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41
Harris Generalized Linear Exponential Distribution and Its Applications
Albin Paula, K.K. Joseb •
aSt.Thomas College Palai, Kerala, India
albinpaul0333@gmail.com bSchool of Mathematics and Statistics, Mahatma Gandhi University, Kottayam, Kerala, India. kkj.smsda.mgu@gmail.com
Abstract
In this paper we consider an extension of the linear exponential distribution based on the Harris generalization method, which includes some of the life time distribution as sub models. Along with the four parameter generalization namely Harris generalized linear exponential distribution, we derive some of its properties such as moments, quantiles and moment generating function. A compound form expression of the density function is given. For the given model the R'enyi entropy, mean residual life and the distribution of order statistics is derived. The problem of estimation of parameters is considered and validated with respect to two real data sets.
Keywords: Harris generalization, Linear exponential distribution, Order Statistics, Moment generating function, Entropy, Mean residual life, Data analysis
1 Introduction
Introduction of new parameters to the well established classical distributions may result in more flexible new families of distributions. Several extended forms of distributions have been studied by many researchers like Azzalini (1985), Marshall and olkin (1997), Ferreira and Steel (2007), Jose et al
(2010), Krishna et al (2013), Jose and Sivdas (2015) etc. Recently Aly and Benkherouf (2011) introduced a method for developing new classes of distributions by adding two new parameters to an existing distribution, which includes the baseline distribution as a special case and gives more flexible models for various types of data. This method is based on probability generating function introduced by Harris (1948). Hence the resulting family of distributions is known as Harris Generalized (HG) family of distributions. This family of distributions can be considered as a generalization of Marshall Olkin family of distributions introduced by Marshall and olkin (1997). Some properties and applications of HG family of distributions are studied by Aly and Benkherouf
(2011), Batsidis and lemonte (2014) and Cordeiro et al (2015) etc. Aly and Benkherouf (2011) derived the general structure of HG family of distributions as follows.
The survival function of HG family of distributions is given by
1
G(x; 8,r) = ( e№\r)r, 0<8 <™, r> 0, 8 = 1-8 (1.1)
where F(.) is the survival function of the baseline distribution. The corresponding density function is
1
g(x) =-9Tf(x) r+1 , x> 0 (1.2)
(i-s(FM)r)[ r >
If r=1 in equations (1.1) and (1.2), reduces to the corresponding survival function and density function of Marshall-Olkin family of distributions. The density function of HG family of distributions can be expressed as a linear combination of exponentiated family of distributions as given in Batsidis and lemonte (2014) and Barreto-Souza et al (2013) as follows.
For 8 E (0,1)
g(x) = f(x)zr=oWi(F(x))ri (1.3)
where Wi = Wi(8,r) = Old1
For 8 > 1
9(x)=f(x)YZoVi(F(x))n (1.4)
where Pi = v^O.r) = (-1)i8-1Y~=1 ( )
From (1.3) and (1.4) it is clear that the HG family of distributions can be expressed as the baseline distribution f(x), multiplied by an infinite power series which differ only for the coefficients.
The aim of this paper is to study a new univariate family of distribution based on the Harris generalization method. The contents are organized as follows. In Section 2 we discuss the Linear exponential distribution. In Section 3 we introduce the Harris generalized linear exponential distribution. In Section 4 we represent the Harris generalized linear exponential distribution as a compound distribution with exponential density. In Section 5 and 6 we evaluate the Entropy and Mean residual life. Section 7 gives the distribution of order statistics. Section 8 and 9 discuss the maximum likelihood estimation of the parameters and an application to a real data set.
2 Linear Exponential Distribution
The Linear Exponential (LE) distribution is an important distribution that has rich variety of applications for modeling life time data. The LE distribution is also known as Linear Failure rate distribution. The LE distribution contains Rayleigh and Exponential distribution as special cases and they are well known in the literature for a variety of applications.
