A new lifetime distribution with applications in inventory and insurance
International Journal of Quality & Reliability Management
ISSN: 0265671X
Publication date: 5 February 2018
Abstract
Purpose
The purpose of this paper is to introduce a new probability density function having both unbounded and bounded support with a wider applicability. While the distribution with bounded support on [0, 1] has applications in insurance and inventory management with ability to fit risk management data on proportions better than existing bounded distributions, the same with unbounded support is used as a lifetime model and is considered as an attractive alternative to some existing models in the reliability literature.
Design/methodology/approach
The new density function, called modified exponentialgeometric distribution is derived from the exponentialgeometric distribution introduced by Adamidis and Loukas (1998). The support of the density function is shown to be both unbounded and bounded depending on the values of one of the shape parameters. Various properties of the density function are studied in detail and the parameters are estimated through maximum likelihood method of estimation. A number of applications related to reliability, insurance and inventory management are exhibited along with some useful data analysis.
Findings
A single probability distribution with both unbounded and bounded support, which does not seem to exist in the reliability literature, is introduced in this paper. The proposed density function exhibits varying shapes including Ushape, and the failure rate also shows increasing, decreasing and bathtub shapes. The Monte Carlo simulation shows that the estimates of the parameters are quite stable with low standard errors. The distribution with unbounded support is shown to have competitive features for lifetime modeling through analysis of two data sets. The distribution with bounded support on [0, 1] is shown to have application in insurance and inventory management and is found to t data on proportions related to risk management better than some existing bounded distributions.
Originality/value
The authors introduce an innovative probability distribution which contributes significantly in insurance and inventory management besides its remarkable statistical and reliability properties.
Keywords
Citation
Chowdhury, S. and Nanda, A. (2018), "A new lifetime distribution with applications in inventory and insurance", International Journal of Quality & Reliability Management, Vol. 35 No. 2, pp. 527544. https://doi.org/10.1108/IJQRM1220160227
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:Emerald Publishing Limited
Copyright © 2018, Emerald Publishing Limited
1. Introduction
Lifetime data modeling is studied extensively by several researchers for different lifetime distributions with unbounded support. These are mainly based on some modifications and generalizations of exponential or Weibull distributions. While this modification is carried out in some of the lifetime distributions through exponentiation and its extension such as exponentiated or generalized exponential distribution (Gupta and Kundu, 1999; Kundu and Gupta, 2011), exponentiated Weibull (EW) distribution (Mudholkar and Srivastava, 1993), there are others where lifetime distributions are compounded with distribution of unknown number of components yielding a new class of lifetime distributions such as exponentialgeometric (EG) distribution (Adamidis and Loukas, 1998), exponentialPoisson (EP) distribution (Kus, 2007), Weibullgeometric (WG) distribution (BarretoSouza et al., 2011), WeibullPoisson (WP) distribution (Hemmati et al., 2011), generalized exponentialPoisson (GEP) distribution (BarretoSouza and CribariNeto, 2009), generalized exponentialgeometric (GEG) distribution (Silva et al., 2010), geometricPoisson distribution (Chowdhury et al., 2017), etc. In reliability optimization and life testing experiments, sometimes the tests are censored or truncated when failure of a device during the warranty period may not be counted or items may be replaced after a certain time under a replacement policy. Moreover, many reliability systems and biological organism including human life span are bounded above because of test conditions, cost or other constraints. These situations result in a data set which is modeled by distributions with finite range (i.e. with bounded support) such as beta, Kumaraswamy, logLindley, etc. (see e.g. Ghitany, 2004; Kumaraswamy, 1980; GόmezDéniz et al. (2014). While lifetime distributions with unbounded support are very large in number, there are few with bounded support, available in the literature of reliability and survival analysis. Moreover, a single probability distribution with both unbounded and bounded support does not seem to exist in the reliability literature. The purpose of the paper is to propose a new probability distribution with both unbounded and bounded support which has a number of applications.
