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Applied Mathematical Sciences, Vol. 13, 2019, no. 1, 33 - 44
HIKARI Ltd, www.m-hikari.com
https://doi.org/10.12988/ams.2019.811179
Record Values from T-X Family of
Pareto-Exponential Distribution with
Properties and Simulations
Noor Waseem
Department of Statistics
Kinnaird College for Women, Lahore, Pakistan
Shakila Bashir
Department of Statistics
Forman Christian College, Lahore, Pakistan
Copyright © 2019 Noor Waseem and Shakila Bashir. This article is distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any
medium, provided the original work is properly cited.
Abstract
In this paper, upper record values from Pareto-Exponential distribution are studied
with its statistical properties and applications. Graphs are also given along with
Probability density function (PDF) and cumulative density function (cdf).
Reliability analysis and various properties which included Survival Function,
Hazard Function, Cumulative Hazard rate, reversed Hazard rate, Geometric mean
and moments are discussed. Moreover, the expressions for Renyi entropy has also
been derived. Finally, Monte Carlo simulation study is carried to generate data of
size 50 with a sample of 15 from Pareto-Exponential distribution and upper records
has been recorded.
Keywords: Record values, Exponential Pareto distribution, Moments, entropy
1. Introduction
Chandler [8] proposed record statistics as a model for ordered random variable.
Theory of record values defined in a sequence of independent and identically
distributed random variable. Ahsanullah [1] studied some characteristics of upper
34 Noor Waseem and Shakila Bashir
record values from the Exponential distribution. Ahsanullah, et. al. [2] established
record values from the classical Pareto distribution. Bashir and Ahmed [5] derived
record values from size-biased Pareto distribution and its characterizations. Bashir
and Akhar [6] introduced record values rising from student’s t Distribution. Paul
and Thomas [13] presented an article on generalized upper (k) record values from
Weibull distribution. Chacko and Muraleedharan [7] developed the lower k-record
values arising from a two parameter generalized exponential distribution and
applied the inference statistics on it. Kumar [11] gave some expressions of
recurrence relations of k-th lower record values from Dagum distribution derived
by single and product moments. Sultan [15] established some new recurrence
relations of record values from the modified Weibull distribution between the single
moments and double moments. Sultan and Moshref [16] derived the exact explicit
expressions for single, double, triple and quadruple moments of the upper record
values from a generalized Pareto distribution. Balakrishnan and Ahsanullah [4]
established recurrence relation of upper record values from the generalized Pareto
distribution. Khan, et.al [10] obtained the relations for moments of k-th record
values from exponential-Weibull lifetime distribution with its characterization.
Dey, et. al. [9] introduced the statistical inference for the generalized Rayleigh
distribution based on upper record values. Minimol and Thomas [12] presented
some properties of Makeham distribution using generalized record values and its
characterization. Saran and Nain [14] developed the Relationships for moments of
kth record values from doubly truncated pth order exponential and generalized
Weibull distributions.
The Pdf of upper record values fn(x) is given by:
n 1
n
R(x)f (x) f (x)
n
, x (1)
The joint pdf of U( j)X and
U(i)X is
i 1 j i 1
i, j
R(x) R(y) R(x)f (x, y) r(x) f (y)
i j i
(2)
, x y j i
Where,
R(x) ln 1 F(x) ,0 1 F(x) 1
2. Record Values from Pareto-Exponential Distribution
Pareto-Exponential distribution (PED) was derived by using the CDF of the T-X family of distribution given by Alzaatreh, et al. [3], where random variable T follows
Record values from T-X family of Pareto-exponential distribution 35
Pareto distribution and X follows the exponential distribution. Following are the
PDF and CDF of Pareto-Exponential distribution
1( ) (1 )f x x , 0x , 1 , 0 (3)
1( ) 1
(1 )F x
x
(4)
Where, is a scale parameter and is a shape parameter.
By substituting eq (3) & (4) in eq (1), the expressions of PDF and CDF of Upper
Record Values arising from Pareto-Exponential (UPE) Distribution are given as
follows
1 11
( ) ln(1 ) (1 ) , 1, 0, n 1, 0.nn
nf x x x xn
(5)
1
( ) , ln 1nF x n xn (6)
Lemma 1: The area under the curve is unity
( ) 1f x dx
Proof:
1 1
0
1( ) ln(1 ) (1 )
nn
nf x x x dxn
, 0x , 1 , 0
let, ln(1 )y x then, (1 )
dy dxx
When 0x then 0y
When xthen y
1ye x
1
0
nn yy e dy
n
1
1
0
yn
ny e dyn
1
nn
nn
( ) 1nf x
Graphical representation of PDF of the UPE Distribution for various parameters
value in figure 1.