From Lai et al (2006) and Zang et al (2005), the LE distribution models phenomenon with increasing failure rate, the accuracy of the statistical procedures depends on the probability model or distributions which are considered for the analysis. Recently there has been a renewed interest in the study of extended versions of conventional classical distributions. Since the real data is affected by various factors, the statistical models derived using the extended form of distribution shows more significant applications in reliability, medical science, finance, economics etc. The LE distribution has many applications in applied statistics, reliability analysis, medical studies (See Carbone et al (1967), Broadbent (1958)). Recently many studies have been done on LE distribution and its generalizations by introducing additional parameters (See Mahmoud and Alam (2010), Cordeiro et al (2015), Nadarajah et al (2014) etc).
The LE distribution with the parameters p1 and p2, (LE(^1, p2)) has the following cumulative distribution function
F(x; Pi,Pz) = 1-exp (-^x-^x2) (2.1)
Albin Paul, K.K. Jose RT&A, No 1 (61) HGLE DISTRIBUTION AND ITS APPLICATIONS_Volume l6, March 2°21
If we put p2=0 in (2.1) we can obtain the exponential distribution with parameter p1 and if we put pr=0 in (2.1) we get the Rayleigh distribution with parameter p2.
The probability density function (pdf) of LE^, p2) distribution is given by
f(x-,p1,p2) = (p1+p2x)exV(-p1x-^fx2), x > 0, ,p2 > 0.
3 Harris Generalized Linear Exponential Distribution
By applying the method given in Aly and Benkherouf (2011) Harris Extended Linear exponential distribution is introduced by Batsidis and lemonte (2014). Here we call this family of distributions as Harris Generalized distribution. By taking LE distribution as the baseline distribution, we get the Harris generalized Linear exponential (HGLE) distribution. The application of this family of distribution in the context of reliability test plan were studied by JoseandPaul(2018).
The substitution of the survival function of LE(^1, p2) distribution in (1.1) gives the HGLE distribution with parameters 9, r, p1, p2 and is denoted as HGLE(0, r, p1, p2), The survival function of the HGLE distribution is obtained as
1-9exp(-r(p1x+^2x
The pdf of HGLE distribution is given by
eexpi-ri^x+^x2))
sv = I A , )L) (3.1)
0r(p1+p2x)exp(-(PiX+^x2) ) g(X) = -V-_1 (3.2)
where x > 0, 9 > 0, r > 0, p1> 0, p2> 0, 9 = 1-9.
The corresponding hazard rate function is given by
h(x) =-fl+Ml--(3.3)
1-eexpi -r(p1x+^2x2) )
By taking r=1 in (3.2) we get the Marshall-Olkin Linear Exponential distribution (MOLE), and when r=1 and 9 = 1, (3.2) reduces to the LE distribution. When p2=0, (3.2) gives the Harris Extended Exponential distribution as given in Pinho et al (2015). In the same way it will reduce to Marshall Olkin Exponential distribution and Exponential distribution when r=1 and r=1, 9 = 1 respectively. If p1 = 0 this distribution reduces to the Harris extended form of Rayleigh distribution and it includes the Marshall-Olkin generalization of Rayleigh distribution (if r=1), Rayleigh distribution (if 9 = 1) as its special cases.
The quantile function of HGLE distribution can be obtained by inverting the cdf given in (2.1), and is obtained as
x = Sl
XP H2
(3.4)
^j1+^llog(9 + 9(1-p)-r)-1 where 0 <p < 1.
Some possible shapes of the pdf of HGLE distribution for different values of parameters are given in Fig. 1.
Some possible shapes of the hazard rate function of the distribution for different values of parameters are given in Fig. 2
Albin Paul, K.K. Jose RT&A, No 1 (61)
HGLE DISTRIBUTION AND ITS APPLICATIONS Volume 16, March 2°21
Figure 1: pdf of HGLE Distribution for different values of parameters.
Figure 2: Hazard rate function of HGLE Distribution for different values of parameters.
Theorem: 1 The pdf of HGLE distribution can be represented as a linear combination of Linear exponential density function as
g(x) = T.T=o VifwifoW
where f{p*p^)(x) follows LE distribution with parameters ft = (ri + 1)ft1 and
01,1
ft = (ri + 1)ft.
2 ( ' <■ + 1)ft2 .