The exponentialgeomeric (EG) distribution with unbounded support was presented by Adamidis and Loukas (1998) as a lifetime model with decreasing failure rate. It is one of the earliest published papers on lifetime distributions of a system with random number of components. Here, the authors assumed lifetime of individual components to follow iid exponential distribution, and model the number of components failed by zerotruncated geometric distribution. While the EG distribution has been proven to be quite effective to model any lifetime behavior with decreasing hazard function, it fails to model lifetime with other forms of hazard rates. The work of Adamidis and Loukas (1998) is being taken a step further in a new direction in this paper to introduce a new probability distribution having both unbounded and bounded support. This distribution, namely modified exponential geometric (MEG) is derived from the EG distribution with the probability density function (PDF) given by:
The proposed MEG distribution exhibits varying shapes of density function including Ushape, and the failure rate also shows increasing, decreasing and bathtub shapes. The bounded support of the MEG distribution is further transformed into the support [0, 1] to have another distribution namely, betaequivalent MEG_{B} which is shown to have application in insurance and inventory management and is used to fit data on proportions.
In insurance, a probability distribution with domain on [0, 1] can be used as a distortion function to define a premium principle. This is why the classical beta distribution has a dominant role in insurance to produce a class of betadistorted premium principles. For detail see Section 2.6 in Denuit et al. (2005). Although many researchers have proposed probability distributions with domain on [0, 1], most of these distributions involved special functions and hence was not probably considered as a distortion function. Recently, GόmezDéniz et al. (2014) use the logLindley distribution as an alternative to the beta distribution to produce a class of distorted premium principle. The MEG_{B} distribution with bounded support on [0, 1] is used as a distortion function in this paper and is shown to fit data on proportions related to risk management better than some existing bounded distributions. It is also proven to be effective for inventory management. One climate data set is well fitted by the MEG distribution with a finite upper limit other than unity. The new distribution with unbounded support is shown to have competitive features for lifetime modeling.
The rest of the paper is organized as follows. In Section 2, the MEG distribution is derived from the EG distribution. The shapes and the hazard rates of the distribution are derived and discussed in detail along with moments, percentiles and coefficients of variation (CV). Parameters of the distribution are estimated by the maximum likelihood method through a simulation study. Two data sets are analyzed with a detailed comparison with some other competitive lifetime distributions. In Section 3, the MEG_{B} distribution is introduced with scaletransformed bounded support [0, 1]. One application in insurance and another in inventory management are shown in detail. It is shown that the MEG_{B} distribution induces a principle whose premium exceeds the net premium (or expected risk) and sometimes is, for appropriate choice of the parameters, less than the dual power premium principle (Wang, 1996). The distribution is found to fit one real data set on proportion of firm’s risk management cost effectiveness quite well as compared to other bounded distributions. Finally, Section 4 concludes the paper.
2. The MEG distribution
The EG distribution with parameters β>0 and 0<p<1, as proposed by Adamidis and Loukas (1998) has the following cumulative distribution function (CDF):
Let us introduce θ∈ℜ=(−∞, ∞) as the third parameter and the result:
Using result (3) in (2), the CDF of the MEG distribution is obtained as follows:
The support of the random variable (RV) X of the MEG distribution in (4) is (0, ∞) when θ⩽0, and [0, (1/θβ)] when θ>0. The PDF of the MEG distribution with parameters θ, β, p denoted by MEG (θ, β, p) is given by:
Following (3), the PDF of the EG distribution as defined in (1) is a limiting special case of the MEG distribution when θ → 0.
Following (3), the PDF of exponential distribution (βe^{−βx}) is a limiting special case of the MEG distribution when θ → 0 and p → 0+.
It is clear that the EG family has been embedded in a larger family, with an additional shape parameter θ. Due to this additional shape parameter, more flexibility is incorporated in the family with a broader class of hazard functions as evident from the theorems in the next subsection. Moreover, the additional parameter also broadens the scope of applications as well as its usefulness for data analysis.
2.1 Statistical and reliability properties
Different properties of the MEG distribution are studied in this section. To be specific, we study the behavior of the density function and the hazard rate function in detail. Moments of the distribution are also derived with some findings on the skewness and the kurtosis. Bowley’s measure of skewness (S_{k}) is also computed with CV. Moreover, expressions of reversed hazard rate function and mean residual function are also derived.
