Afrika Statistika Generalized Stacy-Lindley Mixture ... · Maya R. and Irshad M.R. Afrika...

Afrika Statistika

Vol. 12 (3), 2017, pages 1447–1465.DOI: http://dx.doi.org/10.16929/as/2017.1447.112

Afrika Statistika

ISSN 2316-090X

Generalized Stacy-Lindley MixtureDistribution

R. Maya 1 and M. R. Irshad 2

1Department of Statistics, University College, Trivandrum -695 034, India2Department of Statistics, CUSAT, Kochi -682 022, India

Received June 10, 2017. Accepted October 22, 2017

Copyright c© 2017, Afrika Statistika and The Statistics and Probability African Society(SPAS). All rights reserved

Abstract. . In this paper, we introduce a five parameter extension of mixture of two Stacygamma distributions called generalized Stacy- Lindley mixture distribution. Several statis-tical properties are derived. Two types of estimation techniques are used for estimating theparameters. Asymptotic confidence interval is also calculated for these parameters. Finally,a real data application illustrates the performance of our proposed distribution.

Resume. Dans cet article, nous introduisons une generalisation de la distributions melangeconnue sous le nom de Gamma de Stacy, a une famille de cinq parametres que nous nommonsDistribution melange de Stacy-Lindey. Plusieurs proprietes statistiques sont prouvees. Deuxtypes de techniques d’estimation sont utilises pour estimer les parametres. Les intervalles deconfiances asymptotiques sont egalement fournies. Enfin, une application de donnees reellesillustre la performance de la distribution proposee.

Key words: Lindley distribution; Stacy gamma distribution; Inequality measures; Uncer-tainty measures; Maximum likelihood estimation; Asymptotic confidence interval.AMS 2010 Mathematics Subject Classification : 60E05; 62H12

1. Introduction

Gamma distribution is a widely used distribution in many fields such as lifetime data analy-sis, reliability, hydrology, medicine, meteorology, etc. Many authors generalized the gammadistribution by various ways. Stacy (1962) has proposed a three parameter generalized

∗Corresponding author R. Maya : [email protected]. R. Irshad : [email protected]

Maya R. and Irshad M.R. Afrika Statistika, Vol. 12 (3), 2017, pages 14447 – 1465. GeneralizedStacy- Lindley Mixture Distribution. 1448

gamma distribution. The probability density function of the generalized gamma distribu-tion is given by

f(x; a, d, p) =( pad

)xd−1e−( xa )p

Γ(dp ). (1)

This distribution includes many well known distributions as special cases such as thegamma, the exponential, the Weibull and the half-normal distributions. The generalizedgamma distribution is appropriated for modeling data with dissimilar types of hazard rate.Cox et al. (2007) presented a taxonomy of the hazard functions of the generalized gammafamily and applied the proposed taxonomy to study survival after a diagnosis of clinicalAIDS during different eras of HIV therapy. Khodabin and Ahmadabadi (2010) discussedsome properties of generalized gamma distribution.

The mixture distribution is defined as one of the most crucial ways to obtain new probabilitydistributions in applied probability and several research areas. Lindley (1958) suggestedan one parameter distribution to illustrate the difference between fiducial distribution andposterior distribution and has the following probability density function (pdf),

f(x; θ) =θ2

1 + θ(1 + x)e−θx; x > 0, θ > 0. (2)

This distribution is a mixture of exponential (θ) and gamma (2,θ) distributions.

Ghitany et al. (2008) developed different properties of Lindley distribution and showed thatthe Lindley distribution fits better than the exponential distribution based on the waitingtimes before service of the bank customers. Sankaran (1970) used Lindley distribution asthe mixing distribution of a Poisson parameter and the resulting distribution is known asthe Poisson-Lindley distribution. Zakerzadeh and Dolati (2009) have obtained a generalizedLindley distribution and discussed its various properties and applications.

Ghitany et al. (2013) and Nadarajah et al. (2011) have recently proposed two parameterextensions of the Lindley distribution named as the power Lindley and the the generalizedLindley distributions respectively. A discrete form of Lindley distribution was introduced byGomez and Ojeda (2011) by discretizing the continuous Lindley distribution. Al-Mutairiet al. (2013) mentioned that the Lindley distribution belongs to an exponential family andit can be written as a mixture of an exponential and a gamma distribution with shape pa-rameter 2. Location parameter extension of Lindley distribution is extensively discussed byMonsef (2015). Nedjar and Zeghdoudi (2016) introduced gamma Lindley distribution andstudied some important properties of their proposed generalization. Zeghdoudi and Nedjar(2016) introduced pseudo Lindley distribution and studied some of its important properties.

