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Communication in Physical Sciences, 2022, 8(3):339-354
The Inverse Lomax Chen Distribution: Properties and Applications
Sadiq Muhammed*, Tukur Dahiru and Abubakar Yahaya
Received: 09 February 2022/Accepted 06 June 2022/Published online: 09 June 2022
Abstract: Many researchers in the field of
distribution theory have been expanding or
generalizing existing probability distributions
to improve their modeling flexibility. In this
paper, we introduced a new continuous
probability distribution called the inverse
Lomax Chen distribution with four parameters.
We studied the nature of the proposed
distribution with the help of its mathematical
and statistical properties such as quantile
function, ordinary moments, generating
function and reliability. The distribution of
order statistics for this distribution was also
obtained. Monte Carlo simulation was carried
out to see the performance of MLEs of the
inverse Lomax Chen distribution. We
performed a classical estimation of parameters
by using the technique of maximum likelihood
estimate. The proposed model was applied to
three real datasets and the results show that the
proposed distribution provides a better fit than
its comparators
Keywords: Biases, Glass Fibres, Inverse
Lomax Chen, Maximum Likelihood Estimate,
Mean Square Error, Quantile Function
Sadiq Muhammed*
Department of Statistics, Faculty of Physical
Sciences, Ahmadu Bello University, Zaria,
Nigeria
Email: msadiq071@gmail.com
Orcid id: 0000-0001-8667-3326
Tukur Dahiru
Department of Community Medicine, College
of Health Science, Ahmadu Bello University,
Zaria, Kaduna State, Nigeria
Email: tukurdahiru2012@gmail.com
Orcid id: 0000-0002-1161-8063
Abubakar Yahaya
Department of Statistics, Faculty of Physical
Sciences, Ahmadu Bello University, Zaria,
Kaduna State, Nigeria
Email: ensiliyu2@yahoo.co.uk
Orcid id: 0000-0002-1453-7955
1.0 Introduction
The real-life world phenomena largely describe
the application of statistical distributions in
modeling lifetime data. These distributions are
very helpful and their theory is explored widely
and new distributions are being produced. The
goal of statistical parametric modeling is to
find the best model for a set of data gathered
from experiments, observational studies,
surveys, and other sources.
Most modeling strategies are centered on
determining the most appropriate probability
distribution that explains the data set's
underlying structure. There is, however, no one
probability distribution that corresponds to all
data sets. As a result, there has been a need to
expand or construct new classical distributions
(Nasiru, 2018).
The exponential, gamma and lognormal
distributions may be used to simulate
monotonic hazard rates. These distributions,
however, have several flaws. For starters, none
of their hazard rate functions have bathtub
forms. Only monotonically increasing,
decreasing, or constant hazard rates are seen in
these distributions. The bathtub-shaped hazard
rate is the most realistic.
This happens in almost all real-world systems.
For example, when a population is separated
into subpopulations with early failures, wear-
out failures, and more or less continual failures,
such forms emerge. As a result, a perfect
bathtub is made up of two change points and a
constant component that is contained within the
Communication in Physical Sciences, 2022, 8(3):339-354 340
change points. Bathtub shape's utility is well
known in a variety of fields. To examine real
datasets with bathtub failure rates, many
parametric probability distributions have been
devised.
Chen (2000) presented a bathtub-shaped or
increasing failure rate (IFR) function for a
novel two-parameter lifetime distribution.
Chaubey and Zhang (2015) proposed an
extension of Chen's (2000) family of
distributions based on Lehman alternatives,
Gupta et al., (1998), which was shown to be a
viable alternative to the generalized and
exponentiated Weibull families for modeling
survival data. Khan et al. (2015) introduced a
new distribution called the transmuted
exponentiated Chen (TEC) and looked at some
of its statistical features using survival data.
The density and hazard functions' analytical
shapes were determined by the authors. For
lifetime data, the TEC distribution shows an
increasing and declining hazard function. Khan
et al. (2018) examined various structural
aspects of the Kumaraswamy exponentiated
Chen (KE-CHEN) distribution for modeling a
bathtub-shaped hazard rate function.
