A new lifetime distribution with applications in inventory and insurance

Shovan Chowdhury (Indian Institute of Management Kozhikode, Kozhikode, India)
Asok K. Nanda (Indian Institute of Science Education and Research Kolkata, Kolkata, India)

International Journal of Quality & Reliability Management

ISSN: 0265-671X

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 exponential-geometric distribution is derived from the exponential-geometric 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 U-shape, 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. 527-544. https://doi.org/10.1108/IJQRM-12-2016-0227

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Publisher

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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 exponential-geometric (EG) distribution (Adamidis and Loukas, 1998), exponential-Poisson (EP) distribution (Kus, 2007), Weibull-geometric (WG) distribution (Barreto-Souza et al., 2011), Weibull-Poisson (WP) distribution (Hemmati et al., 2011), generalized exponential-Poisson (GEP) distribution (Barreto-Souza and Cribari-Neto, 2009), generalized exponential-geometric (GEG) distribution (Silva et al., 2010), geometric-Poisson 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, log-Lindley, etc. (see e.g. Ghitany, 2004; Kumaraswamy, 1980; Gόmez-Déniz 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 exponential-geomeric (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 zero-truncated 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:

(1) f EG ( x ; β , p ) = β ( 1 p ) e β x ( 1 p e β x ) 2 ; x > 0 , β > 0 , 0 < p < 1.

The proposed MEG distribution exhibits varying shapes of density function including U-shape, 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, beta-equivalent MEGB 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 beta-distorted 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, Gόmez-Déniz et al. (2014) use the log-Lindley distribution as an alternative to the beta distribution to produce a class of distorted premium principle. The MEGB 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 MEGB distribution is introduced with scale-transformed bounded support [0, 1]. One application in insurance and another in inventory management are shown in detail. It is shown that the MEGB 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):

(2) F EG ( x | β , p ) = ( 1 e β x ) ( 1 p e β x ) 1 ; x > 0 , β > 0 , 0 < p < 1.

Let us introduce θ∈ℜ=(−∞, ∞) as the third parameter and the result:

(3) lim θ 0 ( 1 θ β x ) 1 / θ = e β x .

Using result (3) in (2), the CDF of the MEG distribution is obtained as follows:

(4) F ( x ; θ , β , p ) = { ( 1 ( 1 θ β x ) 1 / θ ) ( 1 p ( 1 θ β x ) 1 / θ ) 1 if θ 0 ( 1 e β x ) ( 1 p e β x ) 1 if θ = 0.

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:

(5) f ( x ; θ , β , p ) = { β ( 1 p ) ( 1 θ β x ) ( 1 / θ ) 1 ( 1 p ( 1 θ β x ) 1 / θ ) 2 if θ 0 β ( 1 p ) e β x ( 1 p e β x ) 2 if θ = 0.
Remark 1.

Following (3), the PDF of the EG distribution as defined in (1) is a limiting special case of the MEG distribution when θ → 0.

Remark 2.

Following (3), the PDF of exponential distribution (βeβx) 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 (Sk) 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 three-parameter 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):

Theorem 1.

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 p ( ( θ 1 ) / ( θ + 1 ) ) , and U-shaped for θ>1 and p ( ( θ 1 ) / ( θ + 1 ) ) .

Proof. Assuming u=(1−θβx)1/θ, the first expression of (5) can be written as follows:

f ( x ) β ( 1 p ) = u 1 θ ( 1 p u ) 2 = a ( u ) , say .

Now, differentiating a(u) with respect to u, we get:

a ( u ) = u θ ( 1 p u ) 2 ( 1 θ + 2 u p ( 1 p u ) 1 ) ,

which gives:

(6) a ( u ) 0 if and only if u θ 1 p ( 1 + θ ) .

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:

u θ 1 p ( 1 + θ ) if and only if x 1 α β [ ( p ( α 1 ) α + 1 ) α 1 ] ,
where α=−θ; α>1. But (((p(α−1))/(α+1))α−1)<0. Hence, a′(u)>0 is not possible.