Proof: Consider the pdf of HGLE distribution given in (3.2), For e < 1,
the expression given by (3.2) can be expanded using the negative binomial power series as
follows,
m ' i + r-1
¿(01 + ftx)exp (- (ftx + f^x2)) ) §iexP (- (?1X + ft
,, I (i + r-1
= ) ((ri + 1)(ft1 + ^exv (-(ri + 1) (ftx + f*2)))
i=0
mel (i + r-1\
Here for 8 < 1, m=—li ) (3.5)
and f(pi,p*)(x) = ((ri + 1)(fi1 + ftx)exp (-(ri + 1) (fax + i2x2)))
where f(pi,p*)(x) follows LE distribution with parameters ft = (ri + 1)ft1 and ft = (r i + 1) ft.
For 8 > 1, let us take 8 = 8-1 so that 0<81<1. On simplification (3.2) reduces to
g{x)= M+My
t -
(1-91(1-yr)) r
where y = exp (- ((ft1x + ^x2))) and 8± = 1- 81. Since the denominator lies in the interval (0,1), we can use the negative binomial power series expansion, then we have
g(x) =81(^1+ ftx)y Zf=0 8J (1 - yr) )
= 81(^1+ ftx)y Zy=0 Z>=0 (-1) (81)jyri ^j1" ^(i ) Interchanging the order of summation and substituting y we get
{(r i + 1)(ft1 + ftx)exp (-(ri + 1) (ftx + yx2))}
which leads to
g(x) = T,T=o VifusifàW where
Vi = 0ir-)(I,м(OlУ(] + r )(]i I) (3.6)
and
f(.Pir2)(x) = ((ri + 1)(ft1 + ft2x)exp (-(ri + 1) (ftx + y x2)))
which is the pdf of LE distribution with parameters ft = (ri + 1)ft and ft = (ri + 1)ft.
Theorem: 2 The moment generating function of HGLE distribution, denoted as M(t) is given by M(t) = tfiWsfi and Msri can be expressed as
as,ri = S™ exp(tx)(ri + l)(ft + ftx)exp (-(ri + 1) (ftx + ^x2)") dx
For 8 > 1, Vi = 0-1-)l,7=o 81 ^ ^i) where 81 =1,0 < 81 < 1.
Proof:
We have M(t) = exp(tx)f(x; 8,r,ft,ft)dx
00 O / \
= J exp(tx) £ V¡(ri + 1)(ft + ftx)exp (-(ri + 1) (ftx + ^x2) ) dx
0 i=o V J
(by using theorem 1)
00
= ^Vif (ri + 1)(ft + ftx)exp(-((ri + 1)ft - t)x)exp (-(ri + 1) (jx2) I dx
i=o 0 ^ '
Albin Paul, K.K. Jose RT&A, No 1 (61) HGLE DISTRIBUTION AND ITS APPLICATIONS_Volume 16, March 2°21
consider
s s
J (ri + 1) ftx2sexp(-((ri + 1)ft - t)x)dx + J (ri + 1)ft2x2s+1exp(-((ri + 1)ft - t)x) dx
00
(r i + 1)ftF(2s + 1) (ri + 1)ftF(2s + 2)
r, \2s+1+ ,, \ 2s+2 ' t — ft1
((ri + 1)ft - t) ((ri + 1)ft - t)
Then
M(t) = LS=L0 mLS=o ^^(-Ds(«^Tpi + f+T^" — 01
where mi can be chosen suitably for 6 — 1 and 8 > 1 as given in equation (3.5) and (3.6). Theorem: 3
If X has HGLE(r, ft, ft, 9) distribution. The the kth moment ofX denoted by nk is given by,
,, _ Vs Vs 0-qif1 + r ^(-(ri+1)Ef) ( PiT(k + 2s+1) , P2T(k + 2s+2) \ = U=° Ls=0 OrO y t ) s] (^D^s^ + ^i+^i)^2)
Proof:
We have,
Hk = f0 xk-^-^fdx J0 xkg(x)dx
(L-eexp(-r(Pi+£2X2j)) r
(1 +r-1\
i )9iexV(-(ri + 1)(ftx + ^x2))dx
Interchanging the integration and summation we get
(1+r-1\
i ) 9r§i fj0 xk(ft + ftx)exp (-(ri + 1) (ftx + ^x2)) dx
1 + -1 s
= LS=o ( i ) ew LS=o (-(ri+sf22 (f0s ftxk+2sexp(-(ri + 1)(01x))dx +
s2
fo ftxk+2s+1exp(-(ri + 1)(jS1x))dx)
'1 + r
_ Vs Vs a-Ei ■ I (-(ri + 1)~r) I PiT(k + 2s+1) , P2T(k + 2s+2)
= Li = 0 LS = 0 I l I ^ ( r, . N(ft+2S+i) + ,, . .„ ,(ft+2S+2)
y / \((n+1)Piy ((n+1)Pi)
This completes the proof of the theorem.
