A new 3-parameter extension of generalized lindley distribution

0.61 MB·

by  Deepesh Bhati

A new 3-parameter extension of generalized lindley
distribution
Deepesh Bhati1, Mohd. Aamir Malik2, and K K Jose3
1Department of Statistics, Central University of Rajasthan, India.
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1 [email protected]
0 2AU Financer(India) Ltd., Jaipur, Rajasthan, India, [email protected]
2
[email protected]

n
a
J
January 7, 2016
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Abstract
S
Here, weintroduceanewclassofLindleygenerateddistributionswhichresultsinmore
.
h flexible model with increasing failure rate (IFR), decreasing failure rate(DFR) and up-
at side down hazard functions for different choices of parametric values. We explore, various
m distributionalpropertiesincludinglimitingdistributionofextremeorderstatisticsexplored.
Maximumlikelihoodestimatorsandtheconfidenceintervalsoftheparametersareobtained.
[
The applicability of the proposed distribution is shown through modelling two sets of real

1 data on bladder cancer patients and waiting time in a queue. Further, we carry out stress-
v strength analysis for applying the model in system reliability studies.
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4 Keyword: Lindley Distribution, Integral Transform, IFR, DFR, upside down hazard function,
0 Entropy, Maximum likelihood Estimator
1
0
AMS 2001 Subject Classification: 60E05
.
1
0
6 1 Introduction
1
:
v Modelling of lifetimes of materials, organisms,devices plays an important role in biological and
i engineering services. Recently, a member of lifetime distributions have been introduced by vari-
X
ous authors Nadarajah et al.(2012), Ristic(2012), Jose et al.(2014). They help in the modelling
r
a of length of life length data from various contents.
Lifetime distribution are used to describe, statistically, the length of the life of a system, a
device, and in general, time-to-event data. Lifetime distributions are frequently used in fields
like reliability, medicine, biology, engineering, insurance etc. Many parametric models such
as exponential, gamma, Weibull have been frequently used in statistical literature to analyse
lifetime data.
Recently, one parameter Lindley distribution has attracted researchers for its potential in
modellinglifetimedata,andithasbeenobservedthatthisdistributionhasperformedexcellently
wellinmanyapplications. TheLindleydistributionwasoriginallyproposedbyLindley[19]inthe
context of Bayesian statistics, as a counter example to fiducial statistics. The distribution can
alsobederivedasamixtureofexp(θ)andgamma(2,θ). MoredetailsontheLindleydistribution
can be found in Ghitany et al.(2008). Nastic et al.(2015) developed auto–generated time series
models with Lindley marginal distribution and applied it to model time series data.
1
Ghitanyetal(2013)introducedapowerLindleydistributionandcarriedoutassociatedinfer-
ences. Azgharzadehetal.(2013)introducedanewlifetimedistributionbycompoundingPoisson-
Lindley distribution. Liyanage and Pararai(2014) introduced an exponential power Lindley dis-
tribution and explore its properties.
A random variable X is said to have the Lindley distribution with parameter θ if its proba-
bility density is defined as:
θ2
r (x;θ)= (1+x)e−θx;x>0,θ >0. (1)
X (θ+1)
The corresponding cumulative distribution function is
e−θx(1+θ+θx)
R(x)=1− ;x∈R+,θ >0,
1+θ
Alzafrehetal(2013,2014)introducedanewclassofdistributionscalledT-Xfamily. Ghitany
et al.(2011) have introduced a two-parameter weighted Lindley distribution and have pointed
out its usefulness, in particular, in modelling biological data from mortality studies. Bakouch
et al.(2012) have introduced extended Lindley (LE) distribution; moreover and Ebatal(2014)
introduced a transmuted Lindley-geometric distribution and by transmuting and compounding
Lindleyandgeometricdistributions. AdamidisandLoukas(1998)haveintroducedanewlifetime
distribution with decreasing failure rate. Shanker et al.(2013) have introduced a two-parameter
Lindleydistribution. Zakerzadehetal.(1998)haveproposedanewtwoparameterlifetimedistri-
bution and studied its properties. Hassan(2014) has introduced convolution of Lindley distribu-
tion. Ghitany et al.(2015) worked on the estimation of the reliability of a stress-strength system
from power Lindley distribution. Elbatal et al.(2013) has proposed a new generalized Lindley
distribution by considering the mixture of two gamma distributions.Ali(2013) considered the
meanresiduallifefunctionandcarriedoutstress-strengthanalysisunderdifferentlossfunctions
for Lindley distribution in the counted of Bayesian Inference.
