record values from t-x family of pareto-exponential ...€¦ · pareto-exponential distribution...

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.


(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


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


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)


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


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).


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

[1] M. Ahsanullah, Some characterizations of exponential distribution by upper

record values, Pakistan Journal of Statistics, 26 (2010), 69-75.

[2] M. Ahsanullah, On the record values of the classical Pareto distribution,

Pakistan Journal of Statististics, 13 (1997), 9-15.

[3] A. Alzaatreh, F. Famoye and C. Lee, Gamma-Pareto distribution and its

applications, Journal of Modern Applied Statistical Methods, 11 (2012), 78-

94. https://doi.org/10.22237/jmasm/1335845160

[4] N. Balakrishnan and M. Ahsanullah, Recurrence relations for single and

product moments of record values from generalized Pareto distribution,

Communications in Statistics-Theory and Methods, 23 (1994), 2841-2852.

https://doi.org/10.1080/03610929408831419

https://doi.org/10.22237/jmasm/1335845160

https://doi.org/10.1080/03610929408831419


[5] S. Bashir and M. Ahmad, Record Values from Size-Biased Pareto

Distribution and a Characterization, International Journal of Engineering

Research and General Science, 2 (2014), 2091-2730.

[6] S. Bashir and K. Akhtar, Record Values on The Size-Biased Student’s t

Distribution, International Journal of Innovative Science, Engineering &

Technology, 1 (2014), 2348-7968.

[7] M. Chacko and L. Muraleedharan, Inference Based on k-Record Values from

Generalized Exponential Distribution, Statistica, 78 (2018), 37-56.

[8] K.N. Chandler, The distribution and frequency of record values, Journal of

Royal Society, Series B (Methodological), 14 (1952), 220-228.

https://doi.org/10.1111/j.2517-6161.1952.tb00115.x

[9] S. Dey, T. Dey and D. Luckett, Statistical inference for the generalized

Rayleigh distribution based on upper record values, Model Assisted Statistics

and Applications, 12 (2017), 1no. 1, 5-29.

https://doi.org/10.3233/mas-160380

[10] R.U. Khan, A. Kulshrestha and M.A. Khan, Relations for moments of k-th

record values from exponential-Weibull lifetime distribution and a

characterization, Journal of the Egyptian Mathematical Society, 23 (2015),

558-562. https://doi.org/10.1016/j.joems.2014.11.003

[11] D. Kumar, K-th record values from Dagum distribution and characterization,

Discussiones Mathematicae Probability and Statistics, 36 (2016), 25-41.

https://doi.org/10.7151/dmps.1183

[12] S. Minimol and P.Y. Thomas, On some properties of Makeham distribution

using generalized record values and its characterization, Brazilian Journal of

Probability and Statistics, 27 (2013), no. 4, 487-501.

https://doi.org/10.1214/11-bjps178

[13] J. Paul and P.Y. Thomas, On generalized upper (k) record values from

Weibull distribution, Statistica, 75 (2015), 313-330.

[14] J. Saran and K. Nain, Relationships for moments of kth record values from

doubly truncated pth order exponential and generalized Weibull distributions,

ProbStat Forum, 5 (2012), 142-149.

[15] K.S. Sultan, Record values from the modified Weibull distribution and

applications, International Mathematical Forum, 2 (2007), 2045-2054.

https://doi.org/10.12988/imf.2007.07184

https://doi.org/10.1111/j.2517-6161.1952.tb00115.x

https://doi.org/10.3233/mas-160380

https://doi.org/10.1016/j.joems.2014.11.003

https://doi.org/10.7151/dmps.1183

https://doi.org/10.1214/11-bjps178

https://doi.org/10.12988/imf.2007.07184


[16] K.S. Sultan and M.E. Moshref, Record values from generalized Pareto

distribution and associated inference, Metrika, 51 (2000), 105-116.

https://doi.org/10.1007/s001840000025

Received: December 4, 2018; Published: January 11, 2019

https://doi.org/10.1007/s001840000025

record values from t-x family of pareto-exponential ...€¦ · pareto-exponential distribution...

Documents