To increase the flexibility for modeling purposes it will be useful to consider furthergeneralizations of the existing models. Hence in this study we introduce a generalizedStacy- Lindley mixture distribution which was obtained by mixing two generalized (Stacy)gamma distributions. Some statistical and reliability properties, inequality measures anduncertainty measures of GSLMD are derived in sections 3, 4 and 5. Different methodsof estimation are carried out in section 6. Section 7 focuses attention on the asymptoticconfidence interval of the parameters. Application of the introduced model is discussed in

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section 8.

2. Generalized Stacy- Lindley Mixture Distribution

Here we introduced a new mixed distribution, namely the Generalized Stacy- Lindley mix-ture distribution (GSLMD), which is obtained by mixing two generalized gamma distri-butions namely generalized gamma (α, β, θ) and generalized gamma (η, β, θ) with mixing

probabilities p1 = θδ

1+θδand p2 = 1

1+θδrespectively, the corresponding pdf has the form

f(x; θ, α, β, η, δ) =βe−θx

β

1 + θδ

{θα+δxαβ−1

Γ(α)+θηxηβ−1

Γ(η)

};x > 0, (3)

θ, α, β, η, δ >0.

The cumulative distribution function (cdf) and survival function of GSLMD are given by

F (x; θ, α, β, η, δ) =1

1 + θδ

{θδ(Γ(α)− Γ(α, θxβ))

Γ(α)+

(Γ(η)− Γ(η, θxβ)

)Γ(η)

}(4)

and

F (x; θ, α, β, η, δ) =1

1 + θδ

{θδΓ(α, θxβ)

Γ(α)+

Γ(η, θxβ)

Γ(η)

}, (5)

where Γ(a, b) =∞∫b

ta−1e−tdt is the upper incomplete gamma function.

3. Properties

If X has the GSLMD with density function given in (3), then the rth raw moment µ′

r aboutorigin is given by, for r = 1, 2, ...

µ′

r =

∞∫0

xrf(x; θ, α, β, η, δ)dx

=θ−

rβ

1 + θδ

{ θδ

Γ(α)Γ(α+

r

β) +

Γ(η + rβ )

Γ(η)

}.

(6)

The mean and variance of GSLMD are respectively given by

µ =θ−

1β

1 + θδ

{ θδ

Γ(α)Γ(α+

1

β) +

Γ(η + 1β )

Γ(η)

}(7)

and



σ2 =θ−2β

1 + θδ

{ θδ

Γ(α)Γ(α+

2

β) +

Γ(η + 2β )

Γ(η)

}−

(θ−

1β

1 + θδ

{ θδ

Γ(α)Γ(α+

1

β) +

Γ(η + 1β )

Γ(η)

})2

. (8)

The rth conditional moment of GSLMD is given by

M′

r = E(Xr|X > x)

=

∞∫x

trf(t; θ, α, β, η, δ)dt

F (x; θ, α, β, η, δ)

=θ−

rβ

{θδΓ(η)Γ(α+ r

β , θxβ) + Γ(α)Γ(η + r

β , θxβ)}

θδΓ(η)Γ(α, θxβ) + Γ(α)Γ(η, θxβ).

(9)

Let X be a continuous random variable with p.d.f. f(x) and c.d.f. F (x). The hazard ratefunction , reversed hazard rate function, vitality function and mean residual life function ofGSLM random variable are respectively defined as

hF (x) =βe−θx

β {θα+δxαβ−1Γ(η) + θηxηβ−1Γ(α)

}θδΓ(η)Γ(α, θxβ) + Γ(α)Γ(η, θxβ)

, (10)

τF (x) =βe−θx

β {θα+δxαβ−1Γ(η) + θηxηβ−1Γ(α)

}θδΓ(η)

(Γ(α)− Γ(α, θxβ)

)+ Γ(α)

(Γ(η)− Γ(η, θxβ)

) , (11)

V (x) = E(X|X > x)

=

∞∫x

tf(t; θ, α, β, η, δ)dt

F (x; θ, α, β, η, δ)

=θ−

1β

{θδΓ(η)Γ(α+ 1

β , θxβ) + Γ(α)Γ(η + 1

β , θxβ)}

θδΓ(η)Γ(α, θxβ) + Γ(α)Γ(η, θxβ),

(12)

which is immediate from (9) and

m(x) = E(X − x|X > x) = E(X|X > x)− x= V (x)− x

=θ−

1β

{θδΓ(η)Γ(α+ 1

β , θxβ) + Γ(α)Γ(η + 1

β , θxβ)}

θδΓ(η)Γ(α, θxβ) + Γ(α)Γ(η, θxβ)− x.

