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
, or inverse power transformation, denoted by
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:
and:
where
and
are respectively the shape and scale parameters. For
, the exponential distribution follows as a special sub-model from the LE model.
Consider the inverse power transformation,
, where
LE
, then the resulting IPLE distribution of
X can be specified by the CDF:
The PDF of the IPLE distribution reduces to:
where
and
are the shape parameters and
is a scale parameter. By setting
, 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
Figure 1 and
Figure 2, to show the flexibility of both functions.
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,
, and different parametric values of
, and
,
,
, and
. We generate
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:
where
.
Table 1,
Table 2,
Table 3,
Table 4 and
Table 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.
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 and
Table 7 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.