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:

u θ 1 p ( 1 + θ ) if and only if x 1 θ β [ 1 ( θ 1 p ( θ + 1 ) ) θ ] = x 0 , say.

Again, we get:

[ 1 ( θ 1 p ( θ + 1 ) ) θ ] 0 , if and only if p θ 1 θ + 1 .
Hence, a(u) is increasing in u, or, equivalently:
(7) f ( x ) is decreasing in x , for p θ 1 θ + 1 , and0 x x 0 .

Similarly, a(u) is decreasing in u if:

1 θ β [ 1 ( θ 1 p ( θ + 1 ) ) θ ] x 1 θ β .

Now, 1 ( ( ( θ 1 ) / ( p ( θ + 1 ) ) ) ) θ > / < 0 when p > / < ( ( θ 1 ) / ( θ + 1 ) ) . Therefore, for p ( ( θ 1 ) / ( θ + 1 ) ) :

(8) f ( x ) is increasing in x x 0
and for p ( ( θ 1 ) / ( θ + 1 ) ) :
(9) f ( x ) is increasing in x 0

Combining (7) and (8), we conclude that f(x) takes U-shape 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)).◼

Remark 3.

From (4), the ξth(ξ∈[0, 1]) order quantile, say xξ, of the MEG distribution can be obtained by solving F(xξ; θ, β, p)=ξ, which gives the ξth order quantile of the MEG distribution as:

(10) x ξ = { 1 θ β [ 1 ( 1 ξ 1 p ξ ) θ ] if θ 0 1 β log ( 1 ξ 1 p ξ ) if θ = 0.

Median and other percentiles of the MEG distribution can be obtained from (11).

The rth order raw moment, μ r , and hence the expectation and the variance of the MEG distribution can be obtained from the following theorem:

Theorem 2.

For θ>0, E ( 1 θ β X ) r = ( ( 1 p ) / ( p ) ) i=1 ( ( i p i ) / ( r θ + i ) ) .

Proof. Using the PDF in (5) of the MEG distribution, we obtain:

E ( 1 θ β X ) r = β ( 1 p ) S ( 1 θ β x ) r + 1 θ 1 ( 1 ( p ( 1 θ β x ) 1 / θ ) 2 d x = 1 p p 0 p t r θ ( 1 t ) 2 d t = 1 p p r θ + 1 0 p ( i = 1 i t i 1 ) t r θ d t = 1 p p r θ + 1 i = 1 i p r θ + i r θ + i = 1 p p i = 1 i p i r θ + i .

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:◼

Remark 4.

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.

Corollary 2.1.

Expectation (μ) and variance (σ2) of the MEG distribution are obtained as follows:

μ = E ( X ) = 1 θ β [ 1 1 p p j = 1 j p j θ + j ] ,
σ 2 = E ( X 2 ) μ 2 = 1 θ 2 β 2 [ 1 p p i = 1 i p i 2 θ + i + 2 θ β μ 1 ] μ 2 .
Remark 5.

The higher-order central moments (μr) can be easily derived from Theorem 2 and hence moment measure of skewness ( γ 1 = ( μ 3 / μ 2 3/2 ) ) and kurtosis ( γ 2 = ( μ 4 / μ 2 2 ) 3 ) along with CV. Table I shows values of moments, variance, γ1,γ2 and CV along with quartiles and Bowley’s measure of skewness for a few choices of parameters of the MEG distribution.

Survival function of the MEG distribution is given by:

F ¯ ( x ; θ , β , p ) = { ( 1 p ) ( 1 θ β x ) 1 / θ ( 1 p ( 1 θ β x ) 1 / θ ) 1 if θ 0 ( 1 p ) e β x ( 1 p e β x ) 1 if θ = 0 ,
with hazard rate:
(11) h ( x ) = { β ( 1 θ β x ) 1 ( 1 p ( 1 θ β x ) 1 / θ ) 1 if θ 0 β ( 1 p e β x ) 1 if θ = 0 ,
and reversed hazard rate:
h ˜ ( x ) = { β ( 1 p ) ( 1 θ β x ) 1 θ 1 [ ( 1 ( 1 θ β x ) 1 / θ ) ( 1 p ( 1 θ β x ) 1 / θ ) ] 1 if θ 0 β ( 1 p ) e β x [ ( 1 e β x ) ( 1 p e β x ) ] 1 if θ = 0.