(13)

The moment generating function and characteristic function of X are respectively given by

MX(t) = E(etX)

=

∞∑r=0

tr

r!

θ−rβ

1 + θδ

{ θδ

Γ(α)Γ(α+

r

β) +

Γ(η + rβ )

Γ(η)

} (14)



and

φX(t) = E(eitX)

=

∞∑r=0

(it)r

r!

θ−rβ

1 + θδ

{ θδ

Γ(α)Γ(α+

r

β) +

Γ(η + rβ )

Γ(η)

}.

(15)

4. Inequality Measures

Bonferroni and Lorenz curves (see, Bonferroni (1930)) have been used in economics to studyincome and poverty. These curves have many applications in other fields such as demography,reliability, insurance and medicine and engineering. Zenga (2007) proposed an alternativecurve based on the ratios of lower and upper means. In this section, we derive the Lorenz,Bonferroni and Zenga curves for GSLM distribution.The Lorenz, Bonferroni and Zenga curves are defined, respectively, by

LF (x) =

x∫0

tf(t)dt

E(X) , BF (x) =

x∫0

tf(t)dt

F (X)E(X)

and AF (x) = 1 − µ−(x)µ+(x) , where µ−(x) =

x∫0

tf(t)dt

F (X) and µ+(x) =

∞∫x

tf(t)dt

F (X)are the lower and

upper means, respectively.

Lorenz, Bonferroni and Zenga curves for GSLM random variable are respectively given by

LF (x) =

x∫0


E(X)

=θδΓ(η)

{Γ(α+ 1

β )− Γ(α+ 1β , θx

β)}

+ Γ(α){

Γ(η + 1β )− Γ(η + 1

β , θxβ)}

Γ(η)θδΓ(α+ 1β ) + Γ(α)Γ(η + 1

β ),

BF (x) =

x∫0


F (X)E(X)

=

(1 + θδ)Γ(α)Γ(η)

(θδΓ(η)

{Γ(α+ 1

β) − Γ(α+ 1

β, θxβ)

}+ Γ(α)

{Γ(η + 1

β) − Γ(η + 1

β, θxβ)

}){

Γ(η)θδΓ(α+ 1β

) + Γ(α)Γ(η + 1β

)}{

Γ(η)θδ(

Γ(α) − Γ(α, θxβ))

+ Γ(α)(

Γ(η) − Γ(η, θxβ))}

and

AF (x) = 1− µ−(x)

µ+(x),



where

µ−(x) =

x∫0

tf(t)dt

F (X)

=

θ−1β

(θδΓ(η)

{Γ(α+ 1

β )− Γ(α+ 1β , θx

β)}

+ Γ(α){

Γ(η + 1β )− Γ(η + 1

β , θxβ)})

Γ(η)θδ(

Γ(α)− Γ(α, θxβ))

+ Γ(α)(

Γ(η)− Γ(η, θxβ))

and

µ+(x) =

∞∫x

tf(t)dt

F (X)

=

θ−1β

(θδΓ(η)Γ(α+ 1

β , θxβ) + Γ(α)Γ(η + 1

β , θxβ)

)Γ(η)θδΓ(α, θxβ) + Γ(α)Γ(η, θxβ)

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(x)

Curves

LF (x)

BF (x)

AF (x)

Fig. 1. Plots of Lorenz, Bonferroni and Zenga curves

5. Uncertainty Measures

The concept of entropy was introduced and extensively studied by Shannon (1948).Let X be a non-negative random variable admitting an absolutely continuous cdfF (x) and with pdf f(x). Then the Shannon’s entropy associated with X is defined as