The following theorem shows that the threeparameter MEG distribution as given in (5) takes various shapes for different choices of θ and p. Figure 1 shows different shapes of the density functions for different choices of the parameters. Below we write f(x) to mean f(x; θ, β, p):
The PDF of MEG (θ, β, p) distribution (a) is strictly decreasing when −1<θ<0 and p∈[0, 1]; (b) does not exist when θ⩽−1 and p∈[0, 1]; (c) is strictly decreasing when 0<θ<1 and p∈[0, 1]; and (d) is strictly increasing when θ>1 and
Proof. Assuming u=(1−θβx)^{1/θ}, the first expression of (5) can be written as follows:
Now, differentiating a(u) with respect to u, we get:
which gives:
Case I: −1<θ<0 – then u>0 and ((θ−1)/(p(1+θ)))<0. Thus, from (6), we see that a′(u)<0 is not possible. This gives that a′(u)⩾0 if and only if u⩾((θ−1)/(1+θ)) (<0), i.e., if u⩾0, proving that a(u) is increasing in u or equivalently, f(x) is decreasing in x, for all x. Therefore, the PDF of the MEG distribution is decreasing for −1<θ<0, for all p∈[0, 1].
Case II: θ<−1 – here, u>0 and ((θ−1)/(p(1+θ)))>0. Now, we get:
Therefore, a′(u)<0 if and only if u<((θ−1)/(p(1+θ)))(>0). So, a(u) is decreasing in u or f(x) is increasing in x, for all x>0. But, the PDF cannot be increasing in an infinite support. Hence, θ <−1 does not make (4) a proper PDF.
Case III: 0<θ<1 – note that θ>0 gives 0⩽x⩽(1/θβ), which, in turn, implies u⩾0 and ((θ−1)/(p(θ+1)))<0. Hence, a′(u)>0 for u>0 implying that a(u) is increasing in u or equivalently, f(x) is decreasing in x. Therefore, the MEG distribution is strictly decreasing for 0<θ<1 and for all p∈[0, 1].
Case IV: let θ>1 – from (6), we have that a(u) is increasing in u if and only if u⩾((θ−1)/(p(1+θ))). Now, we get:
Again, we get:
Similarly, a(u) is decreasing in u if:
Now,
Combining (7) and (8), we conclude that f(x) takes Ushape when θ>1 and p⩾((θ−1)/(θ+1)). Moreover, (9) confirms that f(x) is increasing when θ>1 and p⩽((θ−1)/(θ+1)).◼
From (4), the ξ^{th}(ξ∈[0, 1]) order quantile, say x_{ξ}, of the MEG distribution can be obtained by solving F(x_{ξ}; θ, β, p)=ξ, which gives the ξ^{th} order quantile of the MEG distribution as:
Median and other percentiles of the MEG distribution can be obtained from (11).
The rth order raw moment,
For θ>0,
Proof. Using the PDF in (5) of the MEG distribution, we obtain:
In the first equality, S is support (based on whether θ⩽0 or θ>0) of the RV X, whereas in the second equality, the transformation t=p(1−θβx)^{1/θ} is used:◼
In order for the rth moment of the MEG distribution to exist we must have θ>−(1/r). Hence, for the MEG distribution, all moments exist for θ>0.
Expectation (μ) and variance (σ^{2}) of the MEG distribution are obtained as follows:
The higherorder central moments (μ_{r}) can be easily derived from Theorem 2 and hence moment measure of skewness
Survival function of the MEG distribution is given by:
The following theorem gives us a general result on hazard rate function of the MEG distribution. It shows that the distribution has decreasing, increasing and bathtubshaped failure rates. This fact has been depicted through Figure 2 for different choices of the parameters:
The hazard rate function of MEG(θ, β, p) distribution is: (a) strictly decreasing when −1<θ<0 and p∈[0, 1]; and (b) strictly increasing for θ>0 and p ⩽((θ)/(θ+1)), and bathtubshaped for θ>0 and p⩾((θ)/(θ+1)).
Proof. Assuming u=(1−θβx)^{1/θ} as before, (11) can be written, for θ ≠ 0, as follows:
Then, b′(u)=u^{θ − 1}(θ −pu(1+θ)). Now, we get:
Case I: −1<θ<0 – then u>0 and ((θ)/(p(1+θ)))<0. Thus, from (12), we have that b′(u)>0 is not possible. This gives that b′(u)⩽0 if and only if u⩾((θ)/(p(1+θ)))(<0), i.e., if u⩾0. So, b(u) is decreasing in u, or equivalently, h(x) is decreasing in x⩾0. Therefore, the MEG distribution has decreasing failure rate for −1<θ<0, for all p∈[0, 1].
Case II: θ>0 – from (12), we have that b(u) is increasing in u if and only if u⩽((θ)/(p(1+θ))). Now, we get:
Again,
Similarly, b(u) is increasing in u when
Again, b(u) is decreasing in u when
Combining (14) and (15), we observe that the failure rate of the MEG distribution is bathtubshaped when
Theorem 1(b) proves that the hazard rate function of the MEG distribution does not exist for θ <−1.
