Applied Mathematical Sciences, Vol. 12, 2018, no. 12, 567 - 579

HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2018.8454

A Topp-Leone Generator of Exponentiated Power

Lindley Distribution and Its Application

Sirinapa Aryuyuen

Department of Mathematics and Computer Science, Faculty of Science and

Technology, Rajamangala University of Technology Thanyaburi

Phatum Thani 12110, Thailand

Copyright © 2018 Sirinapa Aryuyuen. 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

A new framework for generating lifetime distributions is proposed, which is

called the Topp-Leone Exponentiated Power Lindley (TL-EPL) distribution. Sub-

models of the TL-EPL distribution, such as the Topp-Leone Power Lindley,

Topp-Leone Generalized Lindley, and Topp-Leone Lindley, are introduced. Some

statistical characteristics of the distributions are investigated (i.e., mean, variance,

and functions of survival, hazard, and quantile). The maximum likelihood

estimation is used to estimate the parameters of each distribution. Some real data

sets are fitted in order to illustrate the usefulness of the proposed distribution.

Keywords: Topp-Leone, exponentiated power Lindley, quantile, hazard,

lifetime data

1 Introduction

The Exponentiated Power Lindley (EPL) distribution with three parameters, as

proposed by Warahena-Liyanage and Pararai [18], will provide many applications

in different fields, such as engineer, biology and others. All parameters of EPL

distribution indexed to this distribution makes it more flexible in describing

different types of real data than its sub-models, i.e., the power Lindley (PL)

distribution [5], the generalized Lindley (GL) distribution [11], and the Lindley

(L) distribution [9]. The EPL distribution, due to its flexibility in accommodating

different forms of the hazard function, seems to be a suitable distribution that can

be applied to a variety of problems in fitting survival data. Ashour and Eltehiwy

568 Sirinapa Aryuyuen

[2] applied the EPL distribution to real data about flood levels and the active

repair times (h) for an airborne communication transceiver. The EPL distribution

provides better fit than the other distributions (i.e., PL, GL, L, exponential,

modified Weibull, Weibull distributions). Hence the data point from the EPL

distribution has better relationship and hence this distribution is good model for

life time data.

The EPL distribution is specified in terms of the probability density function

(pdf), cumulative density function (cdf), and quantile function [18], i.e.,

respectively

12

1( ) (1 ) 1 (1 )e e ,1 1

x xxg x x x

(1)

( ) 1 (1 )e ,1

xxG x

(2)

1/

1/1 1( ) 1 ( 1)(1 )e ,x

GQ u W u

(3)

where 0,x 0, 0 and 0. When u is distributed as the uniform on the

interval (0,1), and W (.) in the Lambert W function [3]. The probability functions

of the sub-models of the EPL distribution are presented. For 1 , we have the

pdf and cdf of the PL distribution with positive parameters and , i.e.,

2

1

1( ) (1 ) e1

xg x x x

and 1( ) 1 (1 )e .1

xG x x

For 1 , we have the pdf and cdf of the GL distribution with positive parameters

and , i.e., respectively,

12

2

(1 )( ) 1 (1 )e e

1 1

x xx xg x

and

2 ( ) 1 (1 )e .1

xG x x

For 1, we have the pdf and cdf of the L distribution with positive

parameter , i.e., respectively,

2

3( ) (1 )e1

xg x x

and 3

1( ) 1 e .

1

xxG x

A Topp-Leone generator of exponentiated power Lindley distribution 569

Statistical distributions are used to model the life of an item in order to study

its important properties. Proper distribution may provide useful information that

result in sound conclusions and decisions. When there is a need for more flexible

distributions, many researchers are about to use the new one with more

generalization. Recently, applying new generators for continuous distributions

became more interesting [14]. This methodology can improve on the goodness of

fit and determine tail properties. These features have been established by the

results of many generators such as the beta distribution and Topp-Leone (TL)

distribution.