ZografosandBalakrishnan(2009),proposedakindofgamma-generatedfamily. Basedonabase-
linecontinuousdistributionF(x)withsurvivalfunctionF¯(x)anddensityf(x),theydefinedthe
cumulative distribution function (cdf) and probability density function (pdf) as
−log(F¯(x))
1 (cid:90)
G (x)= tα−1etdt, x∈R+,α>0,
X Γ(α)
0
The corresponding pdf is obtained as:
g(x)= 1 (cid:0)−logF¯(x)(cid:1)F¯(x)α−1f(x), x∈R+,α>0,
Γ(α)
2 Distributional Properties of EGL Distribution
We consider a new family of distribution generated by an integral transform of the pdf of a
random variable T which follows one parameter Lindley distribution. The distribution function
of this new family is given as:
−log(F¯(x))
(cid:90)
G (x)= r(t)dt, x∈R+,θ >0
X
0
2
substituting r(t) from equation (1), we get the new family of distribution with d.f.
−log(F¯(x))
θ2 (cid:90)
G (x)= (1+t)e−θtdt, x∈R+,θ >0 (2)
X 1+θ
0
where θ >0 and ,the corresponding probability density function (pdf) is given by
g(x)= θ2 (cid:0)1−logF¯(x)(cid:1)F¯(x)θ−1f(x), (3)
1+θ
In this formation, we consider F(x) corresponding to Extended exponential distribution with
survival function e1−(1+λx)α which yields the distribution function of the new distribution as
eθ−θ(1+xλ)α(1+θ(1+xλ)α)
G(x)=1− (4)
1+θ
with corresponding density
αθ2λ(1+xλ)2α−1eθ−θ(1+xλ)α
g(x)= . (5)
1+θ
We refer to random variable with survival function (4) as Extended Generalized Lindley(EGL)
distribution with parameters α,θ and λ and denote it by EGL(λ,θ,α).
Proposition1: IfX ∼EGL(λ,α,θ)thenrandomvariableY =(1+λX)α−1followsLindley(θ).
This paper is arranged as follows. In section 2 we introduce the new Lindley generated
distribution and study its properties. Section 3 deals with maximum likelihood estimation. In
section 4, we conduct the stress- and strength analysis estimation.. Section 5 is devoted for
application on real data set.
2.1 Shape of the density
In this section, we introduce and study the distributional properties of the EGL. In particular,
if X ∼ EGL(λ,θ,α) then the shapes of the density and hazard function, moments, the density
of the rth order statistics, and other important measures of the ELD are derived and studied in
detail.
For the density function of the EGL distribution, the first and the second derivatives of logg(x)
are
d λ(1+α(−2+θ(1+xλ)α))
logg(x)=−
dx 1+xλ
and
d2 λ2(1−2α−(−1+α)αθ(1+xλ)α)
logg(x)=
dx2 (1+xλ)2
Hence the mode of EGL distribution is given by the following theorem.
Theorem 1: The probability density function of EGL(λ,θ,α) is unimodal for α > 1 ∩0 <
2−θ
θ <2 and is given as
(cid:18) (cid:19)1
1 1 2α−1 α
x =− +
0 λ λ αθ
3
Proof: For α > 1, d2 logg(x) < 0 i.e. the density function g(x) is log-concave. Note
dx2
(logg(cid:48))(∞)=−∞ and (logg(cid:48))(∞)=λ(α(2−θ)−1)>0 for α> 1 and 0<θ <2. This im-
2−θ
pliesthatforα> 1 and0<θ <2,g(x)hasuniquemodeatx ,wherex =−1+1 (cid:0)2α−1(cid:1)α1,
2−θ 0 0 λ λ αθ
is the solution of the equation d logg(x) = 0, for θ > 2, d logg(x) < 0 i.e. g(x) is decreasing
dx dx
in x.
Further g(0)= αθ2λ =αλr(0), therefore g(0)<(>)r(0) according as αλ<(>)1.
1+θ
Theorem 2: The hazard function of EGL Distribution is decreasing, upside down and in-
creasing according as (cid:0)0<α< 1(cid:1)(cid:83)(cid:16)1 <α<1(cid:84)θ > 1−2α (cid:17), (cid:16)1 <α<1(cid:84)θ > 1−2α (cid:17) and
2 2 α(α−1) 2 α(α−1)
(α>1)∩(θ >0) respectively.
