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Asian Journal of Probability and Statistics
11(2): 30-46, 2021; Article no.AJPAS.66377
ISSN: 2582-0230
_____________________________________
*Corresponding author: E-mail: bash0140@gmail.com;
A New Extended Generalized Inverse Exponential Distribution:
Properties and Applications
Bashiru Omeiza Sule
1*
1
Department of Mathematical Sciences, Kogi State University, Anyigba, Kogi State, Nigeria.
Author’s contribution
The sole author designed, analyzed, interpreted and prepared the manuscript.
Article Information
DOI: 10.9734/AJPAS/2021/v11i230264
Editor(s):
(1) Dr. Seemon Thomas, Mahatma Gandhi University, India.
Reviewers:
(1) Hazem Al-Mofleh, Tafila Technical University, Jordan.
(2) Melanie G. Gurat, Saint Mary’s University, Philippines.
(3) Chipepa Fastel, Botswana International University of Science and Technology, Botswana.
Complete Peer review History:
http://www.sdiarticle4.com/review-history/66377
Received: 03 January 2021
Accepted: 09 March 2021
Published: 19 March 2021
_______________________________________________________________________________
Abstract
The quest by researchers in the area of distribution theory in proposing new models with greater
flexibility has filled literature. On this note, we proposed a new distribution called the new extended
generalized inverse exponential distribution with five positive parameters, which extends and generalizes
the extended generalized inverse exponential distribution. We derive some mathematical properties of the
proposed model including explicit expressions for the quantile function, moments, generating function,
survival, hazard rate, reversed hazard rate, cumulative hazard rate function and odds functions. The
method of maximum likelihood is used to estimate the parameters of the distribution. We illustrate its
potentiality with applications to three real life data sets which show that the new extended generalized
inverse exponential model provides greater flexibility and better fit than other competing models
considered.
Keywords: Bio-medical analysis; carbon fibers; generalized inverse exponential; survival times; vinyl
chloride.
1 Introduction
Researchers are in the quest of developing and proposing new models in the area of distribution theory by
generalizing the existing ones. The generalization is done by adding more parameters to improve the
Original Research Article
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
31
flexibility. The literature is filled with such distributions that are very worthwhile in predicting and modeling
real life scenario. A number of classical distributions have been used comprehensively over the past decades
for modeling data in several applied areas including bio-medical analysis, reliability engineering, economics,
forecasting, astronomy, demography and insurance.
In generating new distributions, a number of methods have been proposed. A popular and most used method
is the use of family of distributions because it can be used to model a wide variety of random phenomena.
Usually the standard distributions will be mathematically simpler, and often other members of the family can
be constructed from the standard distributions by simple transformations on the underlying standard random
variable. Some recent families of distributions are: Kumaraswamy odd Burr G family of distributions by
Nasir et al. [1], the Marshal-Olkin Odd Lindley-G family of distributions by Jamal et al. [2], The
Exponentiated Kumaraswamy-G family of distributions by Silva et al. [3], the Topp Leone exponentiated G
family of distributions by Ibrahim et al. [4], The Topp Leone Kumaraswamy-G family of distributions by
Ibrahim et al. [5], Odd Chen-G family of distributions by Anzagra et al. [6], Modi family of continuous
probability distributions by Modi et al. [7].
Despite the usage of exponential distribution in Poisson processes, reliability engineering and its attractive
properties, the fact that the exponential distribution has a constant failure rate is a disadvantage because for
that singular reason, the distribution becomes unsuitable for modeling real life situations with bathtub and
inverted bathtub failure rates [8]. This is actually a serious short-coming of the exponential distribution.
Also, the memorylessness is rarely obtainable in real life phenomena. To overcome the limitations of
exponential distribution, Keller and Kamath [9] came up with a modified version of the exponential
distribution, this modification resulted into the inverse exponential distribution and it has also been studied
in details by Lin et al. [10].
Gupta and Kundu [11], generalized the exponential distribution by appending the shape parameter, and
named the distribution as the generalized exponential distribution. Generalized inverted exponential
distribution was first introduced by Abouammoh and Alshingiti [12]. This distribution originated from the
exponentiated Frechet distribution [13].