36 Noor Waseem and Shakila Bashir
(a)
(b)
(c)
(d)
Figure 1
Interpretation: From Fig. 1(a, b, c and d) it can be seen that the pdf of the
proposed model is positivley skewed for various values fo parameters.
Figure 2
CDF plot for α=2, λ=1.5, n=2,3,4
Figure 2 displays the plot of CDF function for various values of n when α and λ
are fixed
Record values from T-X family of Pareto-exponential distribution 37
3. Reliability Analysis
For the UPE distribution several measures of reliability are derived in this section.
Reliability function, hazard rate function, reversed hazard rate function and
cumulative hazard rate function are given as respectively,
1
( ) 1 , ln 1nR x n xn (7)
1 1ln(1 ) (1 )( )
, ln 1
nn
n
x xh x
n n x
(8)
1 1ln(1 ) (1 )( )
, ln 1
nn
n
x xr x
n x
(9)
1
( ) ln 1 , ln 1nH x n xn
(10)
Figure 3
Reliability plot for α=2, λ=1.5, n=2,3,4
38 Noor Waseem and Shakila Bashir
Figure 4
Hazard plot for α=2, λ=1.5, n=2,3,4
4. Statistical properties
In this section, various statistical properties are computed for proposed distribution.
The rth Moment about origin is as follows
( )( ) ( )r r
n nE x x f x dx
( )
0
1( ) 1
nnr kr
n rk
rE x
k k
k (11)
For 1,2,3&4r we get first four raw moments of the upper record values from the
Exponential-Pareto Distribution are as follows respectively,
/
1
11
1
n
n
(12)
/
2 2
12 1
2 1
n n
(13)
/
3 3
1 3 31
3 2 1
n n n
n n n
(14)
0,00E+00
1,00E-01
2,00E-01
3,00E-01
4,00E-01
5,00E-01
6,00E-01
7,00E-01
8,00E-01
9,00E-01
0 1 2 3 4 5 6 7 8 9
10
25
75
12
5
20
0
h(x
)
x
Hazard Function
n=2 n=3 n=4
Record values from T-X family of Pareto-Exponential distribution 39
/
4 4
1 4 6 41
4 3 2 1
n n n n
n n n n
(15)
Mean and Geometric mean is given by respectively
11
1
n
nMean
1 (16)
1
101
1 1. 1
nnnk
n
nki
nG Mkk
k (17)
Variance is as follows
2
2 22
1
2 1
n n
n n
2 (18)
Co-efficient of Variation, skewness and kurtosis are given by respectively
2
22 1
11
n n
n n
n
n
2 (19)
2 3 2
3 2
1 3
22
2
3 3 31
3 2 1 1 1 1
2 1
n n n n n
n n n n n n
n n
n n
3
(20)
2 3 4
2 4
2 22
2
4 6 3
4 1 3 1 2 1
2 1
n n n n
n n n n n n
n n
n n
4
(21)
40 Noor Waseem and Shakila Bashir
5. Measure of inequality and uncertainty
In this section some Measure of inequality and uncertainty are derived. Lorenz and
Bonferroni curves are as follows
2
2
1 1 1( ) , ln(1 ) ( 1) , ln(1 )
11
nn
n n nnn
L x n x n x a
n
(22)
2
2
1 1 1, ln(1 ) ( 1) , ln(1 )
11( )
, ln 1
nn
n nnn
n
n x n x a
B xn x
(23)
Renyi Entropy is given by
1
( ) ln ( )1
v
v f x f x dxv
1
0
1( ) ln ln(1 ) (1 )
1 (1 )
nn
v nf x x x dxv xn
(24)
6. Recurrence Relations for Single and Product Moment of UPE
Distribution
In this section recurrence relations of the single and product moments of the UPE
distribution have been derived. These relations can have used to find moment of the
model in recursive manner. The relation between CDF and PDF of EP distribution given in equation (3) and
(4), is
( ) 1 ( )
1f x F x
x
(25)
Theorem 1: For 𝑛 > 1 𝑎𝑛𝑑 𝑟 = 0,1,2,3, ….