The TL distribution is one of the continuous distributions that is attractive as a

generator. This distribution was proposed by Topp and Leone [16] for empirical

data with J-shaped histograms, such as powered band tool and automatic

calculating machine failures. Let T be a random variable which is distributed as

the TL distribution with parameter , which is denoted by T ~TL( ), the cdf,

and pdf of T respectively,

TL ( ) (2 )F t t t and

1 1

TL

2 (1 )(2 )( ) ,

1 (2 )

t t tf t

t t

(4)

where 0 1t and 0. Solving for the quadratic equation of 2 1/2 0t t u

(see [7]) obtains the quantile function of TL distribution, i.e.,

1 1/

TL TL( ) ( ) 1 1 ,Q t F u u (5)

where u is distributed as the uniform on the interval (0,1).

The TL distributions as a generator for continuous distributions are proposed

such as two-sided generalized Topp and Leone distribution [17], Topp-Leone

generalized exponential distribution [14], Topp-Leone Gumbel distribution [19],

and Generalized Topp-Leone family of distributions [10]. Generating a new

family of distribution requires two principal components, which are a generator

and a parent distribution (see [1, 7, 14]). Indeed, the pdf of a generator is

transformed into a new pdf through the cdf ( )G of a parent distribution. The TL

distribution is a continuous unimodal distribution with a wide range of

applications in reliability fields and is used for modeling lifetime phenomena,

which has a J-shaped density function with a bathtub-shaped hazard function [15].

In this article is proposed a new Topp-Leone generated family distribution

where the parent distribution is the EPL distribution. Sub-models of the proposed

distribution are studied. Some statistical characteristics of the distributions are

investigated. The maximum likelihood estimation (MLE) is used to estimate the

parameters of each distribution. Some real data sets are presented in order to

illustrate that the data fits by using the proposed distribution.

2 A Topp-Leone generated family distribution

Alzaatreh et al. [1] presented a method for generating a new family of distribution

570 Sirinapa Aryuyuen

with the following definition of a random variable 1 2[ , ],T c c

1 2 ,c c

and a random variable X with cdf ( ).G x Let W[ ( )]G x be a function of ( )G x and

satisfy these conditions: a) 1 2W[ ( )] [ , ],G x c c b) W[ ( )]G x is differentiable and

monotonically non-decreasing, and c) W[ ( )]G x a as x and W[ ( )]G x

b as .x Let T be a random variable of a generator distribution with pdf ( )r t

defined as 1 2[ , ].c c Let X be a continuous random variable with cdf ( ).G x Thus,

the cdf and pdf of a new family of distributions are given, respectively, as,

W[ ( )]

TL-G0

( ) ( ) d G x

F t r t t and TL-G ( ) {W[ ( )]} W[ ( )] .

df t r G x G x

dx

If a random variable T is distributed as the TL and bounded on [0,1]. Let X

be a continuous random variable with the TL-G distribution. The cdf can written

as,

TL-G ( ) ( ) [2 ( )] ,F x G x G x (6)

where 0 is a shape parameter. The associated pdf is,

1 1

TL-G ( ) 2 ( )[1 ( )] ( ) [2 ( )] ,f x g x G x G x G x (7)

where ( ) ( ) .g x dG x dx In addition, a TL random variable with finite support has

the same bounds as the cdf ( )G x of any other random variable. Therefore, the

relation of a random variable X having the TL distribution is 1( )X G T where

T ~ TL( ). Let G ( )Q be the quantile function of a parent distribution, by which

can be simulated the TL-G random variate from

1/

G (1 1 ),x Q u (8)

where 1/1 1 u is the quantile function of the TL distribution in (5). The

results obtained in Section 3 can be a new TL-G distribution.

The moment of the TL-G distribution can be computed from the probability

weighted moments order ( ,s r ) of the parent distribution. Let ( )G x be the cdf of

the parent distribution, then the ( , )s r th probability weighted moment of X (see

[12, 14]) will be,

1

, G

0

( ) ( ) ( )d ( ) d .s r s r s r

s r E X G X x G X g x x Q u u x

The moment of the TL-G distribution is,

G , 1

1

E ( ) ( 1) 2 ( ) .s i i

s i

i

X ii

(9)

A Topp-Leone generator of exponentiated power Lindley distribution 571

3 A new TL-G distribution

In this section, we introduce a new distribution, which is called the Topp-Leone

exponential power Lindley (TL-EPL) distribution.