Proof: Considering the hazard rate function (hrf) of the EGL distribution given by
αθ2λ(1+xλ)−1+2α
h(x)= , (6)
1+θ(1+xλ)α
and using theorems of Glaser(1980), we can discuss the shape characteristics of the hrf of EGL
Distribution. The function η(x)=−g(cid:48)(x)/g(x) for EGL Distribution is given by
λ(1+α(−2+θ(1+xλ)α)) λ2
(cid:48)
η(x)= and η (x)= u(x)
1+xλ (1+xλ)2
where
u(x)=(2α−1+(α−1)αθ(1+λx)α) and u(cid:48)(x)=λα2(α−1)θ(1+λx)α−1
For 0 < α < 1, function u(x) < 0,∀λ,θ > 0 hence η”(x) < 0∀x, hence from theorem(b)
2
of Glaser(1980), hazard function is a decreasing function of x. Let us consider the case when
1 <α<1,thenu(0)=2α−1+α(α−1)θandu(∞)=−∞,ifθ > 1−2α ,thenu(x)<0implies
2 α(α−1)
η(cid:48)(x)<0∀x, hence hazard function is decreasing, whereas, for 1 <α <1 and 0<θ ≤ 1−2α ,
2 α(α−1)
u(cid:48)(x) < 0∀x > 0 and u(0) > 0, therefore ∃ a point x such that u(x) > 0 for x ∈ (0,x ) and
0 0
u(x) < 0 for x ∈ [x ,∞) implies η(cid:48)(x) > 0 for x ∈ (0,x ) and η(cid:48)(x) < 0 for x ∈ [x ,∞). Hence
0 0 0
from Glaser(1980) hazard function is upside down shape. Finally when α > 1, both u(x),u(cid:48)(x)
are positive implies u(x) as positively increasing function implies η(cid:48)(x) > 0∀x > 0. Hence
hazard function is increasing, which proves the theorem.
It can also be verified that

0 if 0<α<1
(cid:18) θ2 (cid:19) 
h(0)=αλ and lim h(x)= αλθ if α=1
1+θ x→∞ ∞ if α>1
Hence from the above relation hEGL(0)=αλhL(0).
The pdf and hazard function for different parameter values are shown in the figure 1.
2.2 The Quantile Function of EGL distribution
Thecdf,G (x)=1−G¯(x),isgivenbyusingeq.(4). Further,itcanbenotedthatG iscontin-
X X
uous and strictly increasing so that the quantile function of X is Q (γ) = G−1(γ), 0 < γ < 1.
X X
In the following theorem, we give an explicit expression for Q in terms of the Lambert W
X
function. For more details on Lambert W function we refer the reader to Jo´rda(2010) and also
to Nair et al.(2013) for discussion on quantile functions.
4
Theorem 3: For any θ,λ,α>0, the quantile function of the EGL distribution is
1 (cid:32) W (cid:0)e−θ−1(θ+1)(γ−1)(cid:1)+1(cid:33)1/α 1
x =G−1(γ)= − −1 − , (7)
γ λ θ λ
where W denotes the negative branch of the Lambert W function.
−1
Proof: By assuming p=(1+λx)α the cdf can be written as
eθ(1−p)(1+θp)
G (x)=1−
X 1+θ
For fixed θ,λ,α>0 and γ ∈(0,1), the γth quantile function is obtained by solving F (x)=γ.
X
By re-arranging the above, we obtain
eθ(1−p)(1+θp)=(1−γ)(1+θ)
It can be further written as
−(1+θp)e−(1+θp) =−(1−γ)(1+θ)e−θ−1. (8)
We see that −(1+θp) is the Lambert-W function of real argument −(1−γ)(1+θ)e−θ−1 .
Thus, we have
W (cid:0)−(1−γ)(1+θ)e−θ−1(cid:1)=−(1+θp) (9)
Moreover, for any θ, λ and α > 0 it is immediate that (1+pθ) > 1, and it can also be checked
that (1−γ)(1+θ)e−θ−1 ∈ (−1/e,0) since γ ∈ (0,1). Therefore, by taking into account the
properties of the negative branch of the Lambert W function, we deduce the following.