Next, we derive the expression for mean residual life function of the MEG distribution. The proof is similar to that of Theorem 2:
Mean residual function of the MEG distribution is given by:
2.2 Estimation of the parameters
Here, we consider estimation of the unknown parameters of the MEG distribution by the method of maximum likelihood. Let x_{1}, x_{2}, …, x_{n} be a random sample of size n drawn from (5) with parameters Ψ=(θ, β, p). Then, the loglikelihood function l(Ψ) for the MEG distribution can be written as follows:
Differentiating (16) partially with respect to the parameters, the likelihood equations are obtained as:
Estimation for the case of θ>0 is somewhat different as the support of the distribution is finite and depends on the unknown parameters θ and β. Let us propose a reparametrization of β, θ, p as (α, θ, p) where α=1/θβ. Hence, (5) can be rewritten as follows:
Based on a random sample from (18), the MLEs of (α, θ, p) are obtained by maximizing the loglikelihood function:
The most natural way (Smith, 1985) to estimate the parameters to handle the situation is to estimate α first by its consistent estimator
Likelihood equations from (20) are derived as follows:
2.3 Numerical examples
2.3.1 Data analysis 1
First, we fit the MEG distribution with unbounded support to a real data set from Proschan (1963). The data set consists of the number of successive failures for the air conditioning system of each member in a fleet of 13 Boeing 720 jet airplanes. The pooled data with 214 observations were first analyzed by Proschan (1963) and discussed further by Dahiya and Gurland (1972) and Gleser (1989). To carry out the comparison of the performance of our proposed model, we have considered some alternative models as discussed in the Introduction, namely EG, EP, WG, WP, GEP, GEG and EW along with Weibull distribution. For the data set that we consider here, we derive the MLEs, AIC, BIC, KolmogorovSmirnov (KS) statistic and the corresponding pvalue for each of the distributions. The results of the data analysis are shown in Table III. The results show that the KS test statistic and the pvalue for the proposed MEG model take the smallest and the largest values, respectively, for the data set as compared to the other models, ensuring its applicability in practice. Moreover, the AIC and BIC values for the MEG distribution are found to be least among all the other distributions. The proposed model offers an attractive alternative to these wellestablished models not only to analyze the data set, but also for its flexibility and potentiality with respect to shape and hazard rates.
2.3.2 Data analysis 2
Here, we fit the MEG distribution to a real data set given in Hinkley (1977) which consists of 30 successive values of March precipitation (in inches) in Minneapolis/St Paul. The data are 0.77, 1.74, 0.81, 1.2, 1.95, 1.2, 0.47, 1.43, 3.37, 2.2, 3, 3.09, 1.51, 2.1, 0.52, 1.62, 1.31, 0.32, 0.59, 0.81, 2.81, 1.87, 1.18, 1.35, 4.75, 2.48, 0.96, 1.89, 0.9, 2.05. The data set having maximum value as 4.75 is considered bounded and hence is fitted with the MEG distribution with bounded support. We compare the performance of the model with generalized beta (GB) distribution (McDonald and Xu, 1995) having bounded support. The MLEs of the MEG and the GB distributions are given as (0.4431815, 2.1051, 0.0134) and (2.3045, 4.6472, 4.7500), respectively. The KS test statistics for the MEG and GB models are 0.1524 (pvalue: 0.2113) and 0.2654 (pvalue: 0.0136), respectively, showing that the data set is best fitted by the MEG distribution.
3. Betaequivalent MEG distribution
Here, we transform the MEG distribution to a betaequivalent distribution with support [0, 1] for θ>0, which is named as betaequivalent MEG distribution and is denoted by MEG_{B}. If X follows the MEG distribution as given in (4), then U=Xθβ has the MEG_{B} distribution whose PDF is given by:
Being a twoparameter scaletransformed distribution, all the statistical and the reliability properties of the MEG_{B} distribution related to the shape of the PDF and the hazard rate function remain same as those of the MEG distribution.