3.1 The TL-EPL distribution

From the cdf in (6) and pdf in (7) of the TL-G distribution, let ( )g x and ( )G x be

the pdf and cdf of the EPL distribution (see, [2, 16]). Consequently, a random

variable X of the TL-EPL distribution, X ~TL-EPL( , , , ), has the cdf and

pdf as follows;

TL-EPL ( ) 1 (1 )e 2 1 (1 )e ,1 1

x xx xF x

(10)

1

21

TL-EPL

2( ) (1 ) 1 (1 )e e

1 1

x xxf x x x

1

1 1 (1 )e 2 1 (1 )e .1 1

x xx x

(11)

We define the hazard function of the TL-EPL distribution as follows,

1

TL-EPL( ) 1 (1 )e e 1 1 (1 )e1 1

x x xx xH x

12

1

TL-EPL

2(1 ) 2 1 (1 )e ( ) ,

1 1

xxx x S x

where TL-EPL ( )S x is the survival function of the TL-EPL distribution, i.e.,

TL-EPL ( ) 1 1 (1 )e 2 1 (1 )e .1 1

x xx xS x

The quantile function of the TL-EPL distribution is obtained by substituting

(3) for (8), i.e.,

1/

1/ 1/ 1

TL-EPL

1 1( ) 1 ( 1) 1 (1 1 ) e ,Q u W u

(12)

where u is distributed as the uniform on the interval (0,1), and ( )W is the

Lambert W function [3]. From the moment of the TL-G distribution in (9) and the

572 Sirinapa Aryuyuen

quantile function of the EPL distribution in (3), we obtain the mean and variance

of ,X i.e., respectively,

TL-EPL 1, 1

1

E ( ) ( 1) 2 ( ) ,i i

i

i

X ii

and

2

TL-EPL 2, 1 1, 1

1 1

( ) ( 1) 2 ( ) ( 1) 2 ( ) ,i i i i

i i

i i

V X i ii i

where

1/1

1/ 1

1, 1

0

1 11 ( 1)(1 )e d ,x i

i W u u x

and

2/1

1/ 1

2, 1

0

1 11 ( 1)(1 )e d .x i

i W u u x

Some pdf plots of the TL-EPL distribution are shown in Figure 1 and Figure 2.

Figure 1: Plots of the pdf of the TL-EPL distribution with (a) different values

of and (b) different values of

Figure 2: Plots of the pdf of the TL-EPL distribution with (a) different values

of and (b) different values of

A Topp-Leone generator of exponentiated power Lindley distribution 573

The pdf of the TL-EPL distribution is unimodal. It increases or decreases for

various values of the parameters giving the shapes obtained in Figure 2(a). In

Figure 1(b), for values of 1 and 1 the pdf seems almost symmetric.

3.2 Sub-models

The EPL distribution with three parameters , , and has three sub-models

(see [2, 16]). Thus, the TL-EPL distribution has three sub-models, which are

presented as follows.

3.2.1 The Topp-Leone power Lindley distribution

For X ~TL-EPL( , , , ), when 1 , we obtain the Topp-Leone power

Lindley (TL-PL) distribution, which is denoted by X ~TL-PL( , , ). The TL-

PL distribution has the cdf and pdf, i.e., respectively,

2 2

TL-PL ( ) 1 (1 ) e ,1

xxF x

(13)

12

1 2 2 2

TL-PL

2( ) (1 ) (1 ) 1 (1 ) e e .

1 1 1

x xx xf x x x

(14)

3.2.2 The Topp-Leone generalized Lindley distribution

For X ~TL-EPL( , , , ), when 1 , we obtain the Topp-Leone generalized

Lindley (TL-GL) distribution, which is denoted by X ~ TL-EL ( , , ). The cdf

and pdf of the TL-GL distribution are

TL-GL ( ) 1 (1 )e 2 1 (1 )e ,1 1

x xx xF x

(15)

12

TL-GL

2( ) (1 ) 1 (1 )e e

1 1

x xxf x x

1

1 1 (1 )e 2 1 (1 )e .1 1

x xx x

(16)

3.2.3 The Topp-Leone Lindley distribution

For X ~TL-EPL( , , , ), when 1 , we obtain the Topp-Leone Lindley

(TL-L) distribution, which is denoted by X ~TL-L( , ). The TL-L distribution

has the cdf and pdf in (17) and (18), respectively,

2 2

TL-PL ( ) 1 (1 ) e ,1

xxF x

(17)

574 Sirinapa Aryuyuen

121 2 2 2

TL-PL

2( ) (1 ) (1 ) 1 (1 ) e e .