4 Compounding
Ghitany et al (2005), Ghitany and Kotz (2007) and Krishna et al (2013) expressed Marshall Olkin extended forms of distributions Marshall and olkin (1997) as compound distributions with exponential as mixing density. This gives a new parametric family of distributions in terms of existing ones.
Let F(x/a),x E%a E% be the conditional survival function of a continuous random variable X given . Let follows a distribution with probability density function m(a). A distribution with survival function
F(x) = fS^(x/a)m(a) da, x Eft is called a compound distribution with mixing density m( a).
The following theorem shows that under suitable conditions the HGLE distribution can be obtained as a compound distribution.
Theorem: 4 Let X be a continuous random variable with conditional pdf given by F(x/a) =
Albin Paul, K.K. Jose RT&A, No 1 (61) HGLE DISTRIBUTION AND ITS APPLICATIONS_V°lume 16, March 2021
(exp^-r^ftx + ljx2))) -lja, x,r,ft,ft,a>0.
Let a follows an exponential distribution with with pdf given by m(a) = 8e-ea, 8, a > 0. Then the proportional failure rate model of the compound distribution of X becomes the HGLE (r, 8, p1, ft) distribution.
Proof: For all x > 0, ft, ft, 8 > 0, the unconditional survival function of x is given by F(x) = f™ F(x/a)m(a)da
— ' • "-x + —x2))) — 1)a exp(-a8) da
F(x) —
uexp(-r(p 1x+^x2)) l-aexp^-riPiX+ÎY-x2))
Let us take the proportional failure rate model of F(x), Then
_ _ i G(x) — F(x)r
(uexp( -ripix+^x2) )
-L-
1-uexp( -r(/j1x+^x2
which is the survival function of a random variable with HGLE distribution with parameter
(r.e.ft.ft).
5 Renyi Entropy
Numerous measures of entropy are discussed and studied by many researchers and these Entropy measures has been used in various situations of science and technology especially in communications engineering and information technology. The entropy of a random variable X with density function g(x) is a measure of uncertainty. The Rényi entropy is defined by
Ig(ô) — (1 — ô)-Hog-œgs(x)dx
where 5 > 0 and 5^1
1
-r\PiX+^X2
wexp( —I --
F(x) — ■
(17=0 17=0 ZU 8~r(i — ")k(5)
IJS) — (1- 5)-1log
k=0 ¿,s=0 ¿.1=0 8 8 rsi+y.
f 1 y v_2 ) r(S+2s-t + 1)
y ( 1) ((S+rk)Pif+2S-l+1) J
Table 1 displays the Renyi entropy for HGLE distribution at S = 2.0,2.5,3.0, ft = 0.5, ft = 0.9, r = 1.2 and for different choices of 8 > 1.
u
8 Iq(5)5 — 2.0 Iq(5) 5 — 2.5 Iq(5) 5 — 3.0
1.5 0.8290 0.7549 0.7336
2.0 0.8530 0.8003 0.7783
2.5 0.8661 0.8228 0.7995
3.0 0.8731 0.8342 0.8094
3.5 0.8765 0.8396 0.8134
4.0 0.8777 0.8415 0.8141
Table 1: Renyi Entropy of HGLE Distribution at 5 — 2.0,2.5,3.0, r — 1.2, ft — 0.5, ft — 0.9
6 Mean Residual Life
The expected additional life time given that a component has survived until time t is called mean residual life (MRL). The importance of MRL function in reliability and survival analysis is that it describes the aging process. Also MRL function uniquely determines its distribution function. The MRL of a random variable X representing life of a component is given as follows.