W (cid:0)−(1−γ)(1+θ)e−θ−1(cid:1)=−(1+θp)
−1
Again,solving for x by using p=(1+λx)α, we get
1 (cid:32) W (cid:0)e−θ−1(θ+1)(γ−1)(cid:1)+1(cid:33)1/α 1
x =G−1(γ)= − −1 − (10)
γ λ θ λ
(cid:50)
Further the Median can be obtained by substituting γ = 1 in (10),Thus
2
 W (cid:16)−e−θ−1(θ+1)(cid:17)+11/α
Median(Md)=G−1(1/2)= λ1 − −1 θ2  − λ1 (11)
2.3 Moments
An infinite sum representation is being used to represent rth moment, µ(cid:48) = E[Xr], and conse-
r
quently the first four moments and variance for the EGL Distribution.
The kth raw moment of EGL random variable is given as
1 (cid:90) ∞(cid:16) (cid:17)
E(Xk)= xkeθ(1−(1+xλ)α)αθ2λ(1+xλ)−1+2α dx
1+θ
0
eθαθ2λ(cid:90) ∞
= xke−θ(1+xλ)α(1+xλ)2α−1dx
1+θ
0
eθαθ2λ
E(Xk)= I(k,α,θ) (12)
1+θ
5
where I(k,α,θ)= αλ1k+1 (cid:80)ki=0(cid:0)ki(cid:1)(−1)k−iθ−αi−2Γ(cid:0)αi +2,θ(cid:1) see appendix for detailed proof.
Hence
eθ (cid:18) (cid:18) 1 (cid:19)(cid:19)
E(X)= −Γ(2,θ)+θ−α1Γ 2+ ,θ
λ(1+θ) α
eθ (cid:18) (cid:18) 1 (cid:19) (cid:18) 2 (cid:19)(cid:19)
E(X2)= Γ(2,θ)−2θ−α1Γ 2+ ,θ +θ−α2Γ 2+ ,θ
λ2(1+θ) α α
eθ (cid:18) (cid:18) 1 (cid:19) (cid:18) 2 (cid:19) (cid:18) 3 (cid:19)(cid:19)
E(X3)= −Γ(2,θ)+3θ−α1Γ 2+ ,θ −3θ−α2Γ 2+ ,θ +θ−α3Γ 2+ ,θ
λ3(1+θ) α α α
eθ (cid:18) (cid:18) 1 (cid:19) (cid:18) 2 (cid:19) (cid:18) 3 (cid:19)
E(X4)= Γ(2,θ)−4θ−α1Γ 2+ ,θ +6θ−α2Γ 2+ ,θ −4θ−α3Γ 2+ ,θ +
λ4(1+θ) α α α
(cid:18) (cid:19)(cid:19)
4
θ−α4Γ 2+ ,θ
α
(13)
For lifetime models, it is also of interest to know moment of Future lifetime random variable
Y = X|X > t and its moments. Thus by using the Lemma the kth raw moment of random
variable Y is given as
1 (cid:90) ∞
E(Yk)=E(Xk|X >t)= ukg(u)du (14)
G¯(t)
t
Thus, substituting the value of g(x) and G(x) from equation () and () we get.
λθ2αeθ(1+λt)α (cid:90) ∞
E(Xk|X >t)= uk(1+λu)2α−1e−θ(1+λu)αdu
1+θ(1+λt)α
t
λθ2αeθ(1+λt)α
= L(k,t,α,θ(1+λt)α)
1+θ(1+λt)α
where L(k,t,α,θ(1+λt)α) = αλ1k+1 (cid:80)ki=0(cid:0)ki(cid:1)(−1)k−iθ−αi−2Γ(cid:0)αi +2,θ(1+λt)α(cid:1) see appendix
for detailed proof.
The mean residual lifetime function is E(X|X >t)−t.
2.4 Entropy
Let us now consider the R´enyi entropy which represents a measure of uncertainty of a random
variable and is defined as
 
1 (cid:90)
J(ζ)= log fζ(x)dx, for ζ >1 and ζ (cid:54)=1 (15)
1−ζ
R+
In our case
(cid:90) (cid:18)αλθ2eθ(cid:19)ζ(cid:90)
fζdx= e−ζθ(1+xλ)α(1+xλ)ζ(2α−1)dx
1+θ
R+
R+
substituting t=(1+λx)α the above expression reduces to
(cid:90) (αλ)ζ−1θ2ζeθζ (cid:90) ∞
fζdx= e−γθttαγ(2α−1)+α1−1dt
(1+θ)ζ
1
R+
(αλ)ζ−1θ2ζeθζ
= E (ζθ) (16)
(1+θ)ζ −2ζα+αα+ζ−1
6
where E (z)=(cid:82)∞e−ztt−ndt is known as exponential integral function. For more details
n 1
see http://functions.wolfram.com/06.34.02.0001.01.