The generalized inverse exponential distribution provides many practical applications, including, in horse
racing, queue theory, modeling wind speeds. Ibrahim et al. [14] extended the generalize inverse exponential
distribution by adding one shape parameter to the distribution to make it more flexible in modeling real life
data. Oguntunde and Adejumo [15] have explored the statistical properties of the generalized inverted
generalized exponential distribution and its parameters were estimated at both censored and uncensored
cases using the method of maximum likelihood estimation (MLE). Dey et al. [16] presents some estimation
and prediction of unknown parameters based on progressively censored generalized Inverted Exponential
data.
The probability density and cumulative density function of generalized Inverted exponential distribution
with shape parameter and scale parameter , are given respectively as
(;,)=1−1−
(1)
and ℎ(;,)=
1−
, ≥0. (2)
This paper aims to introduce a new extended version of the generalized inverse exponential distribution
called new extended generalized inverse exponential distribution using the Topp Leone Kumaraswamy-G
(TLK-G) family of distribution. The motivation in proposing this new model is to handle data sets that
exhibit skewness, (left-skewed and right-skewed), symmetric or reversed-J shape and to provide consistently
better fits than other competing distributions. The rest of the paper is outline as follows. In section 2, the new
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
32
extended generalized inverse exponential distribution is defined. Linear representation of the new model is
presented in section 3. Section 4 provides the statistical properties of the new model. In section 5, the
distribution of order statistics is presented. The maximum likelihood estimation is discussed in section 6.
Section 7 presented the application of the new model to real data sets. Finally, concluding remark is given in
section 8.
2. The New Extended Generalized Inverse Exponential (NEGIEx)
Distribution
For an arbitrary baseline cumulative distribution function (cdf) (,), the TLK-G family with three extra
positive shape parameters , and has cdf and probability density function (pdf) for (x > 0) given by
(;,,,)={1−[1−(,)
]
}
(3)
and
(;,,,)=2ℎ(; )(; )
[1−(; )
]
{1−[1−(; )
]
}
(4)
>0 and ,,,>0 respectively.
Where ℎ(;)=
(; )
is the baseline pdf, , and are positive shape parameters.
The cdf of the new model is derived by substituting (1) into (3) as
(;,,,,)=1−1−1−1−
, (5)
the pdf corresponding to (5) is given as
(;,,,,)=2
1−
1−1−
1−1−1−
1 − 1 −1 − 1 −
, (6)
where ≥0, >0 is the scale parameter and ,,,>0 are the shape parameters respectively.
3 Linear Representation of the New Model
Using the series expansion
(1−)
=∑
()
Γ()
!Γ()
∞
(7)
Using the last term in (6) in relation to the expansion in (7), we have
1−1−1−1−
=(−1)
Γ()
!Γ(−)1−1−1−
∞
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
33
1 − 1 − 1 −
()
=(−1)
Γ(2(+1))
!Γ(2(+1)−)1−1−
∞
1 − 1 −
()
=(−1)
Γ((+1))
!Γ((+1)−)1−
∞
1 −
()
=(−1)
Γ((+1))
!Γ((+1)−)
∞
substituting back into (6), we have
()=2
∑ ∑ ∑ ∑
()
Γ()Γ(())Γ(())Γ(())
!!!!Γ()Γ(())Γ(())Γ(())
∞
∞
∞
∞
(8)
Equation (8) is the expansion for the pdf in (6).
