	2.77405	0.99381	0.30362	0.33357
	100	0.56616	2.01473	3.04200	0.15686	0.40511	0.63364	2.37028	0.26999	0.74052	0.31371	0.20255	0.21121
	200	0.51501	2.01103	3.02743	0.08287	0.28648	0.43391	0.01198	0.12981	0.31715	0.16575	0.14324	0.14464
	500	0.50621	2.00366	3.00350	0.05139	0.17942	0.26784	0.00441	0.05089	0.11674	0.10279	0.08971	0.08928
WLS	20	3.88754	2.10038	3.54341	3.55136	0.94658	1.83658	680.43476	1.76249	26.14833	7.10272	0.47329	0.61219
	50	0.59300	2.03469	3.05918	0.20356	0.51373	0.78963	2.39966	0.45358	1.29793	0.40711	0.25687	0.26321
	100	0.52105	2.01904	3.03118	0.10138	0.33352	0.52531	0.01853	0.18159	0.48072	0.20276	0.16676	0.17510
	200	0.50895	2.01393	3.02204	0.06985	0.23696	0.35833	0.00822	0.08908	0.21367	0.13969	0.11848	0.11944
	500	0.50324	2.00546	3.00805	0.04257	0.14463	0.21777	0.00287	0.03274	0.07518	0.08514	0.07232	0.07259

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.

6. 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©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.

We compare the proposed IPLE distribution with some other competing distributions, including the beta exponential (BE) [31], transmuted generalized exponential (TGE) [4], odd inverse Pareto exponential (OIPE) [16], exponentiated exponential (EE) [1], alpha power exponentiated exponential (APEE) [19], generalized odd log-logistic exponential (GLLE) [13], logistic exponential (LE) [21], alpha power exponential (APE) [9], Marshall–Olkin exponential (MOE) [32], Weibull (W), transmuted exponential (TE) [33], and exponential (E) distributions.

The competing models were compared based on some discrimination measures namely the Akaike information (AI-C) Akaike [34], consistent Akaike information (CAI-C) Sugiura [35], Hannan–Quinn information (HQI-C) [36], and the Bayesian information (BI-C) [37] criteria. Further discrimination measures include the Cramér–von Mises (CVM), Anderson–Darling (AD) [38], and Kolmogorov–Smirnov (K-S) with its p-value. The formulae of these measures were mentioned in [39].

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.

Table 6. Discrimination measures of the IPLE model and other fitted models. CAI-C, consistent Akaike information criterion; BI-C, Bayesian information criterion; HQI-C, Hannan–Quinn information criterion; BE, beta exponential; TGE, transmuted generalized exponential; OIPE, odd inverse Pareto exponential; GLLE, generalized odd log-logistic exponential; APEE, alpha power exponentiated exponential; MOE, Marshall–Olkin exponential; TE, transmuted exponential.

Model	-L	AI-C	CAI-C	BI-C	HQI-C	Estimates
IPLE	180.351	366.702	367.559	371.099	368.164	$\widehat{\alpha }=1.52792\left(0.77685\right)$
						$\widehat{\beta }=3.15494\left(1.42388\right)$
						$\widehat{\lambda }=2.65845\times {10}^7(2.08706\times {10}^8)$
BE	181.626	369.252	370.109	373.649	370.71	$\widehat{\lambda }=0.03202\left(0.000038\right)$
						$\widehat{a}=510.847\left(0.08904\right)$
						$\widehat{b}=0.39293\left(4.87011\right)$
TGE	181.851	369.702	370.559	374.099	371.159	$\widehat{\alpha }=35.9735\left(20.016\right)$
						$\widehat{\lambda }=0.74507\left(0.29061\right)$
						$\widehat{\theta }=0.01357\left(0.00255\right)$
OIPE	182.661	371.321	372.178	375.718	372.779	$\widehat{\alpha }=2004.08(19,007.7)$
						$\widehat{\lambda }=0.01671\left(0.00262\right)$
						$\widehat{\beta }=0.02631\left(0.25009\right)$
EE	182.724	369.447	369.861	372.379	370.419	$\widehat{\alpha }=56.611\left(0.01696\right)$
						$\widehat{\lambda }=2.65845\left(0.00253\right)$
APEE	182.876	371.752	372.609	376.149	373.21	$\widehat{\alpha }=1.90787\times {10}^{11}(2.27727\times {10}^{12})$
						$\widehat{a}=0.01729\left(0.00261\right)$
						$\widehat{c}=2.41738\left(1.91861\right)$
GLLE	183.422	370.845	371.258	373.776	371.816	$\widehat{\alpha }=4.6185\left(0.68532\right)$
						$\widehat{\lambda }=0.00268\left(0.00013\right)$
LE	183.422	370.845	371.258	373.776	371.816	$\widehat{\alpha }=0.00268\left(0.68532\right)$
						$\widehat{\lambda }=2.65845\left(0.00013\right)$
APE	183.787	371.574	371.987	374.505	372.545	$\widehat{\alpha }=1.2042\times {10}^{12}(3.72879\times {10}^{12})$
						$\widehat{a}=0.01407\left(0.00088\right)$
MOE	187.319	378.638	379.052	381.57	379.61	$\widehat{\alpha }=340.079\left(319.528\right)$
						$\widehat{a}=0.02237\left(0.00349\right)$
W	194.425	392.85	393.264	395.782	393.822	$\widehat{a}=2.46021\left(0.27047\right)$
						$\widehat{b}=309.814\left(23.7393\right)$
TE	202.028	408.056	408.47	410.988	409.028	$\widehat{\beta }=0.00528\left(0.00099\right)$
						$\widehat{\lambda }=-1.0000\left(0.78162\right)$
E	211.894	425.787	425.921	427.253	426.273	$\widehat{\lambda }=0.00362\left(0.00064\right)$

and

Table 7. The Anderson–Darling (AD), Cramér–von Mises (CVM), K-S, and p-value of the IPLE model and other fitted models.

Model	AD	CVM	K-S	p-Value
IPLE	0.32689	0.03905	0.08389	0.97794
BE	0.49015	0.05744	0.10498	0.87228
TGE	0.47873	0.05966	0.11582	0.78399
OIPE	0.56721	0.07179	0.12591	0.69073
EE	0.56576	0.07133	0.12591	0.69072
APEE	0.57419	0.07242	0.12691	0.68125
GLLE	0.60687	0.070002	0.11695	0.77389
LE	0.60687	0.07000	0.11695	0.77389
APE	0.94303	0.13285	0.13769	0.57886
MOE	0.96658	0.11006	0.14402	0.52043
W	2.84585	0.45267	0.23395	0.06022
TE	5.46613	1.06381	0.37487	0.00025
E	8.12694	1.71592	0.46956	1.48825$\times {10}^{-6}$

report the analytical measures, MLEs, and their standard errors. The results in Table 6 and Table 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.

The fitted PDF, CDF, SF, and P-Pplots of the IPLP distribution for the analyzed dataset are depicted in Figure 4.

7. 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.