We intend to find out the type of Pearsonian system of curves the MEG_{B} distribution belongs to. For this purpose, we have computed b_{0}, b_{1} and b_{2}, and hence
3.1 Data analysis
We fit the MEG_{B} distribution with a real data set on firm’s risk management cost effectiveness which is available in the personal web page of Professor E. Frees (Wisconsin School of Business Research). The data are defined as the total property and casualty premiums and uninsured losses as a percentage of the total assets leading to a bounded set on [0, 1]. The collection of data was obtained from a questionnaire that was sent to 374 risk managers (73 observations) of large USbased organizations. The purpose of the study was to relate cost effectiveness to managements’ philosophy of controlling the company’s exposure to various property and casualty losses, after adjusting for company effects such as size and industry type. While Schmit and Roth (1990) used beta distribution to fit the data set, GόmezDéniz et al. (2014) used the logLindley distribution. We compare the fit of the proposed distribution with beta, Kumaraswamy and logLindley distributions which also have support on [0, 1]. Table V presents the MLEs of the parameters together with the AIC, BIC, KS statistics and pvalues. It is evident from the table that the MEG_{B} distribution outperforms the other bounded distributions in terms of low KS, AIC, BIC values and high pvalue.
3.2 Application of MEG_{B} distribution
In this subsection, some further applications relating to inventory management and insurance are introduced.
3.2.1 Application in inventory management
Lariviere and Porteus (2001) define the generalized failure rate of a RV X as g(x)=xr(x) where r(x) is the hazard rate (or failure rate) of X. The RV X is said to have an increasing generalized failure rate (IGFR) if g(x) is nondecreasing. IGFR distributions have useful applications in the context of inventory management, pricing and supply chain contracting problems (see e.g. Ziya et al., 2004; Lariviere and Porteus, 2001; Lariviere, 2006), where demand distribution is required to have the IGFR property. In some types of inventory models, the demand is assumed to vary between [0, 1] and Lariviere (2006) shows that all beta distributions are IGFR. The following result shows that the MEG_{B} distribution may be a good alternative in this context:
The MEG_{B} distribution has IGFR property.
Proof. The hazard rate function of the MEG_{B} distribution is derived as follows:
Assuming y=(1−x), we have to show that yr(y) is decreasing in y.
Now, we get:
Differentiating g(y), for all y and θ, we get:
Data analysis
Here, we analyze one simulated realistic data on demand (in suitable unit) which follows the MEG_{B} distribution. The data on daily demand for 48 days are furnished as shown in Table VI.
The data set yields the estimated values of the parameters as θ=0.82364 and p=0.2416 with respective standard errors as 0.022 and 0.018. The hazard rates, r(x), are estimated for each observation and for the estimated parameters. The estimated values of xr(x) are plotted in Figure 3 which clearly shows that xr(x) is increasing in x and hence the MEG_{B} distribution has IGFR property.
3.2.2 Application in insurance
Traditionally, an insurance risk X is defined as a nonnegative loss RV with CDF G_{X} and survival function (also known as decumulative distribution function in the actuarial literature) S_{X}, and a premium calculation principle refers to a functional ρ: X → [0, ∞). The premium principle ρ(X) gives the premium associated with the contract providing coverage against X. For an overview of premium principle, see Denuit et al. (2005). In general, for a risk X, the expected loss can be evaluated directly from its survival function as
A concave distortion function gives more weightage to the higher risk events. For instance, Wang (1996) suggests to use ρ_{h}(X) as a premium principle; for insurance premium purpose, ρ_{h}(X) must be at least equal to
Here, we use the CDF of the MEG_{B} distribution to distort the survival function S_{X} of any loss RV to offer a premium with nonnegative loading. Next, we intend to find out another riskadjusted premium principle, known as dual power premium principle (Wang, 1996). It can be easily shown that the concave distortion function h(x)=(1−(1−x)^{1/θ})/(1−p(1−x)^{1/θ}) (see Theorem 6) transforms the survival function S_{X}(x) of loss RV into the survival function 1−[G_{X}(x)]^{n} which corresponds to the survival function of the RV X_{n:n}, the nth order statistic, where X_{i}, i=1, 2, …, n are iid RVs. So, the corresponding riskadjusted premium is
The distortion function, the CDF of the MEG_{B} distribution in the present case, is said to follow the dual power premium principle if
The CDF of the MEG_{B} distribution (given in (23)) is concave for θ⩽1.