Fig. 1. Plots of pdf of New Extended Generalized Inverse Exponential distribution with different
parameter values
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
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4 Statistical Properties
In this section, some of the statistical properties of the new extended generalized inverse exponential
distribution will be obtained as follows:
4.1 Moments
Moments function is used to study many important properties of distribution such as dispersion,
tendency, skewness and kurtosis. The r
th
moments of the new extended generalized inverse exponential
distribution is obtained as follow:
(
)=∫
∞
() (9)
Using (6), we have,
(
)=∫
∞
2
∑ ∑ ∑ ∑
()
Γ()Γ(())Γ(())Γ(())
!!!!Γ()Γ(())Γ(())Γ(())
∞
∞
∞
∞
(10)
(
)=2∑ ∑ ∑ ∑
()
Γ()Γ(())Γ(())Γ(())
!!!!Γ()Γ(())Γ(())Γ(())
∫
∞
∞
∞
∞
∞
(11)
(
)=2
∑ ∑ ∑ ∑
()
Γ()Γ()Γ()Γ()
()
Γ()
!!!!Γ()Γ(())Γ(())Γ(())
∞
∞
∞
∞
(12)
To obtain the mean, we set =1 in (12)
4.2 Moment generating function
The moment generating function (mgf) of can be obtained using the equation
()=(
)=∫
()
∞
(13)
=∑
!
∞
(14)
()=2
∑ ∑ ∑ ∑ ∑
()
Γ()Γ()Γ()Γ()
()
Γ()
!!!!!Γ()Γ(())Γ(())Γ(())
∞
∞
∞
∞
∞
(15)
4.3 Quantile function
The Quantile function is given by;
()=
() (16)
Therefore, the corresponding quantile function for the extended generalized inverse exponenetial model is
given by;
=()=⎣⎢⎢⎢⎡−⎩
⎪
⎨
⎪
⎧1−1−1−1−
⎭
⎪
⎬
⎪
⎫⎦⎥⎥⎥⎤
(17)
where has the uniform (0,1) distribution.
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4.4 Median
The median of the extended generalized inverse exponential distribution is obtained by setting =0.5 in
(17) to obtain,
=(0.5)=⎣⎢⎢⎢⎡−⎩
⎪
⎨
⎪
⎧1−1−1−1−0.5
⎭
⎪
⎬
⎪
⎫⎦⎥⎥⎥⎤
(18)
4.5 Survival function
The survival function, which is the probability of an item not failing prior to some time. It can be defined as
()=1−() (19)
The survival function the NEGIEx distribution is given as
()=1−1−1−1−1−
(20)
4.6 Hazard rate function
The hazard rate function is given as
()=
()
()
=
()
()
(21)
()=
(22)
4.7 Odds function
The odds function is obtained using the relation
()=
()
()
(23)
()=
(24)
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Fig. 2. Plots of hrf of New Extended Generalized Inverse Exponential distribution with
different parameter values
From Fig. 2, it can be seen that the new model exhibits different shapes such as increasing, decreasing, J-
shape and reversed J-shape.
4.8 Reversed hazard rate function
The reverse hazard rate function of the new extended generalized inverse exponential distribution is given as
∅()=
()
()
(25)
∅()=
(26)
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4.9 Cumulative hazard function
()= −ln (()) (27)
Then, the cumulative hazard function of the NEGIEx distributions is given as
()= −ln 1 − 1 − 1 − 1 − 1 −
(28)
5 Distribution of Order Statistics
Let
,
,
,…,
be a random sample and its ordered values are denoted as
()
,
()
,
()
,…,
()
. The
pdf of order statistics is obtained using the below function
:
()=
(,)
()[()]
[1−()]
(29)
5.1 Minimum order statistics
The minimum order statistics is obtained by setting =1 in (29) as
:
()=()[1−()]
(30)
Then the minimum order statistics of the new extended generalized inverse exponential distribution is given
as
:
()=
2 ∑ ∑ ∑ ∑ ∑
()
Γ()Γ()Γ()Γ()Γ()
!!!!!Γ()Γ(())Γ(())Γ(())Γ(())
∞
∞
∞
∞
∞
(31)
5.2 Maximum order statistics
The maximum order statistics is obtained by setting = in (29) as
:
()=()[()]
(32)
Then the maximum order statistics of the new extended generalized inverse exponential distribution is given
as
:
()=2∑ ∑ ∑ ∑
()
Γ(())Γ(())Γ(())Γ(())
!!!!Γ(())Γ(())Γ(())Γ(())
∞
∞
∞
∞
(33)
6 Maximum Likelihood Estimates
Since maximum likelihood estimators give the maximum information about the population parameters,
therefore this section presents the maximum likelihood estimates (MLEs) of the parameters that are inherent
within the new extended generalized inverse exponential distribution function given by the following: Let
,
,
,…,
be random variables of the extended generalized inverse exponential distribution of size n.