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 bathtub-shaped failure rates. This fact has been depicted through Figure 2 for different choices of the parameters:

Theorem 3.

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 bathtub-shaped 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:

[ h ( x ) β ] 1 = u θ ( 1 p u ) = b ( u ) , say .

Then, b′(u)=uθ − 1(θpu(1+θ)). Now, we get:

(12) b ( u ) 0 if and only if u θ p ( 1 + θ ) .

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:

u θ p ( 1 + θ ) if and only if x 0 = 1 θ β [ 1 ( θ p ( θ + 1 ) ) θ ] x 1 θ β .

Again, 1 ( ( θ ) / ( p ( θ + 1 ) ) ) θ 0 if and only if p ( ( θ ) / ( θ + 1 ) ) . Hence, b(u) is increasing in u, or equivalently:

(13) h ( x ) is increasing in x , when p θ θ + 1 .

Similarly, b(u) is increasing in u when 1 ( ( ( θ ) / ( p ( θ + 1) ) ) ) θ 0 , i.e., when p ( ( θ ) / ( θ + 1 ) ) . Therefore:

(14) h ( x ) is increasing in x , for p θ θ + 1 , and x 0 x 1 θ β .

Again, b(u) is decreasing in u when u ( ( θ ) / ( p ( θ + 1 ) ) ) , i.e., when x x 0 . However, as we have seen from the previous discussion, x0⩾0 if p ( ( θ ) / ( θ + 1 ) ) . Therefore:

(15) h ( x ) is decreasing in x , for0 x x 0 , when p θ θ + 1 .

Combining (14) and (15), we observe that the failure rate of the MEG distribution is bathtub-shaped when p ( ( θ ) / ( θ + 1 ) ) . Also, (13) ensures that the distribution has increasing failure rate when p ( ( θ ) / ( θ + 1 ) ) .◼

Remark 6.

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:

Theorem 4.

Mean residual function of the MEG distribution is given by:

E ( X x 0 | X x 0 ) = 1 p θ F ¯ ( x 0 ) [ 1 1 β p i = 1 i p i ( 1 θ β x 0 ) i θ + 1 θ + i θ ( 1 θ β x 0 ) 1 θ ] .

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 x1, x2, …, xn be a random sample of size n drawn from (5) with parameters Ψ=(θ, β, p). Then, the log-likelihood function l(Ψ) for the MEG distribution can be written as follows:

(16) l ( Ψ ) = n log β + n log ( 1 p ) + ( 1 θ 1 ) i = 1 n log ( 1 θ β x i ) 2 i = 1 n log ( 1 p ( 1 θ β x i ) 1 / θ ) .

Differentiating (16) partially with respect to the parameters, the likelihood equations are obtained as:

(17) l β = n β ( 1 θ ) i = 1 n x i A i 2 p i = 1 n C i x i A i ( 1 p C i ) , l θ = 1 / θ 2 i = 1 n B i β ( 1 θ 1 ) i = 1 n x i A i + 2 p θ i = 1 n [ C i 1 p C i ( B i θ + β x i A i ) ] , l p = n 1 p + 2 i = 1 n C i 1 p C i , }
where A i = 1 θ β x i , B i = log A i , C i = A i 1 / θ . The maximum likelihood estimator (MLE) of Ψ, say Ψ ˆ , is the simultaneous solutions of the likelihood equation (17) when θ<0.

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:

(18) f ( x ; θ , α , p ) = 1 p θ α ( 1 x α ) ( 1 / θ ) 1 ( 1 p ( 1 x α ) 1 / θ ) 2 ; 0 < x < α .