Proof. Differentiating (23) twice, we get:
For x∈[0, 1], 0<p<1 and θ>0, it is obvious that F′′(x)<0 proving that the CDF of the MEG_{B} distribution is concave for 0<θ ⩽ 1.◼
Data analysis
Here, we analyze one simulated realistic data on loss (in suitable unit) incurred by an insurance company to show the dual premium principle. The following data are shown to be well fitted by exponential distribution with parameter λ=0.7372 with standard error as 0.134 (Table IX).
The expected loss is estimated from the data as
4. Conclusion
In this paper, a new PDF with both unbounded and bounded support is proposed, which exhibits a variety of shapes of PDF and hazard functions. The new distribution, called modified EG distribution is derived from the exponentialgeomeric distribution, introduced by Adamidis and Loukas (1998). The parameters of the proposed distribution are estimated using the maximum likelihood method through Monte Carlo simulation. The distribution with scaletransformed bounded support on [0, 1] known as betaequivalent MEG distribution is shown to have applications in insurance and inventory management. It fits one real data set from risk management better than other bounded distributions. The proposed distribution with unbounded support is considered as a competitive lifetime model with respect to some wellestablished lifetime models.
Figures
Moments and quartiles of the MEG distribution for some choices of (θ, β, p)
(θ, β, p)  μ  μ_{2}  CV  γ_{1}  γ_{2}  Q_{1}  Q_{2}  Q_{3}  S_{k} 

(−0.10, 2, 0.2)  0.493  0.338  1.179  3.014  16.900  0.120  0.303  0.651  0.311 
(−0.10, 2, 0.6)  0.332  0.216  1.400  3.785  26.209  0.063  0.171  0.410  0.378 
(−0.24, 2, 0.2)  0.578  0.709  1.456  6.743  513.567  0.122  0.316  0.711  0.341 
(−0.24, 2, 0.6)  0.380  0.422  1.710  8.151  736.727  0.064  0.175  0.434  0.400 
(0.4, 0.5, 0.2)  1.298  1.067  0.795  0.212  −0.076  0.451  1.048  1.935  0.195 
(0.4, 0.5, 0.6)  0.935  0.827  0.973  1.393  1.570  0.244  0.630  1.352  0.303 
(0.4, 2.0, 0.2)  0.325  0.067  0.795  0.881  0.063  0.113  0.262  0.484  0.197 
(0.4, 2.0, 0.6)  0.234  0.052  0.972  1.393  1.570  0.061  0.157  0.338  0.307 
(2.0, 0.5, 0.2)  0.629  0.093  0.486  −0.039  −1.487  0.377  0.691  0.913  −0.172 
(2.0, 0.5, 0.6)  0.505  0.099  0.622  0.010  −1.110  0.221  0.490  0.793  0.059 
(2.0, 2.0, 0.2)  0.157  0.006  0.486  −0.461  −1.084  0.094  0.173  0.228  −0.179 
(2.0, 2.0, 0.6)  0.126  0.006  0.622  0.064  −1.317  0.055  0.122  0.198  0.063 
MLEs and SD of parameters of the MEG Distribution based on Monte Carlo simulation
(θ, β, p)  n  MLE  SD  

(−0.5, 0.5, 0.5)  20  0.55857  −0.55471  0.59873  0.0044687  0.0040145  0.0074738 
50  0.54524  −0.57388  0.56295  0.0005057  0.0010094  0.0005307  
100  0.52891  −0.54595  0.54339  0.0004016  0.0004605  0.0002835  
500  0.49229  −0.52165  0.53313  0.0003393  0.0002993  0.0002079  
(−5.0, 0.5, 0.5)  20  5.59075  −0.57812  0.57266  0.0446627  0.0045076  0.0163101 
50  5.49697  −0.60776  0.55305  0.0032612  0.0023078  0.0006664  
100  5.50101  −0.62647  0.51458  0.0026782  0.0016946  0.0004795  