Tarvirdizade and Ahmadpour (2019) created
the Weibull–Chen (W–C) distribution, which is
constructed by compounding the Weibull and
Chen distributions and has growing,
decreasing, and bathtub-shaped hazard rate
functions. The new distribution is more
versatile in terms of modeling bathtub-shaped
hazard rate data, and its hazard rate function is
straightforward. Quantiles, moments, order
statistics, and Renyi entropy were among the
statistical properties explored by the authors.
Recent research in this area has focused on
expanding existing probability distributions to
improve their modeling flexibility. Some
families of distributions proposed in the
literature include Inverse Lomax-G by Falgore
and Doguwa (2020), Topp Leone
exponentiated-G by Ibrahim et al. (2020a),
Topp Leone Kumaraswamy-G by Ibrahim et al.
(2020b), The Kumaraswamy-G by Cordeiro
and DeCastro (2011), Modi family of
continuous probability distributions by Modi et
al., (2020), Odd Chen-G by El-Morshedy et al.,
(2020).
In this context, we proposed a generalization of
the Chen distribution based on the inverse
Lomax-G family of distributions proposed by
Falgore and Doguwa (2020), which stems from
the following general construction: if G
denotes a random variable's baseline
cumulative function, then a generalized class of
distributions can be defined by
(1)
The pdf corresponding to (1) is
(2)
where is the cdf of the baseline
distribution with parameter vector .
for , where equations (1) and
(2) are the cdf and pdf of the IL-G family of
distributions.
The cdf and pdf of the Chen distribution are
given by
(3)
(4)
.
2. 0 The Inverse Lomax Chen (ILC)
Distribution
This section defines a new continuous
distribution called ILC distribution and provide
some plots of its pdf, cdf and hazard rate
function (hrf). The cdf of the ILC distribution
is obtained by inserting (3) into (1) given as:
(1 ( ; ))
( ; , , ) 1 ( ; )
Gx
Fx Gx
−
−
=+
1
2
( ; ) (1 ( ; ))
( ; , , ) 1
( ; ) ( ; )
g x G x
fx G x G x
−−
−
=+
( ; )Gx
0, , , 0x
(1 )
( ; , ) 1 b
x
e
G x b e
−
=−
1 (1 )
( ; , ) b
bx
b x e
g x b bx e e
−−
=
0, , 0xb
Communication in Physical Sciences, 2022, 8(3):339-354 341
(5)
(6)
For .
where is the scale parameter and are the shape parameters respectively.
(1 )
(1 )
()
( ; , , , ) 1 1
b
x
b
x
e
e
e
F x b e
−
−
−
=+
−
( )
1
1 (1 ) (1 )
2(1 )
(1 )
()
( ; , , , ) 1 1
1
bb
b x x
b
x
b
x
b x e e
e
e
bx e e e
f x b e
e
−−
− − −
−
−
=+
−
−
0, , , , 0xb
b
,,
Communication in Physical Sciences, 2022, 8(3):339-354 342
Fig. 1: Plots of pdf of the ILC distribution for different parameter values.
Fig. 2: Plots of cdf of the ILC distribution for different parameter values.
Communication in Physical Sciences, 2022, 8(3):339-354 343
3. 0 Important Representation.
This section provides an expansion for (6) using the generalized binomial expansion given as
(7)
Using the last term in the equation (6) in relation to equation (7), we have
(8)
(9)
The substitution of equation (9) into equation (6) yields equation 10
(10)
Also, the expansion of the last term in equation (10 leads to equation 11 as follows
(11)
Equation 11 is also substituted into equation 10 to obtained equation 12 and upon expansion,
equation 13 was obtained ,
(12)
(13)
Equation 14 was obtained from the expansion of the last term in equation (13), while equation 15
was obtained by the substitution of equations 13 and 14 into equation 12
(14)
(15)
Equation (15) is the important representation of the pdf of Inverse Lomax Chen distribution from
which we can obtain some of the properties of the distribution.