Based on a random sample from (18), the MLEs of (α, θ, p) are obtained by maximizing the log-likelihood function:

(19) l 1 ( α , θ , p ) = n log ( 1 p ) n log α n log θ + ( 1 θ 1 ) i = 1 n log ( 1 x ( i ) α ) 2 i = 1 n log ( 1 p ( 1 x ( i ) α ) 1 / θ ) .
Here, x(i) denotes the ith ordered observation. It is obvious from (19) that, for 0<p<1, as αx(n) (the largest observation among xi, i=1, 2, …, n), L1(α, θ, p) → ∞ (resp. −∞) when θ>1 (resp. θ∈[0, 1]), resulting in the non-existence of MLEs.

The most natural way (Smith, 1985) to estimate the parameters to handle the situation is to estimate α first by its consistent estimator α ˆ = x ( n ) . The modified log-likelihood function based on the remaining (n−1) observations after ignoring x(n) and substituting α as x(n) is given as follows:

(20) l 2 ( θ , p ) = ( n 1 ) log ( 1 p ) ( n 1 ) log x ( n ) ( n 1 ) log θ + ( 1 θ 1 ) i = 1 n 1 log ( 1 x ( i ) x ( n ) ) 2 i = 1 n 1 log ( 1 p ( 1 x ( i ) x ( n ) ) 1 / θ ) .

Likelihood equations from (20) are derived as follows:

(21) l 2 θ = ( n 1 ) / θ 1 / θ 2 i = 1 n 1 E i 2 p / θ 2 i = 1 n 1 E i F i 1 p F i , l 2 p = ( n 1 ) 1 p + 2 i = 1 n 1 F i 1 p F i , }
where D i = 1 ( x ( i ) / x ( n ) )   , E i = log D i , F i = D i 1 / θ . Since no closed form expressions for the MLEs of Ψ are available from the expressions (17) and (21), we go for detailed simulation studies to capture the means (μ) and the standard deviations of the MLEs of Ψ. In total, 10,000 replicates of Monte Carlo experiments of size 20, 50, 100 and 500 are considered in the present investigation for each of the six sets of Ψ, namely (−0.5, 0.5, 0.5), (−5.0, 0.5, 0.5), (0.5, 0.5, 0.5), (0.5, 5.0, 0.5), (5, 0.5, 0.5), (5.0, 5.0, 0.5). These estimates are obtained by using the function optim from the statistical software R (version R.3.0.1). In the current context, we recommend to use quasi-Newton algorithms, namely the BFGS algorithm for numerical maximization of log-likelihood functions. The results from simulated data sets are reported in Table II. The results show that the estimates are quite stable around the assumed values of Ψ and moreover, standard errors of the MLEs decrease when sample size increases, which is expected.

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, Kolmogorov-Smirnov (K-S) statistic and the corresponding p-value for each of the distributions. The results of the data analysis are shown in Table III. The results show that the K-S test statistic and the p-value 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 well-established 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 K-S test statistics for the MEG and GB models are 0.1524 (p-value: 0.2113) and 0.2654 (p-value: 0.0136), respectively, showing that the data set is best fitted by the MEG distribution.

3. Beta-equivalent MEG distribution

Here, we transform the MEG distribution to a beta-equivalent distribution with support [0, 1] for θ>0, which is named as beta-equivalent MEG distribution and is denoted by MEGB. If X follows the MEG distribution as given in (4), then U=Xθβ has the MEGB distribution whose PDF is given by:

(22) f U ( x | θ , p ) = 1 p θ ( 1 x ) 1 θ 1 ( 1 p ( 1 x ) 1 / θ ) 2 ; 0 < x < 1 , θ > 0 , 0 < p < 1 ,
with CDF:
(23) F U ( x | θ , p ) = ( 1 ( 1 x ) 1 / θ ) ( 1 p ( 1 x ) 1 / θ ) 1 .
Remark 7.

Being a two-parameter scale-transformed distribution, all the statistical and the reliability properties of the MEGB distribution related to the shape of the PDF and the hazard rate function remain same as those of the MEG distribution.

Remark 8.

We intend to find out the type of Pearsonian system of curves the MEGB distribution belongs to. For this purpose, we have computed b0, b1 and b2, and hence κ = ( b 1 2 / 4 b 0 b 2 ) for different choices of the parameters and are shown in Table IV. Details on Pearsonian system of curves can be obtained in Elderton and Johnson (1969). The κ-criterion suggests that the distribution belongs to Pearson’s Type I system of curves where beta distribution also belongs to.