MR(t)^-^Jt°° G(x)dx, t>0
The MRL function of a lifetime random variable X with HGLE distribution is given by
i i
f^gexJr^+^fly f 9exjr(pix+?±x2f\y
MR(t) = I -^-^ ) f I -^-) dx
Table 2 displays MRL function for HGLE distribution at point t=0.2, 0.5, 0.9, r = 1.2, ft = 0.5, ft = 0.9 and for different choices of parameter 8.
8 MRL,t=0.2 MRL,t=0.5 MRL,t=0.9
1.0 0.8087 0.6906 0.5735
1.5 0.9191 0.7676 0.6143
2.0 1.0034 0.8302 0.6502
2.5 1.0716 0.8829 0.6823
3.0 1.1288 0.9284 0.7114
3.5 1.1781 0.9685 0.7379
Table 2: Mean residual life of HGLE Distribution at r = 1.2, ft = 0.5, ft = 0.9
7 Order Statistics
Let X1,X2,...,Xn be a random sample taken from HGLE distribution and X1.n,X2.n,...,Xn.n be the corresponding order statistics. We derive the pdf of ith order statistics XL.n which is denoted as 9i-.n(x), and express it as a linear combination of HGLE density function. We have the general formula for the pdf of the h order statistics as follows
3i.n(x) = ii-1)n^n-i)]9(x){G(x))n~L{1 - G(x))l~1 By using Binomial expansion, we get
at.n(x) = y-^gCx) X(-1) C -1) (G~(x))n+J-L
(i — iy.(n—iy. ¿-ij=°
On simplification gim(x) reduces to
1 I Ä«^
i - 1
(i-1).(n-i).^i=0 \ Î J (n+j-i+1)'
exp(-(n+j-L + 1)(ß1x+^2x2)r)
(n+j-i + l)(ßi+ß2x)
1 T P2aJ ^(n+j-i+1)
1X-<—2 '
[l-9exp(-(ßix+^2x2)r) f i-l
= ( i-l)!(n-i)! ( 7 ) J+jli + 1) 9n,j,i(x)
where gn,j,i(x) is the density function of HGLE distribution with parameters r(n + j — i + 1)-1, 8, (n+j-i + 1)ft, (n + j — i + 1)ft.
The kth moment of ith order statistic of HGLE distribution can be derived by using Theorem 3 as the distribution of order statistics can be expressed as the linear combination of HGLE density function. Consider the asymptotic distribution of the first order statistic X1:n and nth order statistic Xn:n. y using the asymptotic results for X1:n and Xn:n (Arnold et al (1992), Kotz and Nadarajah (2001)),
we can find the limiting distribution of extreme order statistics. We have
limP(X1.n <a*n + b*nt) = 1- exp(-t-a), t>0, a > 0, (7.1)
of Weibull type, where a*n = G-1(0) and b*n = G-1 Q - G-1(0) if and only if G-1(0) is finite and, for all > 0 and > 0,
l imG(°-1™+et] = ta. (7.2)
6^0+ G(G-1(0) + 6) V '
For the maximal order statistics Xn.n, we have
limP(Xnn < an + bnt) = exp(-e-t), -™<t<™ (7.3)
n^™
of extreme value type, where a„ = G-1(1 — - ) and b„ = —1— if
Jr n \ nj n ng(an)
lim — = 0. (7.4)
x^™ dx \h(x)/
The selection of the norming constants an and bn are not unique but depend on G. In general we use the result given in (Arnold et al (1992)) and the same used in (Ghitany and Kotz (2007)). The following theorem gives the limiting distributions of the smallest and largest order statistics from the HGLE distribution. Theorem: 5
Let X1:n and Xn.n be, respectively, the smallest and largest order statisticsfrom HGLE(8, r, ft, ft) distribution. Then
(i). limP[Xln < b^t] = 1- exp(-t), t > 0, where b*n = G-1 (-) and G-1(.) is given by (3.4).