4.0 Properties of ILC Distribution
Some of the mathematical and statistical properties of the ILC distribution such as the quantile
function, moments, moment generating function, reliability measure and order statistics are
presented in this section as follows
4.1 Moments
The rth moment of is obtained as
0
1bi
i
b
zz
i
−
=
−
+=
1
(1 ) (1 )
(1 ) (1 )
0
1
( ) ( )
11
11
bb
xx
bb
xx
i
ee
ee
i
ee
ee
−−
−−
−−
=
−−
+=
−
−−
1
(1 ) (1 ) (1 )
(1 ) 0
1
()
1 ( ) 1
1
1
b
xbb
xx
b
x
eii
i e e
ei
eee
e
−−
−−
−−
−=
−−
+ = −
−
−
2
1 (1 ) (1 ) (1 )
0
1
( ; , , , ) ( ) 1
b b b
b x x x
ii
b x e i e e
i
f x b bx e e e e
i
−−
− − − −
=
−−
=−
−
2
(1 ) (1 )
0
2
1 ( 1) ( )
bb
xx
ij
e j e
j
i
ee
j
−−
−−
=
−−
− = −
−
1
1 (1 )
00
21
( ; , , , ) ( 1) ( )
b
bx
ij
b x j i e
ij
i
f x b bx e e
ji
++
−−
==
− − − −
=−
−−
1
(1 )
0
1
( ) ( 1) (1 )
b
xb
ij
e k k x k
k
ij
ee
k
++
−
=
++
= − −
0
(1 ) ( 1)
bb
l
x k l x
l
k
ee
l
=
− = −
1
1 1 1
0 0 0 0
2 1 1
( ; , , , ) ( 1) bl
i k j k l b x
i j k l
i i j k
f x b b x e
j i k l
+
+ + + + −
= = = =
− − − − + +
=−
−−
x
Communication in Physical Sciences, 2022, 8(3):339-354 344
(16)
The rth moments of the ILC distribution are obtained as
(17)
The solution to equation 17 are as follow
Let
Therefore, the moment of inverse Lomax Chen distribution is given by equation 18
(18)
Equation (18) is the rth moment of the ILC distribution. The mean of the distribution will be
obtained by setting r=1 in (18).
4.2 Moment generating function(MGF)
The mgf of can be obtained using the equation
(19)
(20)
(21)
Following the process of moments above, we have the MGF given as
(22)
0
( ) ( )
rr
E X x f x dx
=
1
1 1 1
00 0 0 0
2 1 1
( ) ( 1) bl
r r i k j k l b x
i j k l
i i j k
E X x b x e dx
j i k l
+
+ + + + −
= = = =
− − − − + +
=−
−−
1
1 1 1
0
0 0 0 0
2 1 1
( ) ( 1) bl
r i k j k l r b x
i j k l
i i j k
E X b x e dx
j i k l
+
+ + + + + −
= = = =
− − − − + +
=−
−−
( 1)bl
yx
+
=
( 1) 1
( 1) bl
dy b l x
dx +−
=+
1( 1) 1
0( 1)
r b y bl
dy
xe
b l x
+−
+−
+
( )
( )
12
2
( 1) 0
( 1)
( 1)
rb r bl y
bl
bl y e dy
bl
+− −−
+
+
+
2
2
0
21
2
r bl yr bl
y e dy
−−
−−
= +
( )
( )
1
11 ( 1)
0 0 0 0
2 1 1 ( 1) 2
( ) ( 1) 1
2
( 1)
rb
r i k j k l bl
i j k l
i i j k bl r bl
E X b j i k l bl
+−
+ + + +
+
= = = =
− − − − + + +
−−
= − +
−−
+
X
0
( ) ( )
tx tx
E e e f x dx
=
1
1 1 1
00 0 0 0
2 1 1
( ) ( 1) bl
tx tx i k j k l b x
i j k l
i i j k
E e e b x e dx
j i k l
+
+ + + + −
= = = =
− − − − + +
=−
−−
0!