1 1 1

x xx xf x x x

(18)

3.4 Maximum likelihood estimation

In this section, we describe the MLE procedure to obtain the estimated value of

the parameters of the TL-EPL based on the random sample 1 2( , ,..., )nx x x x of

size .n Let iX for 1,2,...,i n be independent and identically distributed. The

log-likelihood function of iX ~TL-EPL( , , , ), on the observed sample x is

log ( , , , | ) ( | )L x x given by,

1 1 1

( | ) log(1 ) ( 1) log log 2 log logn n n

i i i

i i i

x x x x n n n

( , ; )

1

2 log log log( 1) ( 1) log 2i

n

x

i

n n n

( , ; ) ( , ; )

1 1

( 1) log log 1 ,i i

n n

x x

i i

(19)

where ( , ; ) 1 (1 ( 1))e .i

i

x

x ix

By differentiating ( | ),x the partial derivatives of ( | )x with respect to

, , and are given by,

( , ; ) ( , ; )

1 1

( | )log 2 log ,

i i

n n

x x

i i

x n

(20)

( , ; )

1 1

( | ) 2( 1) log 2

1 i

n n

x i

i i

x n nx

( , ; ) ( , ; )

1 1

log 1 ( 1) log ,i i

n n

x x

i i

(21)

1 1 1

( | )log(1 ) log

n n n

i i i

i i i

xx x x

( , ; ) ( , ; )

1 1

( 1) log log 1 ,i i

n n

x x

i i

n

(22)

( , ; ) ( , ; )

1 1

( | )log log 1

i i

n n

x x

i i

x

( , ; )

1

( 1) log 2 .i

n

x

i

n

(23)

A Topp-Leone generator of exponentiated power Lindley distribution 575

The expression of these differential equations in (19)-(23) are not in the closed

form. In this study, the parameter estimates of ˆ ˆˆ , , and ˆ , can be obtained by

using the numerical optimization with the nlm function in the R language [13],

which R code for the MLE of the TL-EPL distribution are as follows:

#The TL exponentiated power Lindley (TL-EPL) distribution x <- c(...,…,…) logL_TL_EPL <- function(theta0,x) { beta <- (theta0[1]) lambda <-(theta0[2]) omega <- (theta0[3]) alpha <- (theta0[4]) G <- (1-(1+(beta*x^lambda)/(beta+1))*exp(-beta*x^lambda))^omega g1 <- ((lambda*beta^2*omega)/(beta+1))*(1+x^lambda) *(x^(lambda-1))*exp(-beta*x^lambda) g2 <- (1-(1+(beta*x^lambda)/(beta+1)) *exp(-beta*x^lambda))^(omega-1) g <- g1*g2 logL <- -log(2*alpha)-log(g)-log(1-G)-(alpha-1)*log(G)-(alpha-1) *log(2-G) return(sum(logL)) } theta0 <- c(...,...,...,...) Est<-nlm(logL_TL_EPL,theta0,x) 3.5 Application study In this section we provide a data analysis in order to assess the goodness-of-fit of

the TL-EPL model with two real data sets. In addition, the sub-models of the TL-

EPL distribution (i.e., TL-PL, TL-GL, and TL-L distributions), and the EPL

distribution and its sub-models (e.g., PL, GL, L distribution) are considered. The

parameter (s) of each distribution are estimated by the MLE method. To verify

which distribution fits better with real data sets, the Komogorov-Smirnov test (KS

test) will be employed. Other criteria including the Akaike Information Criterion

(AIC) and Bayesian Information Criterion (BIC) are considered, i.e., AIC

2 2log L p and BIC 2log( ) log( ),L p n where n is the sample size and

p is the number of parameters of each distribution.