(ii). limP{Xnn < an + bnt] = exp(-e-t), —x> < t < x>, where an = G-1 (1 — i), bn = —,
n^™ \ n) ng(an)
and g(.) and G-1(.), respectively are given by (3.2) and (3.4). Proof:
( i). For HGLE distribution G-1(0) = 0 which is finite and by using L Hospitals rule
G(G-1(0) + 6t) g(6t)
lim ———r = t lim —- = t.
6^o+ g(f-1(o)+6) 6^o+ g(t)
From (7.2) we have a=1 and the asymptotic distribution of X1:n is of Weibull type, where
ъn = G-^Q=-lll+l-i,og(e- + e(l-nГ)-l
KnJ P2
Hence ( ) follows from (7.1) and (7.2).
(ii). For HGLE distribution, by using Von Mises sufficient condition for the weak convergence and the properties given in ([?]), we get,
lim M
x^G-1(l) dx \h(x)/ = \im
(r(l — в)exp (—r (ß1x + ^x2))^
(l—(1—e)exp(—r(ßlx+£2x2j)ß2)
y (ßl+ß2x)2 y
=О
where an = G 1(1 — n -), G ) given in (3.4) and bn = —-—-, where bn can be obtained by using
n3(an)
an and (3.2). Hence, statement (ii) follows from (7.3) and (7.4).
8 Estimation
In this section we consider maximum likelihood estimation for a given sample X1,X2,.. .,Xn. Then the log likelihood function is given by
logL = niog0 + Tn=0 logtfi + ftXi) — m=0 (ti* +yxí) — (l+1) Yn=0 log(l — ~eexp (—r (ftXi + ^x2))
The maximum likelihood estimates can be obtained by solving the equations oiogL = o, = o, = o, = 0. These equations are non-linear and can be solved
0r d-1 d-2 09 n
iteratively using nlm program in R software.
9 Data Analysis
Example 1
Consider the data set which gives the survival times for 121 breast cancer patients treated over the period 1929-1938 (Boag (1984), Lawless (2003)). We compare the HGLE distribution with two other distributions LE distribution with cdf given in (2.1) and MOLE distribution. The survival function of MOLE distribution can be derived by setting r=1 in (3.1). For the given data set, we estimate the unknown parameter of each distribution by the maximum likelihood method, with these obtained estimates we obtain the values of Kolmogrov Smirnov (K-S) statistics and p value. From the values given in Table 3, we observe that HGLE distribution is a competitive distribution compared with other two distributions.
Model Parameters Estimates Log likelihood K-S Statistics p value
LE ß1 0.0155 579.72 0.0826 0.3816
ß2 0.000186
MOLE ß1 0.0029963
ß2 0.0000612 579.456 0.0563 0.8354
Q 2.360
HGLE r 3.800
ß1 0.0199 578.780 0.0542 0.8698
ß2 0.000135
Q 2.099
Table 3: Fitting for the LE, MOLE and HGLE Distribution Example 2
Consider the data set given in Chhikara and Folks (1989) and Lawless (2003), the data on repair times (in hours) for 46 failures of an airbone communications receiver and here we compare the HGLE distribution with MOLE distribution. Table 4 gives the MLE's of the fitted models to the current data with the K-S statistics and p- value. From the given table we can conclude that the HGlE distribution better fits the given data than the MOLE distribution.
Model Parameters Estimates Log likelihood K-S Statistics p value
MOLE ß1 0.00535
ß2 0.00324 103.5308 0.0955 0.785
Q 0.0163
HGLE r 2.1513
ß1 0.000401 102.176 0.0676 0.9827
ß2 0.00721
Q 0.00687
Table 4: Fitting for the MOLE and HGLE Distribution
Figure 3: P P Plot for LE distribution, MOLE distribution and HGLE
Figure 4: P P Plot for MOLE distribution and HGLE distribution
10 Conclusion
In this paper we have shown a generalization of LE distribution namely HGLE distribution. We have studied some of the statistical properties of the distribution such as probability density function, hazard rate function, moment generating function, distribution of order statistics, asymptotic distribution of extreme order statistics, Renyi entropy etc. The method of maximum likelihood estimation is also derived and also described two cases of real data application to show how HGLE distribution performs better than its baseline distributions.
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