mm
tx
m
tx
em
=
=
( )
( )
1
11 ( 1)
0 0 0 0 0
2 1 1 ( 1) 2
( ) ( 1) 1
!2
( 1)
mb
m
tx i k j k l bl
i j k l m
i i j k bl
t m bl
E e b j i k l mbl
+−
+ + + +
+
= = = = =
− − − − + + +
−−
= − +
−−
+
Communication in Physical Sciences, 2022, 8(3):339-354 345
4.3 Reliability function
The reliability function is also known as survival function, which is the probability of an item not
failing prior to some time. It can be defined as
(23)
(24)
4.4 Hazard rate function
(25)
(26)
4.5 Quantile function
The quantile function is defined as the inverse of the cdf and it is given as: . Using
the cdf of ILC distribution in (3.65), we have
( ; , , , ) ( ) 1 ( ; , , , )R x b P X x F x b
= = −
(1 )
(1 )
()
( ; , , , ) 1 1 1
b
x
b
x
e
e
e
R x b e
−
−
−
= − +
−
( ; , , , )
( ; , , , ) ( ; , , , )
f x b
xb
R x b
=
( )
1
1 (1 ) (1 )
2(1 )
(1 )
(1 )
(1 )
()
11
1
( ; , , , ) ()
11
1
bb
b x x
b
x
b
x
b
x
b
x
b x e e
e
e
e
e
bx e e e
e
e
xb e
e
−−
− − −
−
−
−
−
−
+
−
−
=
−+
−
1
( ) ( )Q u F u
−
=
(1 )
(1 )
()
( ; , , , ) 1 1
b
x
b
x
e
e
e
F x b u
e
−
−
−
= + =
−
1(1 )
(1 )
()
11
b
x
b
x
e
e
e
ue
−
−
−
=+ −
1(1 )
(1 )
()
11
b
x
b
x
e
e
e
ue
−
−
−
−= −
( )
1
(1 ) (1 )
( ) 1 1
bb
xx
ee
e u e
−−
−
= − −
11
(1 ) (1 ) (1 )
( ) 1
b b b
x x x
e e e
e u e u e
− − −
−−
+ = − +
11
(1 ) (1 ) (1 )
( ) 1
b b b
x x x
e e e
e u e e u
− − −
−−
+ − = −
11
(1 ) 11
b
x
e
e u u
−−−
+ − = −
Communication in Physical Sciences, 2022, 8(3):339-354 346
Fig. 2: Plots of hazard rate function of the ILC distribution for different parameter values.
Communication in Physical Sciences, 2022, 8(3):339-354 347
(27)
The median of the ILC distribution can be
derived by substituting in (27) as
follows:
(28)
5.0 Order Statistics
Let be independent random
variable from the ILC distributions and let
be their corresponding
order statistic. Let and ,
denote the cdf and pdf of the rth
order statistics respectively. The pdf of
the rth order statistics of is given as
(29)
Using the cdf and pdf of ILC distribution, we have
(30)
1
(1 )
1
1
1
b
x
eu
e
u
−
−
−
−
=
+−
1
1
1
(1 )
1
b
xu
e log
u
−
−
−
−=
+−
1
1
1
1
1b
x
u
log
u
e
−
−
−
+−
−=
1
1
1
1
1
b
x
u
log
u
e
−
−
−
+−
=−
1
1
1
1
1
b
u
log
u
x log
−
−
−
+−
=−
1
1
1
1
1
1
b
u
log
u
x log
−
−
−
+−
=−
0.5u=
1
1
1
10.5
0.5 1
1
b
log
x log
−
−
−
+−
=−
12
, ,..., n
X X X
n
(1) (2) ( )
... n
X X X
:()
rn
Fx
:()
rn
fx
1,2,3,...rn=
:rn
X
:rn
X
1
:0
1
( ; , , , ) ( 1) [ ( ; , , , )] ( ; , , , )
( , 1)
nr i r i
rn i
f x b F x b f x b
B r n r
−+−
=
=−
−+
11 1
1
:0 0 0 0 0
( 1) 2 1
( ; , , , ) ( 1)
( , 1)
b
jl nr m
i k l m b x
rn i j k l m
r i j j k l
b
f x b x e
j k l m
B r n r
++ − +
+ + + −
= = = = =
− + + − − + +
=−
−+
Communication in Physical Sciences, 2022, 8(3):339-354 348
Equation (30) is the rth order statistics of the ILC
distribution.
Therefore, the pdf of the minimum and
maximum order statistics of the ILC distribution
are obtained by setting and
respectively in (30).