Thus according to (15) the R´enyi entropy of EGL(θ,λ,α) distribution given as
1 (cid:18)(αλ)ζ−1θ2ζeθζ (cid:19)
J(ζ)= log E (ζθ) . (17)
1−ζ (1+θ)ζ −2ζα+αα+ζ−1
Moreover, The Shannon entropy defined by E[log(f(x))] is a special case derived from limJ(ζ)
ζ→1
2.5 Order Statistics
LetX ,X ,··· ,X bearandomsamplefromtheEGL(λ,θ,α)distribution, andletX denote
1 2 n i:n
the ith order statistic. Assuming p = (1+λx)α, the pdf of the ith order statistic X is given
i:n
by (see [7])
n!
g (x)= g(x)Gi−1(x)(1−G(x))n−i
i:n (i−1)!(n−i)!
n! αθ2λ (cid:18) eθ(1−p)(1+θp)(cid:19)i−1
= eθ(1−p)(1+n−i)p2−α1(1+θp)n−i 1−
(i−1)!(n−i)!(1+θ)1+n−i (1+θ)
= n! αθ2λ (cid:88)i−1(cid:18)i−1(cid:19) (−1)j p2−α1(1+θp)n−i+jeθ(1−p)(1+n−i+j)
(i−1)!(n−i)!(1+θ)1+n−i j (1+θ)j
j=0
Substituting back the value of p=(1+λx)α, we get
g (x)= n! αθ2λ (cid:88)i−1(cid:18)i−1(cid:19) (−1)j (1+λx)2α−1(1+θ(1+λx)α)n−i+j
i:n (i−1)!(n−i)!(1+θ)1+n−i j (1+θ)j
j=0
(cid:16) (cid:17)
× eθ(1−(1+λx)α)(1+n−i+j)
Thus the moments of X can be expressed as
i:n
n! αθ2λ (cid:88)i−1n(cid:88)−i+j(cid:18)i−1(cid:19)(cid:18)n−i+j(cid:19) (−1)j
E(Xq )= eθ(1+n−i+j)
i:n (i−1)!(n−i)!(1+θ)1+n−i j k (1+θ)j
j=0 k=0
(cid:90) ∞
× xq(1+λx)α(k+2)−1e−θ(1+n−i+j)(1+λx)αdx
0
n! αθ2λ (cid:88)i−1n(cid:88)−i+j(cid:18)i−1(cid:19)(cid:18)n−i+j(cid:19) (−1)j
= eθ(1+n−i+j)
(i−1)!(n−i)!(1+θ)1+n−i j k (1+θ)j
j=0 k=0
q (cid:18) (cid:19) (cid:18) (cid:19)
× 1 (cid:88) q (−1)q−l(θ(n+1−i+j))−(k+2+αl)Γ k+2+ l ,θ(n+1−i+j)
αλq+1 l α
l=0
2.6 Limiting Distribution of Sample Minimum and Maximum
Usually,wemaybeinterestedintheasymptoticbehaviourofsampleminimaX and/orsample
1:n
maxima X . Using, theorem 8.3.6 of Arnold et al., it follows that the asymptotic distribution
n:n
of X follows exponential whereas the X follows
1:n 1:n
Consideringcdfgiveninequation()andstrictlypositivefunctiong(t)asg(t)= 1(1+λt)1−α,t>
λ
0. It can be seen that
7
F(tx)
lim =x,∀x>0
t→0+ F(t)
and
1−F(t+xg(t)) e−θ(1+λt+λxg(t))α(1+θ(1+λt+λxg(t))α)
lim = lim
t→∞ 1−F(t) t→∞ 1+θ(1+λt)α
e−θ(1+λt)α((λ1x+gλ(tt)+1)α−1)(θ(1+λ+λxg(t))α+1)
= lim
t→∞ 1+θ(1+λt)α
e−θ(1+λt)α[(α1λ+xgλ(tt))+···](θ(1+λt+λxg(t))α+1)
= lim
t→∞ 1+θ(1+λt)α
substituting the value of g(t) and taking limit, we obtain
lim 1−F(t+xg(t)) = lim (cid:0)θ(cid:0)1+λt+x(λt+1)1−α(cid:1)α+1(cid:1)e−αxθ((λλtt++11))αα+···
t→∞ 1−F(t) t→∞ 1+θ(λt+1)α
=e−θαx
so it follows from the theorem 1.6.2 in Leaderbetter that there must be norming constants
a >0 and b such that a =[g(γ )]−1 , b =γ where γ =F−1(1− 1)
n n n n n n n n
3 Maximum Likelihood Estimators
In this section, we shall discuss the point and interval estimation of the parameters of EGL