3.1 Data analysis

We fit the MEGB 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 US-based 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, Gόmez-Déniz et al. (2014) used the log-Lindley distribution. We compare the fit of the proposed distribution with beta, Kumaraswamy and log-Lindley distributions which also have support on [0, 1]. Table V presents the MLEs of the parameters together with the AIC, BIC, K-S statistics and p-values. It is evident from the table that the MEGB distribution outperforms the other bounded distributions in terms of low K-S, AIC, BIC values and high p-value.

3.2 Application of MEGB 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 non-decreasing. 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 MEGB distribution may be a good alternative in this context:

Theorem 5.

The MEGB distribution has IGFR property.

Proof. The hazard rate function of the MEGB distribution is derived as follows:

r ( x ) = ( ( 1 p ) / ( θ ) ) ( 1 x ) 1 θ 1 ( 1 p ( 1 x ) 1 / θ ) 2 1 [ ( 1 ( 1 x ) 1 / θ ) ( 1 p ( 1 x ) 1 / θ ) 1 ] .

Assuming y=(1−x), we have to show that yr(y) is decreasing in y.

Now, we get:

y r ( y ) = sign 1 y y ( 1 p y 1 / θ ) = g ( y ) , say .

Differentiating g(y), for all y and θ, we get:

g ( y ) = sign y 1 / θ [ p θ ( 1 y ) + p ] 1
and:
g ( y ) = sign p θ ( 1 + 1 θ ) [ y 1 / θ 1 ( 1 y ) ] 0.
Hence, g′(y) is increasing in y. Now, g′(0)=−1<0 and g′(1)=p−1<0 showing that g′(y)<0∀y which, in turn, implies that g(y) is decreasing in y and hence xr(x) is increasing in x proving that the MEGB distribution has IGFR property.◼

Data analysis

Here, we analyze one simulated realistic data on demand (in suitable unit) which follows the MEGB 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 MEGB distribution has IGFR property.

3.2.2 Application in insurance

Traditionally, an insurance risk X is defined as a non-negative loss RV with CDF GX and survival function (also known as decumulative distribution function in the actuarial literature) SX, 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 ρ 1 * ( X ) = E ( X ) = 0 S X ( x ) d x and is commonly applied when decision makers agree on the risk distribution. Note that ρ 1 * ( X ) is the simplest premium principle and is known as the net premium. As there does not exist any unique risk distribution, insurers add a loading to X that reflects the danger associated with the risk. Premium principle by Wang (1996) suggests to transform the survival function by a continuous and non-decreasing distortion function h: [0, 1] → [0, 1] with h(0)=0 and h(1)=1. The distortion function h(SX(.)) can be thought of as a risk-adjusted survival function of the RV Xh (say). A distortion risk measure associated with distortion function h, for a random loss X, is given by ρ h ( X ) = E h ( X ) = 0 h ( S X ( x ) ) d x . The distortion risk measure adjusts the true probability measure to give more weight to higher risk events. Both of the quantile and the conditional tail expectation risk measures fall into the class of distortion risk measures. They are by far the most commonly used distortion measures for capital adequacy, but others are also seen in practice, particularly for premium setting in property and casualty insurance.

A concave distortion function gives more weightage to the higher risk events. For instance, Wang (1996) suggests to use ρh(X) as a premium principle; for insurance premium purpose, ρh(X) must be at least equal to ρ 1 * ( X ) and such is the case when h is concave. Moreover, as h is increasing and concave, h(x)⩾x for all x∈[0, 1] which ensures that h(S(x))⩾S(x) for all x, or equivalently, XstXh, resulting in ρ h ( X ) ρ 1 * ( X ) . It is to be mentioned here that for two RVs X and Y with respective survival functions F ¯ and G ¯ , X is said to be smaller that Y in usual stochastic order, written as XstY, if F ¯ ( x ) G ¯ ( x ) , for all x. Hence, the premium principle contains a non-negative loading. In fact, the premium principle with concave distortion function satisfies some desirable properties of premium functional, namely non-ripoff, positive homogeneity, comonotonicity and subadditivity. These four axioms make the concave distortion risk measure coherent; for more details, see Denuit et al. (2005).