H(X) = −∞∫0

f(x) ln f(x) dx. It gives the expected uncertainty contained in f(x) about

the predictability of an outcome of X. The Shannon’s entropy finds immense applicationsin several branches of learning. In Communication theory, an aspect of interest is the flowof information in some network where information is carried from a transmitter to receiver.Another field of application of Shannon’s entropy is Economics. Hart (1971) discussed theentropy and other measures of concentration in the context of Economics and Business



concentration. The usefulness of this concept in statistical problems has been examined byseveral authors, including Kullback and Leibler (1951)and Lindley (1957). Kapur (1968)used the technique of dynamic programming to solve some optimization problems relatingto entropy in operation research. In data mining, the entropy is used to define an errorfunction as part of the learning of weights in multilayer preceptrons in neural networks. Theentropy error function is then minimized to determine the optimal weights (see, Giudici(2005)).

Renyi’s entropy

Several generalizations of Shannon’s entropy have been put forward by researchers. Ageneralization which has received much attention subsequently is due to Renyi (1959). The

Renyi’s entropy of order ν is defined as Hν(X) = 11−ν ln

∞∫0

fν(x) dx, for ν > 0, ν 6= 1.

Renyi entropy for GSLM random variable is given by

Hν(X) =1

(1− ν)ln

∞∫0

fν(x; θ, α, β, η, δ)dx; for ν > 0, ν 6= 1.

Now

∞∫0

fν(x; θ, α, β, η, δ)dx =

(β

1 + θδ

)ν (θα+δ

Γ(α)

)ν ν∑j=0

1

β

(ν

j

)(θηΓ(α)

θα+δΓ(η)

)jΓ(ν(αβ−1)+jβ(η−α)+1

β

)(θν)

ν(αβ−1)+jβ(η−α)+1β

.

(16)

Therefore,

Hν(X) =1

1− ν

{ln

(βν−1ν θ

1β−

1βν+δν

1β−

1βν−α

(1 + θδ)Γ(α)

)ν+

ln

ν∑j=0

(ν

j

)(θ−δΓ(α)να−η

Γ(η)

)jΓ(ν(αβ − 1) + jβ(η − α) + 1

β

) .

Havrda-Charvat-Tsallis entropy

Another important generalization of Shannon’s entropy is the Havrda-Charvat-Tsallis(HCT) entropy. It was introduced by Havrda and Charvat (1967) and furtherdeveloped by Tsallis (1988) and Gell-Mann and Tsallis (2004) and is defined as

Hρ(X) = 1ρ−1

(1−

∞∫0

fρ(x) dx

), for ρ > 0, ρ 6= 1.



Havrda-Charvat-Tsallis entropy for GSLM random variable is given by

Hρ(X) =1

(ρ− 1)

1−∞∫

0

fρ(x; θ, α, β, η, δ)dx

; for ρ > 0, ρ 6= 1.

From (16)

∞∫0

fρ(x; θ, α, β, η, δ)dx =

(βρ−1ρ θ

1β−

1βρ+δρ

1β−

1βρ−α

(1 + θδ)Γ(α)

)ρ ρ∑j=0

(ρ

j

)(θ−δΓ(α)ρα−η

Γ(η)

)j

×Γ(ρ(αβ − 1) + jβ(η − α) + 1

β

).

Therefore

Hρ(X) =1

(ρ− 1)

{1−

(βρ−1ρ θ

1β−

1βρ+δρ

1β−

1βρ−α

(1 + θδ)Γ(α)

)ρ ρ∑j=0

(ρ

j

)(θ−δΓ(α)ρα−η

Γ(η)

)j

× Γ(ρ(αβ − 1) + jβ(η − α) + 1

β

)}.

6. Different Methods of Estimation

Method of Moments

From (6), the rth raw moment about origin for the GSLM random variable is given by

µ′

r =θ−

rβ

1 + θδ

{ θδ

Γ(α)Γ(α+

r

β) +

Γ(η + rβ )

Γ(η)

}.