(λ,θ,α). The log-likelihood function l(Θ) of single observation (say x ) for the vector of param-
i
eter Θ=(θ,λ,α)(cid:62) is
n n
l =nlog(α)+nlogλ+2nlog(θ)−nlog(1+θ)+θ(cid:88)(1−(1+x λ)α)+(2α−1)(cid:88)log(1+x λ)
n i i
i=1 i=1
The associated score function is given by Un =(cid:0)∂∂lθn,∂∂lλn,∂∂lαn(cid:1)(cid:62), where
n
∂ln = 2n − n +(cid:88)(1−(1+λx )α) (18)
∂θ θ 1+θ i
i=1
n n
∂ln = n +(2α−1)(cid:88) xi −θ(cid:88)αx (1+λx )α−1. (19)
∂λ λ 1+λx i i
i
i=1 i=1
n n
∂ln = n +2(cid:88)log(1+λx )−θ(cid:88)(1+λx )αlog(1+λx ) (20)
∂α α i i i
i=1 i=1
As we know the expected value of score function equals to zero, i.e.E(U(Θ)), which implies
E((cid:80)n (1−(1+λx )α))= 1 − 2.
i=1 i θ+1 θ
Thetotallog-likelihoodoftherandomsamplex=(x ,··· ,x )(cid:62) ofsizenfromX isgivenby
1 n
n n
l =(cid:80)l(i) andthtotalscorefunctionisgivenbyU = (cid:80)U(i),wherel(i) isthelog-likelihoodof
n n
1 i=1
ith observation and U(i) as given above. The maximum likelihood estimator Θˆ of Θ is obtained
by solving equations (18) and (19) numerically or this can also obtained easily by using nlm()
8
function in R. The initial guess for the estimators were obtained from the inner region of 3D
contourplotoflog-likelihoodfunctionforagivensample. Forexample,inFigure(3),thecontour
plotoflog-likelihoodfunctionfordifferentθ andλ,theinitialestimatesweretakenfrominterior.
The associated Fisher information matrix is given by
 
κ κ κ
θ,θ θ,λ θ,α
K =Kn(Θ)=nκλ,θ κλ,λ κλ,α (21)
κ κ κ
α,θ α,λ α,α
where
n 2n
κ = −
θ,θ (θ+1)2 θ2
κ =− n +αθ(1+α)E(cid:0)X2(λX+1)α−2(cid:1)+(1−2α)E(cid:18) X2 (cid:19)
λ,λ λ2 (λX+1)2
κα,α =−αn2 −θE(cid:0)(λX+1)αlog(1+λX)2(cid:1) (22)
κ =κ =−αE(cid:0)X(λX+1)α−1(cid:1)
λ,θ θ,λ
κ =κ =E((λX+1)α(−log(λX+1)))
α,θ θ,α
(cid:18) (cid:19)
κ =κ =E(cid:0)−θX(λX+1)α−1−αθX(λX+1)α−1log(λX+1)(cid:1)+2E X
α,λ λ,α λX+1
Theaboveexpressionsdependonsomeexpectationswhichcanbeeasilycomputedusingnumer-
ical integration. Under the usual regularity conditions, the asymptotic distribution of
√ (cid:16) (cid:17)
n Θˆ −Θ is N (0,K(Θ)−1) (23)
2
where lim =K (Θ)−1 =K(Θ)−1. The asymptotic multivariate normal N (0,K(Θ)−1) distri-
n 2
n→∞
butionofΘˆ canbeusedtoconstructapproximateconfidenceintervals. Anasymptoticconfidence
interval with significance level γ for each parameter θ,α and λ is
√ √
ACI(θ,100(1−γ)%)≡(θˆ−z κ ,θˆ+z κ )
γ/2 (θ,θ) γ/2 (θ,θ)
√ √
ACI(λ,100(1−γ)%)≡(λˆ−z κ ,λˆ+z κ ) (24)
γ/2 (λ,λ) γ/2 (λ,λ)
√ √
ACI(α,100(1−γ)%)≡(αˆ−z κ ,αˆ+z κ )
γ/2 (α,α) γ/2 (α,α)
where z denotes 1−γ/2 quantile of standard normal random variable.