6.0 Parameter Estimation
In this section, we estimate the parameters of
the ILC distribution using maximum likelihood
estimation (MLE). For a random sample,
of size from the ILC
, the log-likelihood function L
of (6) is given as
(31)
The components of the score vector, say .
Differentiating (31) with respect to each parameter, we have
(32)
(33)
(34)
(35)
Now, equations (32), (33), (34) and (35) do not
have a simple form and are therefore
intractable. As a result, we have to resort to
non-linear estimation of the parameters using
iterative procedures.
1r=
rn=
12
, ,..., n
X X X
n
( , , , )b
( , , , )b
(1 )
(1 )
(1 )
1 1 1 1 1
()
( ) ( ) ( ) ( ) ( ) ( 1) ( ) ( ) (1 ) 2 1 ( 1) 1 1
b
xi
b
bx
i
ib
xi
e
n n n n n
x
be
ii e
i i i i i
e
nlog nlog nlog nlog b b log x x e e log e
−
−
−
= = = = =
= − + + + − + + − − − − − +
−
l
( ) ( ) ( ) ( )
( ) , , ,
log log log logb
=
llll
Vl
(1 )
(1 )
1
( ) ( )
10
1
b
xi
b
xi
e
n
e
i
log n e
log e
−
−
=
= − + =
−
l
(1 )
(1 )
(1 )
1
(1 )
()
1
() ( 1) 0
()
11
b
xi
b
xi
b
xi
b
xi
e
e
n
e
i
e
e
e
log n
e
e
−
−
−
=
−
−
= − − =
+
−
l
( )
(1 ) (1 ) (1 )
2
(1 )
(1 )
(1 ) (1 )
11
(1 )
( )(1 )(1 ) ( )
1
( ) ( )(1 )
(1 ) 2 ( 1)
1 ( ) ()
11
b b b
x b x x
i i i
i
b
xi
b
xb
ii
b
ibb
xx
ii
b
xi
x
e e e
e
x
e
nn
x
ee
ii
e
e e e e
e
log n e e
eee
e
− − −
−
−
−−
==
−
− − −
−
−
= + − − − −
−
+
−
l
10
n
i=
=
( )
( )
(1 ) (1 )
2
(1 )
(1 )
(1 ) (1 )
1 1 1
(1 )
( )( ) 1
1
( )( )
() ( ) ( ) 2 ( 1)2
1()
11
bb
b x b x
ii
ii
b
xi
b
bx
i
i
bb
xx
ii
b
xi
xx
b e e
ii
e
bxbe
n n n ii
ii ee
i i i
i
e
e x log x e e e
e
e x log x e
log n log x X log x
bb ee
e
−−
−
−
−−
= = =
−
−−
−
= + + − − −
−
+−
l
10
n
i=
=
Communication in Physical Sciences, 2022, 8(3):339-354 349
7. 0 Simulation Study
In this section, we perform the simulation study
to see the performance of MLEs of ILC
distribution. The random number generation is
obtained with its quantile function. We note
that the uth quantile function of the ILC
distribution is given in (27). Hence, if U has a
uniform random variable on (0, 1), then x has
the ILC random variable.
We generated N=10000 samples of sizes
n=20, 50, 100, 250 and 500 from ILC
distribution with its quantile function. Then
we computed the empirical means, biases and
mean squared errors (MSE) of the MLEs with
(36)
and
, (37)
for
To examine the performance of the MLEs for
the ILC distribution, we perform a simulation
study as follows:
1. Generate N samples of size n from the
ILC distribution with its quantile
function.
2. Compute the MLEs for the N samples,
say , for
3. Compute the MLEs for N samples
4. Compute the biases and mean squared
errors MSE given in (35) and (36).
We repeat these steps for N= 10000 and n = 20,
50, 100, 250 and 500 with different values of
. Table 1 shows how the biases
and MSE vary with n. As expected, the Biases
and MSEs of the estimated parameters converge
to zero as n increases which proves the
consistency of the estimators.
Table 1: Biases and MSE of the ILC distribution for selected parameter values.