Put r = 1, 2, 3, 4 and 5, the first five raw moments are obtained. Equating this raw moments

to the corresponding sample moments; say m′

1, m′

2, m′

3, m′

4 and m′

5, where m′

r = 1n

n∑i=1

Xri ,

we get

θ−1β

1 + θδ

{ θδ

Γ(α)Γ(α+

1

β) +

Γ(η + 1β )

Γ(η)

}= m

′

1, (17)

θ−2β

1 + θδ

{ θδ

Γ(α)Γ(α+

2

β) +

Γ(η + 2β )

Γ(η)

}= m

′

2, (18)

θ−3β

1 + θδ

{ θδ

Γ(α)Γ(α+

3

β) +

Γ(η + 3β )

Γ(η)

}= m

′

3, (19)



θ−4β

1 + θδ

{ θδ

Γ(α)Γ(α+

4

β) +

Γ(η + 4β )

Γ(η)

}= m

′

4(20)

and

θ−5β

1 + θδ

{ θδ

Γ(α)Γ(α+

5

β) +

Γ(η + 5β )

Γ(η)

}= m

′

5. (21)

Since the above system of equations are nonlinear, the numerical solutions are obtainedusing iteration procedure. From the practical point of view, we consider a real life dataset which is given in section 8 and find out first five sample moments. These momentsare equated to the corresponding moments of the population and solve these system ofequations iteratively using statistical softwares like MATHCAD, MATHEMATICA and R,we get the moment estimators of the parameters of GSLMD and are given in Table 1.

Method of Maximum Likelihood

Let X1, X2, ..., Xn be a random sample of size n from GSLMD with unknown parameter

vector H© =(θ, α, β, η, δ

). The likelihood function for H© is

l(

H©)

=

n∏i=1

fi

(x; θ, α, β, η, δ

)=

(β

(1 + θδ)Γ(α)Γ(β)

)ne−θ

n∑i=1

xβin∏i=1

{θα+δΓ(η)xαβ−1

i + θηΓ(α)xηβ−1i

}.

The partial derivatives of log l(

H©)

with respect to the parameters are

∂ log l

∂θ=−nδθδ−1

(1 + θδ)−

n∑i=1

xβi +

n∑i=1

((α+ δ)θα+δ−1Γ(η)xαβ−1

i + ηθη−1Γ(α)xηβ−1i

θα+δΓ(η)xαβ−1i + θηΓ(α)xηβ−1

i

),

(22)

∂ log l

∂α= −nψ(α) +

n∑i=1

θα+δΓ(η)xαβ−1i log(θxβi ) + θηΓ

′(α)xηβ−1

i


i

, (23)

∂ log l

∂β=n

β− θ

n∑i=1

xβi log(xi) +

n∑i=1

(θα+δΓ(η)xαβ−1

i log(xαi ) + θηΓ(α)xηβ−1i log(xηi )


i

),

(24)



∂ log l

∂η= −nψ(η) +

n∑i=1

(θα+δΓ

′(η)xαβ−1

i + θηΓ(α)xηβ−1i log(θxβi )


i

)(25)

and

∂ log l

∂δ=−nθδ log(θ)

1 + θδ+

n∑i=1

(θα+δ log(θ)Γ(η)xαβ−1

i


i

). (26)

As in the case of solving moment equations. The MLE of the parameters H© =(θ, α, β, η, δ

)say H© =

(θ, α, β, η, δ

)are obtained by solving the equations ∂ log l

∂θ = 0, ∂ log l∂α = 0, ∂ log l

∂β = 0,∂ log l∂η = 0 and ∂ log l

∂δ = 0. Here first we set moment estimates obtained from the real life dataset as the initial solution of the parameters and then solving the above nonlinear system oflikelihood equations using statistical softwares like MATHCAD, MATHEMATICA and R,we get the maximum likelihood estimates of the parameters of GSLMD and are given inTable 1. From this table, it is observed that chi-square statistic for the GSLMD is lowerthan those of competing models showing that our model satisfactorily fits better.

7. Asymptotic Confidence Interval

In this section, we present the asymptotic confidence intervals for the parameters

of the GSLMD. Let H© =(θ, α, β, η, δ

)be the maximum likelihood estimator of

H© =(θ, α, β, η, δ

). To obtain the asymptotic confidence intervals for these parameters, we

can use the large sample property of MLE. Therefore(θ− θ, α− α, β − β, η − η, δ − δ

)τis asymptotically normally distributed with mean vector 0 =

(0, 0, 0, 0, 0

)τand estimated

variance covariance matrix ∆−1, where ∆ is the observed Fisher’s information matrix givenby