Then sample likelihood function of extended generalized inverse exponential distribution is obtained as
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
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()=(2)
∏
1−
1−1−
1−1−1−
1−
1
−
1
−
1
−
−
2
−
1
(34)
Log-likelihood function () is
()=2+++++++∑
−∑
+(−1)∑1−
−
+
−
1=11
−
1
−
−
+(2
−
1)=11
−
1
−
1
−
−
+(
−
1)=11
−
1
−
1
−
1
−
−
2
(35)
Therefore, The MLE's of parameters ,,,, which maximize the above log-likelihood function must
satisfy the normal equations. We take the first derivative of the above log-likelihood equation with respect to
each parameter and equate to zero respectively.
\
()
=
+∑
1−1−
+(2−1)∑
+(−
1)∑
(36)
()
=
−∑
+(−1)∑
+(−1)∑
+(2−
1)∑
+(−
1)∑
(37)
()
=
+∑1−
+(−1)∑
+(2−
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1)∑
+(−
1)∑
(38)
()
=
+2∑
1−1−1−
+(−1)∑1−1−1−1−
(39)
()
=
+∑1−1−1−1−
(40)
Since the above derived equations (36), (37), (38), (39) and (40) are in the complex form, therefore the exact
solution of ML estimator for unknown parameters is not possible. So it is convenient to use nonlinear
Newton Raphson algorithm for exact numerically solution to maximize the above likelihood function.
7 Applications
In this section, we applied two data sets to illustrate the usefulness of the proposed model and observe its
flexibility over some existing models. The models considered are:
Extended generalized inverse exponential (EGIEx) distribution
=2
1−
1−1−
1
−1−1−
1−1−1−1−
Inverse exponential (IEx) distribution
()=
Exponentiated generalized inverse exponential (ExGIEx) distribution
()=
1−
1−1−
Generalized Inverse exponential (GIEx) distribution
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
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()=
1−
The statistics used in comparing the fit of the models is Akaike Information Criteria (AIC). The model with
the lowest AIC is considered the best with regard to the data set considered.
7.1 Data set 1
The first data set represents the breaking stress of carbon fibers of 50 mm length (GPa) was reported by
Nicholas and Padgett [17]. This data was used by Yousof et al. [18]. The data are:
0.39, 0.85, 1.08, 1.25, 1.47, 1.57, 1.61, 1.61, 1.69, 1.80, 1.84, 1.87, 1.89, 2.03, 2.03, 2.05, 2.12, 2.35, 2.41,
2.43, 2.48, 2.50, 2.53, 2.55, 2.55, 2.56, 2.59, 2.67, 2.73, 2.74, 2.79, 2.81, 2.82, 2.85, 2.87, 2.88, 2.93, 2.95,
2.96, 2.97, 3.09, 3.11, 3.11, 3.15, 3.15, 3.19, 3.22, 3.22, 3.27, 3.28, 3.31, 3.31, 3.33, 3.39, 3.39, 3.56, 3.60,
3.65, 3.68, 3.70, 3.75, 4.20, 4.38, 4.42, 4.70, 4.90.
Table 1. MLEs and selection criteria for data set 1
Models
−
AIC
NEGIEx 0.0039 10.1562 13.6247 79.2456 0.2025 86.0794 182.1587
EGIEx
1.18861
13.0965
0.3690
19.5331
-
93.9751
195.9502
IEx - 2.2992 - - - 136.0285 274.0570
ExGIEx 39.7875 13.8792 0.4454 - - 94.0114 196.0228
GIEx - 13.2879 - 7.6019 - 99.6202 203.2403
Table 1 shows the result of the analysis of data set representing the breaking stress of carbon fibers of 50
mm length (GPa). It can be seen from the result in the table that the NEGIEx distribution has the lowest AIC
which makes it fits better and appropriate in this data set than the other competing models considered.