Here, we use the CDF of the MEGB distribution to distort the survival function SX of any loss RV to offer a premium with non-negative loading. Next, we intend to find out another risk-adjusted 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 SX(x) of loss RV into the survival function 1−[GX(x)]n which corresponds to the survival function of the RV Xn:n, the nth order statistic, where Xi, i=1, 2, …, n are iid RVs. So, the corresponding risk-adjusted premium is ρ n * ( X ) = E ( X n : n ) . It is obvious that ρ 1 * ( X ) ρ n * ( X ) .

The distortion function, the CDF of the MEGB distribution in the present case, is said to follow the dual power premium principle if ρ h ( X ) ρ n * ( X ) for some relationship between the parameters of the distortion function, the risk distribution and the sample size n. The results in Table VII confirm that the premium obtained by distorting original loss distribution by the MEGB distribution lies between the net premium and the dual power premium ( ρ 1 * ( X ) ρ h ( X ) ρ n * ( X ) ) for selected choices of the parameters. However, it is also observed from Table VIII that the same distortion function does not satisfy the dual power premium principle for some other choices of the parameters with the same sample size. We have considered the exponential, the Weibull, the lognormal and the inverse Gaussian distributions as the original loss distribution with different choices of the parameters. Hence, the upper bound of the distorted premium principle with CDF of the MEGB distribution as the distortion function may not always be the dual premium principle. We conclude this section by showing that the CDF of the MEGB distribution is concave:

Theorem 6.

The CDF of the MEGB distribution (given in (23)) is concave for θ⩽1.

Proof. Differentiating (23) twice, we get:

F ( x ) = ( 1 p θ ) ( 1 p ( 1 x ) 1 / θ ) 3 ( 1 x ) 1 θ 2 [ p ( 1 x ) 1 / θ ( 1 θ + 1 ) + ( 1 θ 1 ) ] .

For x∈[0, 1], 0<p<1 and θ>0, it is obvious that F′′(x)<0 proving that the CDF of the MEGB 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 ρ 1 * ( X ) = E ( X ) = 0.7372 . Using the CDF of the MEGB distribution with parameters θ=β=0.5 as distortion function, the premium principle (distorted expected risk) is obtained as ρh(X)=0.9265. It is seen that ρ 1 * ( X ) < ρ h ( X ) .

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 exponential-geomeric 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 scale-transformed bounded support on [0, 1] known as beta-equivalent 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 well-established lifetime models.

Figures

Probability density functions of the MEG distributions

Figure 1

Probability density functions of the MEG distributions

Hazard functions of the MEG distributions

Figure 2

Hazard functions of the MEG distributions

Figure showing IGFR propert of the MEGB distribution

Figure 3

Figure showing IGFR propert of the MEGB distribution

Moments and quartiles of the MEG distribution for some choices of (θ, β, p)

(θ, β, p) μ μ2 CV γ1 γ2 Q1 Q2 Q3 Sk
(−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, K-S statistics and p-values for the Proschan data

Distribution Estimates AIC BIC K-S p-value
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 (γ1, γ2) values of the MEGB 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, K-S statistics and p-values for the risk management data

Distribution Estimates AIC BIC K-S p-value
MEGB (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
Log-Lindley (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

ρ 1 * ( X ) , ρh(X) and ρ n * ( X ) for different loss PDFs with varying choices of parameters

Premium ρ 1 * ( X ) ρh(X) ρ n * ( X )
Parameters of MEGb 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

ρ 1 * ( X ) , ρh(X) and ρ n * ( X ) for different loss PDFs with varying choices of parameters

Premium ρ 1 * ( X ) ρh(X) ρ n * ( X )
Parameters of MEGb 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.

Corresponding author

Shovan Chowdhury can be contacted at: meetshovan@gmail.com