( 1) ( )(1 ) (1 ) 1r r
U n U n
rE X E X
(26)
Proof: From equation (2) and, for 𝑛 > 1 𝑎𝑛𝑑 𝑟 = 0,1,2,3, … ..
( ) ( )0
(1 ) (1 ) ( )r r
U n U n nE X X f x dx
Record values from T-X family of Pareto-exponential distribution 41
1
( )0
1(1 ) (1 ) ln 1 ( ) 1 ( )
( 1)! 1
nr r
U nE X x F x F x dxn x
1
1
( )0
(1 ) (1 ) ln 1 ( ) 1 ( )( 1)!
nr r
U nE X x F x F x dxn
After some simplification we get the result in equation (26).
Theorem 5.2: For 𝑖 < 𝑚 < 𝑛 − 2 𝑎𝑛𝑑 𝑟, 𝑠 = 0,1,2,3 ….
( ) ( 1) ( ) ( )(1 ) (1 ) (1 ) (1 ) 1r s r s
U i U j U i U j
sE X X E X X
(27)
for 𝑗 = 𝑖 + 1 𝑎𝑛𝑑 𝑟, 𝑠 = 0,1,2,3 … ….
( ) ( 1) ( ) ( ) ( 1)(1 ) (1 ) (1 ) (1 ) (1 )r s r s r s
U i U i U i U i U i
sE X X E X E X X
(28)
Proof: Let 𝑋𝑈(𝑖) 𝑎𝑛𝑑 𝑋𝑈(𝑗) are from UEP distribution in equation (5) and,
𝑎𝑛𝑑 𝑟, 𝑠 = 0,1,2,3 … ….
1
( ) ( ) ( )0
1 ( )(1 ) (1 ) (1 ) ln 1 ( ) ( )
( 1)!( 1)! 1 ( )
ir s r
U i U j U i
f xE X X X F x I x dx
i j i F x
(29)
where,
1
( ) (1 ) ln(1 ( )) ln(1 ( )) ( )j is
xI x y F y F x f y dy
1
( ) (1 ) ln(1 ( )) ln(1 ( )) 1 ( )(1 )
j is
xI x y F y F x F y dy
y
11( ) (1 ) ln(1 ( )) ln(1 ( )) 1 ( )
j is
xI x y F y F x F y dy
1
2
(1 ) (1 )( ) ln(1 ( )) ln(1 ( )) ( )
( 1) ln(1 ( )) ln(1 ( )) ( )
s sj i
x x
j i
y yI x F y F x f y dy
s s
j i F y F x f y dy
Substitute the above equation in equation (29), we get the result in equation (27)
and for j=i+1, the result in (28).
7. Simulation
Monte Carlo simulation study is carried out using the R software to generate data
from the Pareto-Exponential distribution. F(x)-u=0 equation is used where F(x) is
the CDF of the distribution and u is an observation from uniform distribution (0,1).
42 Noor Waseem and Shakila Bashir
We simulate 15 samples each of size 50 from the Pareto-Exponential distribution
with the specified values of parameters taking α=3 and λ=2. For upper record values
from Pareto-Exponential distribution we consider the upper record from each
sample of size 50.
Descriptive Statistics
Mean Geometric
Mean
Harmonic
Mean Median Variance
Standard
Deviation Skewness Kurtosis Minimum Maximum
1.935 1.807 1.678 1.807 0.515 0.717 0.29 -1.13 0.89 3.21
8. Conclusion
In this article we considered the Pareto-Exponential distribution derived by T-X
family of distributions and introduced the upper record values from Pareto-
Exponential (UPE) distribution. The UPE distribution is a two parameters
positively skewed continuous distribution. Various properties of the UPE
distribution have been derived. From the Fig. 4, it can be seen that the hazard
function of the UPE distribution is showing firstly increasing (IHR) and then
decreasing (DHR) trend. Measure of inequality and uncertainty named Renyi
Entropy, Lorenz and Bonferroni curves for UPE distribution have been derived.
Recurrence relation for the single and product moments of UPE distribution have
been derived, the recurrence relations can be used to find moments in a recursive
manner. Finally, a Monte Carlo simulation has been done by generating data of size
50 with sample size 15 and upper record has been noted. Descriptive measures of
the UPE distribution have calculated.
References
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44 Noor Waseem and Shakila Bashir
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Received: December 4, 2018; Published: January 11, 2019