1−γ/2
4 Application to Real Datasets
In this section, we illustrate, the applicability of Extended Generalized Lindley Distribution by
considering two different datasets used by different researchers. We also fit Extended Gener-
alized Lindley distribution, Lindley-Exponential Distribution proposed by Bhati et al.(2015),
Power-Lindley distribution (2015), New Generalized Lindley Distribution(2013), Lindley Distri-
bution(1958) and Exponential distribution. Namely
(i) Lindley-Exponential Distribution (L-E(θ,λ))
θ2λe−λx(cid:0)1−e−λx(cid:1)θ−1(cid:0)1−log(cid:0)1−e−λx(cid:1)(cid:1)
f(x)= x,θ,λ>0. (25)
(1+θ)
9
(ii) Power-Lindley distribution (PL(α,β)):
αβ2
f (x)= (1+xα)xα−1e−βxα, x,α,β >0.
1 1+β
(iii) New Generalized Lindley distribution (NGLD(α,β,θ)):
e−θx (cid:18)θα+1xα−1 θβxβ−1(cid:19)
f (x)= + , x,α,θ,β >0.
2 1+θ Γ(α) Γ(β)
(iv) Lindley Distribution (L(θ))
θ2
f (x)= (1+x)e−θx, x,α,β >0.
3 1+θ
Foreachofthesedistributions,theparametersareestimatedbyusingthemaximumlikelihood
method, and for comparison we use Negative LogLikelihood values (−LL), the Akaike informa-
tion criterion (AIC) and Bayesian information criterion (BIC) which are defined by −2LL+2q
and −2LL + qlog(n), respectively, where q is the number of parameters estimated and n is
the sample size. Further K-S statistic (Kolmogorov-Smirnov test)=sup |F (x)−F(x)|, where
x n
n
F (x)= 1 (cid:80)I is empirical distribution function and F(x) is cumulative distribution func-
n n xi≤x
i=1
tion is calculated and shown for all the datasets.
4.1 Illustration 1: Application to bladder cancer patients
We consider an uncensored data set corresponding to remission times (in months) of a random
sample of 128 bladder cancer patients (Lee and Wang(2003)) as presented in Appendix A.1 in
Table(9). The results for these data are presented in Table (1). We observe that the EGL
distribution is a competitive distribution compared with other distributions. In fact, based
on the values of the AIC and BIC criteria and K-S test statistic, we observe that the EGL
distribution provides the best for these data among all the models considered. The probability
densityfunctionandempiricaldistributionfunctionarepresentedinfigure(8)forallconsidered
distributions for these data.
Table 1: The estimates of parameters and goodness-of-fit statistics for Illustration 1.
Model Parameter -LL AIC BIC K-S statistic
EGL λˆ= 0.9360, θˆ=0.5878,αˆ=0.6457 401.254 796.508 796.186 0.0387
L-E λˆ= 0.0962, θˆ=1.229 401.78 807.564 807.780 0.0454
PL θˆ=0.385, βˆ=0.744 402.24 808.474 808.688 0.0446
L θˆ=0.196 419.52 841.040 843.892 0.0740
NGLD θˆ=0.180, αˆ=4.679, βˆ=1.324 412.75 831.501 840.057 0.1160
4.2 Illustration 2: Application to waiting times in a queue
As second example, we consider 100 observations on waiting time (in minutes) before the cus-
tomer received service in a bank (see Ghitany et al.(2008)). The data sets are presented in
appendix A.2 as Table (10). The results for these data are presented in Table(2). From these
results we can observe that EGL distribution provide smallest K-S test statistics values as com-
pare to Lindley-Exponential, Power lindley, new generalized Lindley distribution, Lindley and
Exponential and hence best fits the data among all the models considered. The results are
presented in Table (2) and probability density function and empirical distribution function are
shown in figure (9).
10