Initial
values
Bias and
MSE
Sample sizes
n=20
n=50
n=100
n=250
n=500
=0.5
Bias
4.8589
-0.0119
-0.0271
-0.0220
-
0.0151
MSE
987.3814
0.2446
0.0360
0.0201
0.0121
=0.7
Bias
0.4689
0.2312
0.1300
0.0834
0.0513
MSE
0.6061
0.1563
0.0809
0.0502
0.0282
=0.5
Bias
1.9754
-0.0424
-0.0060
-0.0067
-
0.0071
MSE
1.9754
0.3178
0.2594
0.1550
0.0924
=0.6
Bias
2.2144
0.5163
0.2968
0.1297
0.0552
MSE
405.2246
11.1458
1.9379
0.7012
0.2424
8. 0 Real-life Application
In this section, we fit the ILC distribution to data set 1, data set 2 and data set 3 and for illustrative
purposes also present a comparative study with the fits of TEC, EC and C models. These
applications prove empirically the flexibility of the proposed distributions in modeling real life
data sets. All the computations are performed using the R software.
The first data set represents the lifetime data relating to relief times (in minutes) of patients
receiving an analgesic. The data set was given by Gross and Clark (1975). The data set consists of
twenty (20) observations and it is as follows:
1.1, 1.4, 1.3, 1.7, 1.9, 1.8, 1.6, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.2, 1.4, 3, 1.7, 2.3, 1.6, 2.
( )
ˆ1
1ˆ
N
ii
i
Bias N
=
=−
( )
2
ˆ1
1ˆ
N
ii
i
MSE N
=
=−
( , , , )b
=
ˆ
ˆˆ
ˆ
( , , , )b
1,2,...,iN=
( , , , )b
=
b
Communication in Physical Sciences, 2022, 8(3):339-354 350
The second data set was given by Lee (1992) and it represents the survival times of one hundred
and twenty-one (121) patients with breast cancer obtained from a large hospital a period from
1929 to 1938. It has also been applied by Ramos et al., (2013). 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, 1.0, 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.
The third data set represents the breaking strength of 100 Yarn as reported by Gomes-Silva et al.,
(2017). The data set consists of 63 measurements of the strengths of 1.5 cm glass fibres, which
were initially collected by United Kingdom National Physical Laboratory staff. The data is
presented below:
0.55, 0.74, 0.77, 0.81, 0.84, 1.24, 0.93, 1.04, 1.11, 1.13, 1.30, 1.25, 1.27, 1.28, 1.29, 1.48, 1.36,
1.39, 1.42, 1.48, 1.51, 1.49, 1.49, 1.50, 1.50, 1.55, 1.52, 1.53, 1.54, 1.55, 1.61, 1.58, 1.59, 1.60,
1.61, 1.63, 1.61, 1.61, 1.62, 1.62, 1.67, 1.64, 1.66, 1.66, 1.66, 1.70, 1.68, 1.68, 1.69, 1.70, 1.78,
1.73, 1.76, 1.76, 1.77, 1.89, 1.81, 1.82, 1.84, 1.84, 2.00, 2.01, 2.24.
The pdf of the comparators considered are:
• Transmuted Exponentiated Chen (TEC) Distribution (Khan et al. (2016)).
(38)
• Extented Chen (EC) distribution (Chaubey and Zang (2015)).
(39)
The model selection is carried out using the AIC
(Akaike information criterion) and the CAIC
(consistent Akaike information criteria).
(40)
(41)
where denotes the log-likelihood function
evaluated at the maximum likelihood estimates,
is the number of parameters, and is the
sample size.
The model with a minimum value of AIC or
CAIC is chosen as the best model to fit the data
sets considered.