Fig. 3. Histogram and fitted models breaking stress of carbon fibers of 50 mm length (GPa) data
Sule; AJPAS, 11(2): 30-46, 2021; Article no.AJPAS.66377
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Fig. 4. Plots for the fitted pdf, cdf, Q-Q plot and P-P plot for data set 1
7.2 Data set 2
The second data set was given by Lee [19] and it represents the survival times of one hundred and twenty-
one (121) patients with breast cancer obtained from a large hospital in a period from 1929 to 1938. It has
also been applied by Ramos et al. [20]. The data set is as follows:
0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8,
15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6,
24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0,
40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0,
51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0,
69.0, 78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0,
126.0, 127.0, 129.0, 129.0, 139.0, 154.0.
Table 2 shows the result of the analysis of data set representing the survival times of one hundred and
twenty-one (121) patients with breast cancer. It can be seen from the result in the table that the NEGIEx
distribution has the lowest AIC which makes it fits better and appropriate in this data set than the other
competing models considered.
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Table 2. MLEs and selection criteria for data set 2
Models
−
AIC
NEGIEx 7.1153 0.0017 7.5251 0.1364 9.9600 609.4956 1228.9910
EGIEx 63.7813 0.0040 2.2264 0.5013 - 622.6595 1253.3190
IEx
-
10.3215
-
-
-
677.2791
1357.5580
ExGIEx 0.2142 0.0021 4.6702 - - 743.3310 1492.6620
GIEx - 0.5595 - 6.4034 - 664.1239 1332.2480
Fig. 5. Histogram and fitted models from survival times of one hundred and twenty-one (121) patients
with breast cancer data
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Fig. 6. Plots for the fitted pdf, cdf, Q-Q plot and P-P plot for data set 2
7.3 Data set 3
The third data set represents the vinyl chloride (in mg/l) that was obtained from clean up gradient monitoring
wells. It has previously been used by Bhaumik et al. [21]. The data set has 34 observations and are presented
below:
5.1, 1.2, 1.3, 0.6, 0.5, 2.4, 0.5, 1.1, 8, 0.8, 0.4, 0.6, 0.9, 0.4, 2, 0.5, 5.3, 3.2, 2.7, 2.9, 2.5, 2.3, 1, 0.2, 0.1, 0.1,
1.8, 0.9, 2, 4, 6.8, 1.2, 0.4, 0.2.
Table 3. MLEs and selection criteria for data set 1
Models
−
AIC
NEGIEx 16.4598 0.0286 0.3223 0.3290 20.7196 54.5905 119.1810
EGIEx
2.5246
1.2638
0.1792
0.7188
-
58.4206
124.8411
IEx - 0.5725 - - - 59.1930 120.3860
ExGIEx
0.8710
0.0056
59.1930
-
-
58.6960
123.3920
GIEx - 0.9214 - 0.5415 - 59.1285 122.2569
Table 3 shows the result of the analysis of data set representing the vinyl chloride (in mg/l). It can be seen
from the result in the table that the NEGIEx distribution has the lowest AIC which makes it fits better and
appropriate in this data set than the other competing models considered.
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Fig. 7. Histogram and fitted models from vinyl chloride data
Fig. 8. Plots for the fitted pdf, cdf, Q-Q plot and P-P plot for data set 3
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8 Conclusion
In this paper, a five parameter life time model called the new extended generalized inverse exponential
distribution which generalizes the extended generalized inverse exponential distribution has been studied.
This new model is capable of modeling different kinds of data with different shapes as shown in Fig. 1 and
2. We derived explicit expressions for some of its statistical and mathematical properties including the
moments, generating function, quantile function, survival function, hazard rate function, reversed hazard rate
function, cumulative hazard rate function and odds function. The minimum and maximum distributions of
order statistics of the new model were derived. The model parameters were estimated by using maximum
likelihood method based on complete sample. We observed from the analysis that the new extended
generalized inverse exponential distribution provides better fits than the competing distributions considered
on the three real life data sets based on the value of AIC and also from the histograms and the fitted plots of
the pdfs.
Competing Interests
Author has declared that no competing interests exist.
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