The first dataset consists of 20 observations with the respect to maximum

flood level data to see how the new model works in practice. The data has been

obtained from Dumonceaux and Antle (see [2, 4]), as shown in Table 1. For the

results of Table 2 , the KS test indicates that the TL-EPL distribution is a strong

competitor compared to other distributions.

576 Sirinapa Aryuyuen

Table 1. Maximum flood levels data from Dumonceaux and Antle [4].

0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.3235

0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265

Table 2. Values of parameter estimates and Statistical criteria concerning

maximum flood level data.

Distribution Parameter estimate values Statistical values

-Log AIC BIC KS p-value

TL-EPL 4.5801 7.7322 0.6639 21.0802 16.1471 40.2942 47.0497 0.1171 0.9466

TL-PL 547.9893 6.7696 0.6472 - 16.2989 38.5978 43.6644 0.1247 0.9148

TL-GL 109.269 6.0083 - 0.6412 16.1725 38.3450 43.4116 0.1171 0.9467

TL-L 53.0705 6.1417 - - 16.1462 36.2924 39.6702 0.1228 0.9236

EPL - 13.2699 0.6129 847.2759 16.3058 38.6116 43.6782 0.1284 0.8968

PL - 15.2883 3.5207 - 13.2487 30.4974 33.8752 0.1991 0.4060

GL - 11.6817 - 54.9888 16.1547 36.3094 39.6872 0.1223 0.9258

L - 2.9602 - - 2.1792 6.3584 8.0473 0.4532 0.0005

For a second application, we analyze a real data set on the active repair times

(h) for an airborne communication transceiver. The data is given in Table 3, and

its source is Jorgensen (see [2, 8]). For the KS test in Table 4, the TL-EPL

distribution is the best distribution corresponding to a high p-value of the KS test.

Table 3. Active repair time (h) for an airborne communication transceiver [8].

0.50, 0.60, 0.60, 0.70, 0.70, 0.70, 0.80, 0.80, 1.00, 1.00, 1.00, 1.00, 1.10, 1.30,

1.50, 1.50, 1.50, 1.50, 2.00, 2.00, 2.20, 2.50, 2.70, 3.00, 3.00, 3.30, 4.00, 4.00,

4.50, 4.70, 5.00, 5.40, 5.40, 7.00, 7.50, 8.80, 9.00, 10.20, 22.00, 24.50

Table 4. Values of parameter estimates and Statistical criteria of the active repair

time data.

Distribution Parameter estimate values Statistical values

-Log AIC BIC KS p-value

TL-EPL 9.1321 7.0096 0.0966 300.5382 89.6808 187.3616 194.1171 0.0950 0.8630

TL-PL 2,755.4010 4.3244 0.1390 - 89.5073 185.0146 190.0812 0.0955 0.8592

TL-GL 1,025.1370 0.1403 - 0.0109 94.0351 194.0702 199.1368 0.1525 0.3097

TL-L 0.6692 0.2074 - - 98.9111 201.8222 205.2000 0.1678 0.2100

EPL - 8.9099 0.1293 6,196.2090 89.4702 184.9404 190.007 0.0959 0.8552

PL - 0.5867 0.7988 - 95.9425 195.885 199.2628 0.1346 0.4637

GL - 0.3588 - 0.7460 97.9109 199.8218 203.1996 0.166 0.2201

L - 0.4242 - - 98.7913 199.5826 201.2715 0.2157 0.0484

A Topp-Leone generator of exponentiated power Lindley distribution 577

4 Conclusion

The Topp-Leone Exponentiated Power Lindley (TL-EPL) distribution is

proposed, which has the Topp-Leone Power Lindley, Topp-Leone Generalized

Lindley, and Topp-Leone Lindley, are sub-model. Some statistical characteristics

of the distributions are investigated. The maximum likelihood estimation is used

to estimate the parameters of each distribution. We provide a data analysis in

order to assess the goodness-of-fit of the TL-EPL model with two real data sets.

The results of KS test indicates that the TL-EPL distribution is a strong

competitor compared to other distributions (i.e., TL-PL, TL-GL, TL-L, PL, GL,

and L distributions).

Acknowledgements. We wish to gratefully acknowledge the referee of this paper

who helped to clarify and improve its presentation.

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Received: April 16, 2018; Published: May 23, 2018