Table 2: The MLEs and Information Criteria of the models based on data set 1
Models
AIC
CAIC
ILC
311.3293
138.8220
5.2802
-
0.2187
-15.5453
39.0906
41.7573
TEC
1.1603
0.5039
0.5869
25.8577
-
-16.6857
41.3714
44.0381
EC
2.9249
0.3325
-
671.5116
-
-16.6857
39.7436
42.4104
C
-
-
0.9523
-
0.1369
-24.5700
53.1401
53.8460
1
111
1
( ) 1 1 2 1
xxx
xe ee
f x x e e e
−
+− −−
−
= − + − −
1
11
1
( ) 1
xx
xe e
f x x e e
−
+− −
−
=−
AIC= -2 +2kl
2 ( 1)
( 1)
kk
CAIC AIC nk
+
=+
−−
l
k
n
ˆ
ˆ
ˆ
ˆ
ˆ
b
l
Communication in Physical Sciences, 2022, 8(3):339-354 351
Fig. 3: Histogram and fitted pdfs for the ILC, TEC, EC and C models to the data set 1
Table 3: The MLEs and Information Criteria of the models based on data set 2
Models
AIC
CAIC
ILC
1.8449
3.0331
0.1825
-
0.2506
-579.3084
1166.6170
1166.9620
TEC
0.0142
0.3512
0.0146
0.9951
-
-581.7857
1167.5710
1167.6930
EC
0.0859
0.2817
-
0.9951
-
-580.8960
1167.792
1167.9041
C
-
-
0.3389
-
0.0214
-581.7857
1167.571
1167.6730
Fig. 4: Histogram and fitted pdfs for the ILC, TEC, EC and C models to the data set 2.
ˆ
ˆ
ˆ
ˆ
ˆ
b
l
Communication in Physical Sciences, 2022, 8(3):339-354 352
Table 4: The MLEs and Information Criteria of the models based on data set 3
Models
AIC
CAIC
ILC
1.5719
3.6272
0.3462
-
1.4863
-13.0803
34.1605
34.8502
TEC
0.2372
1.5861
0.6292
1.7796
-
-13.7696
35.4191
36.1088
EC
0.1726
1.6831
-
1.9494
-
-14.2733
34.5465
34.9533
C
-
-
1.9603
-
0.0721
-16.4613
36.9447
37.1227
Fig. 4: Histogram and fitted pdfs for the ILC, TEC, EC and C models to the data set 3
Figs. 3 and 4 present the shapes, fit and
flexibility of the new model about the data sets
considered. The black line represents the new
model, the red line represents the baseline
distribution, the green line represents the TEC
and the blue line represents the EC
distributions. It can be seen from the histogram
and fitted plots that the black line which
represents the proposed distribution fits better
in the three data sets considered.
9 Conclusion
This paper has derived a new distribution
called the inverse Lomax Chen distribution
with four parameters that extends the Chen
distribution. Some properties of the new
distribution were derived such as the survival
function, hazard rate function, quantile
function, median and order statistics. The
shapes of the proposed distribution were shown
by plotting the graphs of the pdf and hazard rate
function. It can be seen from the hazard rate
plots that the shape of the new distribution has
increased, decreasing, constant and bathtub
shapes. The estimation of the model parameters
by the method of the maximum likelihood was
carried out using a package in R known as
AdequacyModel. Monte Carlo simulation was
carried out to see the performance of MLEs of
the inverse Lomax Chen distribution and as
expected, the Biases and MSEs of the estimated
parameters converge to zero as n increases
which proves the consistency of the estimators.
Application of the new distribution to three real
data sets was carried out and the results are
ˆ
ˆ
ˆ
ˆ
ˆ
b
l
Communication in Physical Sciences, 2022, 8(3):339-354 353
presented in Table 1, Table 2 and Table 3. The
results indicate that the inverse Lomax Chen
distribution is quite effective and superior in
fitting the three data sets considered. Also, the
flexibility of the proposed distribution can be
seen from the histogram and fitted pdf plots for
the three data sets and it is evident that the new
model fits the three data sets better than the
competing distributions considered.
8.0 References
Bourguignon, M., Silva, R. B. & Cordeiro, G.
M. (2014). The weibull-G family of
probability distributions. Journal of Data
Science, 12, 1, pp. 53-8.
Chaubey, Y. P. &Zhang, R. (2015). An
extension of chen’s family of survival
distributions with bathtub shape or
increasing hazard rate function.
Communications in Statistics-Theory and
Methods, 44, 19, pp. 4049-404.
Chen, Z. (2000). A new two-parameter lifetime
distribution with bathtub shape or
increasing failure rate function. Statistics
and Probability Letters, 49, 2, pp. 155–161.