500  5.33426  −0.64013  0.48321  0.0020705  0.0008048  0.0001576  
(0.5, 0.5, 0.5)  20  0.61561  0.58036  0.63011  0.0225243  0.0898215  0.0211134 
50  0.58632  0.54686  0.60020  0.0086874  0.0436351  0.0056752  
100  0.55655  0.51928  0.56231  0.0009265  0.0052417  0.0007985  
500  0.51100  0.49685  0.52332  0.0005563  0.0007372  0.0002322  
(0.5, 5.0, 0.5)  20  5.10213  0.65178  0.58390  0.0721342  0.0560551  0.0261122 
50  5.03214  0.59113  0.56662  0.0083615  0.0053224  0.0063531  
100  4.98201  0.49174  0.50888  0.0034747  0.0002398  0.0003215  
500  4.91201  0.46961  0.51030  0.0002983  0.0001531  0.0001555  
(5.0, 0.5, 0.5)  20  0.70077  5.01690  0.51395  0.0483501  0.0029333  0.0175577 
50  0.61212  5.10325  0.52101  0.0232521  0.0012035  0.0096321  
100  0.54331  5.00692  0.51115  0.0088524  0.0005664  0.0005036  
500  0.51655  4.95983  0.50601  0.0008871  0.0003134  0.0002691  
(5.0, 5.0, 0.5)  20  6.76976  5.16878  0.52539  0.4781179  0.0066869  0.0186187 
50  5.01394  4.97019  0.59852  0.0024631  0.0012543  0.0005525  
100  4.90802  4.94170  0.60559  0.0017124  0.0006001  0.0004561  
500  4.99433  4.91830  0.61991  0.0006238  0.0003339  0.0001983 
ML estimates, KS statistics and pvalues for the Proschan data
Distribution  Estimates  AIC  BIC  KS  pvalue 

MEG  (−0.1003; 0.0106; 0.1936)  98.3524  101.3622  0.0416  0.8546 
WG  (0.0051; 1.1843; 0.7695)  302.2654  306.6520  0.0658  0.5021 
WP  (0.0100; 0.8202; 0.9741)  232.3614  237.5735  0.0591  0.6840 
GEG  (0.3263; 0.2854; 0.9931)  310.6924  313.3222  0.0711  0.4988 
GEP  (0.8915; 0.0100; 1.1436)  296.6354  299.9587  0.0602  0.5501 
EW  (0.5712; 0.0023; 0.0268)  1009.3631  1013.3632  0.8524  0.0010 
EG  (0.0082; 0.4088)  246.9521  249.8675  0.0522  0.6302 
EP  (0.0079; 1.2011)  201.6374  205.3575  0.0471  0.7623 
Weibull  (0.0012; 0.9240)  241.3524  245.5864  0.0509  0.6393 
κ (γ_{1}, γ_{2}) values of the MEG_{B} distribution for different choices of parameters (θ, p)
p  0.1  0.3  0.5  0.7  0.9 

θ  
0.1  −2.23 (1.6, 3.1)  −2.81 (1.8, 4.0)  −3.85 (2.1, 5.5)  −6.31 (2.6, 8.8)  −20.25 (4.0, 22.8) 
0.5  −0.16 (0.6, −0.5)  −0.25 (0.8, −1.0)  −0.43 (1.1, 0.4)  −0.85 (1.5, 1.7)  −2.83 (2.6, 7.5) 
0.9  −0.01 (0.2, −1.1)  −0.04 (0.3, −0.2)  −0.11 (0.6, −0.8)  −0.31 (0.9, −0.1)  −1.37 (1.9, 3.5) 
2.0  −0.09 (−0.5, −0.9)  −0.04 (−0.4, −1.2)  −0.02 (−0.1, −1.3)  −0.02 (0.3, −1.2)  −0.43 (1.2, 0.3) 
5.0  −0.48 (−1.3, 0.4)  −0.26 (−0.9, −0.4)  −0.08 (−0.5, −1.1)  −0.85 (1.5, 1.7)  −0.03 (0.4, −1.2) 
ML estimates, KS statistics and pvalues for the risk management data
Distribution  Estimates  AIC  BIC  KS  pvalue 

MEG_{B}  (1.2341; 0.9471)  107.6347  112.9664  0.1212  0.2334 
Beta  (0.6125; 3.7979)  143.6541  148.2350  0.1805  0.0171 
Kumaraswamy  (0.6648; 3.4407)  126.3884  130.0232  0.1535  0.0642 
LogLindley  (0.3647; 12347)  133.6853  137.7578  0.1689  0.0323 
Simulated realistic data on demand (in suitable unit)
0.01997  0.28627  0.36028  0.42148  0.63753  0.45707  0.00711  0.01307 
0.28551  0.00129  0.27055  0.21997  0.20966  0.17605  0.62444  0.44407 
0.75669  0.95088  0.44320  0.31903  0.50700  0.12810  0.56246  0.14788 
0.24947  0.62110  0.69374  0.08485  0.00604  0.24458  0.53885  0.50839 
0.26701  0.91673  0.51382  0.16999  0.77082  0.17371  0.55187  0.53570 
0.04705  0.47392  0.54239  0.78548  0.14416  0.87739  0.69808  0.00725 
Premium 

ρ_{h}(X) 



Parameters of MEG_{b}  θ=0.2  θ=0.4  θ=0.6  θ=0.8  n=15  n=25  
p=0.1  p=0.5  p=0.7  p=0.9  
Loss density  
Exponential  
λ=0.5  0.500  1.176  1.066  1.073  1.379  1.659  1.908 
λ=1.0  1.000  2.352  2.132  2.146  2.759  3.318  3.816 
λ=3.0  3.000  7.056  6.396  6.439  8.279  9.955  11.447 
Weibull  
λ=0.5, γ=0.5  1.000  3.524  3.112  3.221  4.959  6.296  8.084 
λ=1.0, γ=1.5  0.903  1.719  1.591  1.589  1.896  2.192  2.414 
λ=2.0, γ=2.5  1.774  2.729  2.576  2.876  2.876  3.180  3.375 
λ=2.5, γ=3.0  2.232  3.229  3.084  3.069  3.368  3.674  3.863 
lognormal  
μ=0.5, σ=0.50  1.868  3.188  2.981  2.990  3.565  4.085  4.556 
μ=1.0, σ=1.00  2.718  11.516  10.387  10.598  14.836  18.192  22.369 
μ=2.0, σ=1.50  22.760  78.227  69.495  72.426  115.999  146.135  195.068 
μ=2.5, σ=0.75  16.139  34.304  31.401  31.734  41.100  49.035  57.515 
Inverse Gaussian  
μ=0.5, σ=0.5  0.500  1.156  1.050  1.063  1.407  1.703  2.011 
μ=1.0, σ=1.0  1.000  2.313  2.100  2.126  2.185  3.406  4.021 
μ=2.0, σ=1.5  2.000  4.982  4.495  4.569  6.221  7.614  9.133 
μ=2.5, σ=2.0  2.500  6.124  5.533  5.620  7.602  9.282  11.093 
Premium 

ρ_{h}(X) 



Parameters of MEG_{b}  θ=0.1  θ=0.2  n=15  n=25  
Loss density  p=0.8  p=0.9  
Loss density  
Exponential  
λ=0.5  0.500  2.007  2.038  1.659  1.908 
λ=1.0  1.000  4.154  4.076  3.318  3.816 
λ=3.0  3.000  12.462  12.229  9.955  11.447 
Weibull  
λ=0.5,γ=0.5  9.786  9.578  3.112  6.296  8.084 
λ=1.0,γ=1.5  2.546  2.507  1.591  2.192  2.414 
λ=2.0,γ=2.5  3.478  3.440  2.576  3.180  3.375 
λ=2.5,γ=3.0  3.958  3.921  3.368  3.674  3.863 
Lognormal  
μ=0.5,σ=0.50  1.868  4.900  4.829  4.085  4.556 
μ=1.0,σ=1.00  2.718  26.387  25.886  18.192  22.369 
μ=2.0,σ=1.50  241.679  236.984  69.495  146.135  195.068 
μ=2.5,σ=0.75  16.139  64.662  63.517  49.035  57.515 
Inverse Gaussian  
μ=0.5,σ=0.5  0.500  2.248  2.205  1.703  2.011 
μ=1.0,σ=1.0  1.000  4.496  4.410  3.406  4.021 
μ=2.0,σ=1.5  2.000  10.345  10.139  7.614  9.133 
μ=2.5,σ=2.0  2.500  12.528  12.280  9.282  11.093 
Simulated realistic data on loss (in suitable unit)
1.31209  0.61040  0.50733  2.17203  1.78809  0.60867  2.64002  0.57412 
0.45361  0.42831  0.34411  0.23693  0.07457  0.18864  0.38920  0.18449 
0.76830  0.29148  0.67360  2.06885  0.51296  0.16361  0.13590  0.56586 
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Acknowledgements
The authors are thankful to the reviewers for their constructive comments and suggestions. Shovan Chowdhury acknowledges the financial support from Indian Institute of Management, Kozhikode, Kerala, India (Grant No. IIMK/SGRP/2014/72). The financial support from NBHM, Govt. of India (Grant No. 2/48(25)/2014/NBHM(R.P.)/R & D II/1393 dt. 03.02.2015) is acknowledged by Asok K. Nanda.