Dey, S., Kumar, D., Ramos, P. L. & Louzada,
F. (2017). Exponentiated Chen distribution:
Properties and estimation.
Communications in Statistics-Simulation
and Computation, 46, 10, pp. 8118–8139.
Dimitrakopoulou, T., Adamidis, K., & Loukas,
S. (2007). A lifetime distribution with an
upside-down bathtub-shaped hazard
function. IEEE Transactions on Reliability,
56, 2, pp. 308–311.
El-Morshedy, M., Eliwa, M. & Afify, A.
(2020). The odd chen generator of
distributions: Properties and estimation
methods with applications in medicine and
engineering. Journal of the national
science foundation of Sri Lanka, 48, 2, pp.
113–130.
Falgore, J. Y. and Doguwa, S. I. (2020). The
Inverse Lomax-G family with application
to breaking strength data. Asian Journal of
Probability and Statistics, pp. 49–60.
Gomes-Silva, F., Percontini, A., Brito, E.,
Ramos, M. W., Silva, R. V. & Cordeiro, G.
M. (2017), The odd Lindley-G family of
distributions, Austrian Journal of Statistics,
46, pp. 65-87.
Gross, A. J. and Clark, V. A. (1975). Survival
distributions: reliability applications in the
biometrical sciences, John Wiley and Sons,
Inc., New York.
Khan, M. S., King, R. & Hudson, I. L. (2015).
Transmuted exponentiated chen
distribution with application to survival
data. ANZIAM Journal, 57, pp. C268–
C290.
Khan, M. S., King, R., & Hudson, I. L. (2018).
Kumaraswamy exponentiated chen
distribution for modelling lifetime data.
Applied Mathematics, 12, 3, pp. 617–623.
Ibrahim, S., Doguwa, S.I., Audu, I. & Jibril,
H.M., (2020a). On the Topp Leone
exponentiated-G Family of Distributions:
Properties and Applications, Asian Journal
of Probability and Statistics; 7, 1, pp. 1-15.
Ibrahim, S., Doguwa S. I., Audu, I. & Jibril, H.
M., (2020b). The Topp Leone
Kumaraswamy-G Family of Distributions
with Applications to Cancer Disease Data,
Journal of Biostatistics and Epidemiology,
6, 1, pp. 37-48.
Lee, E. T. (1992). Statistical methods for
survival data analysis (2nd Edition), John
Wiley and Sons Inc., New York, USA, 156
Pages.
Marshall, A. W. & Olkin, I. (1997). A new
method for adding a parameter to a family
of distributions with application to the
exponential and weibull families.
Biometrika, 84, 3, pp. 641–652.
Mudholkar, G. S. and Hutson, A. D. (1996).
The exponentiated weibull family: some
properties and a flood data application.
Communications in Statistics–Theory and
Methods, 25, 12, pp. 3059–3083.
Nasiru, S. (2018). Extended Odd Frechet-G
family of Distributions, Journal of
Communication in Physical Sciences, 2022, 8(3):339-354 354
Probability and Statistics, ,
doi.org/10.1155/2018/2932326
Ramos, M. A., Cordeiro, G. M., Marinho, P.
D., Dias, C. B. & Hamadani, G. G. (2013).
The zografos-balakrishman log-logistic
distribution: properties and applications,
Journal of Statistical Theory and
Applications, 12, 3, pp. 225-244.
Tarvirdizade, B. & Ahmadpour, M. (2019). A
new extension of chen distribution with
applications to lifetime data.
Communications in Mathematics and
Statistics, pp. 1–16.
Consent for publication
Not Applicable.
Availability of data and materials
The publisher has the right to make the data
public.
Competing interests
The authors declared no conflict of interest.
This work was carried out in collaboration
among all authors.
Funding
There is no source of external funding.
Authors' contribution.
Sadiq Muhammed designed the study,
performed the statistical analysis, wrote the
protocol and wrote the first draft of the
manuscript. Authors Tukur Dahiru and
Abubakar Yahaya managed the analyses of the
study and the literature searches. All authors
read and approved the final manuscript.