∆ =

− ∂2 log l

∂θ2− ∂

2 log l∂θ∂α − ∂

2 log l∂θ∂β − ∂

2 log l∂θ∂η − ∂

2 log l∂θ∂δ

− ∂2 log l∂α∂θ − ∂

2 log l

∂α2 − ∂2 log l∂α∂β − ∂

2 log l∂α∂η − ∂

2 log l∂α∂δ

− ∂2 log l∂β∂θ − ∂

2 log l∂β∂α − ∂

2 log l

∂β2− ∂

2 log l∂β∂η − ∂

2 log l∂β∂δ

− ∂2 log l∂η∂θ − ∂

2 log l∂η∂α − ∂

2 log l∂η∂β − ∂

2 log l

∂η2− ∂

2 log l∂η∂δ

− ∂2 log l∂δ∂θ − ∂

2 log l∂δ∂α − ∂

2 log l∂δ∂β − ∂

2 log l∂δ∂η − ∂

2 log l

∂δ2

The elements of ∆ are given by

∆11 =nδθδ−2(δ − 1− θδ)

(1 + θδ)2− 1

θ2

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

((α+ δ)(α+ δ − 1)Ui

+ η(η − 1)Vi

)−(

(α+ δ)Ui + ηVi

)2},

(27)



∆12 = ∆21 =− 1

θ

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{Ui

(1 + (α+ δ) log(θxβi )

)+ ηψ(α)Vi

}−(

(α+ δ)Ui + ηVi

)(Ui log(θxβi ) + Viψ(α)

)},

(28)

∆13 = ∆31 =

n∑i=1

xβi log xi −1

θ

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{(α+ δ) log(xαi )Ui

+ η log(xηi )Vi

}−(

(α+ δ)Ui + ηVi

)(Ui log(xαi ) + Vi log(xηi )

)},

(29)

∆14 = ∆41 =− 1

θ

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{(α+ δ)ψ(η)Ui + (1 + η log(θxβi ))Vi

}−(

(α+ δ)Ui + ηVi

)(ψ(η)Ui + log(θxβi )Vi

)},

(30)

∆15 = ∆51 =nθδ−1(1 + δ log θ + θδ)

(1 + θδ)2− 1

θ

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

(1 + (α+ δ)

log(θ)Ui

)−(

(α+ δ)Ui + ηVi

)Ui log(θ)

},

(31)

∆22 =n

(Γ(α))2

{Γ(α)Γ

′′(α)− (Γ

′(α))2

}−

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

(Ui(log(θxβi ))2 +

Γ′′(α)

Γ(α)Vi

)−(Ui log(θxβi ) + ψ(α)Vi

)2},

(32)

∆23 = ∆32 =−n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{Ui log(xi)

(1 + α log(θxβi )

)+

Viψ(α) log(xηi )}−(Ui log(θxβi ) + ψ(α)Vi

)(Ui log(xαi ) + Vi log(xηi )

)},

(33)

∆24 = ∆42 =−n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{ψ(η)Ui log(θxβi ) + ψ(α)Vi

(log(θ) + log(xβi )

)}

−(Ui log(θxβi ) + ψ(α)Vi

)(ψ(η)Ui + Vi

(log(θ) + log(xβi )

))},

(34)

∆25 = ∆52 = − log(θ)

n∑i=1

1

(Ui + Vi)2

{Ui(Ui + Vi) log(θxβi )− Ui

(Ui log(θxβi ) + ψ(α)Vi

)},

(35)



∆33 =n

β2+ θ

n∑i=1

xβi

(log(xi)

)2

−n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

(Ui

(log(xαi )

)2

+ Vi

(log(xηi )

)2)−(Ui log(xαi ) + Vi(log(xηi )

)2},

(36)

∆34 = ∆43 =−n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

{ψ(η) log(xαi )Ui + Vi log(xi)(

1 + η log(θ) + η log(xβi ))}−(Ui log(xαi ) + Vi log(xηi )

)(ψ(η)Ui + Vi

(log(xβi ) + log(θ)

))},

(37)

∆35 = ∆53 = − log(θ)

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)Ui log(xαi )− Ui

(Ui log(xαi ) + Vi log(xηi )

)},

(38)

∆44 =n

(Γ(η))2

{Γ(η)Γ

′′(η)− (Γ

′(η))2

}−

n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

(Γ′′(η)

Γ(η)Ui

+ Vi log(θxβi )(

log(θ) + log(xβi )))−(ψ(η)Ui + Vi log(θxβi )

)(ψ(η)Ui

+ Vi

(log(θ) + log(θxβi )

))},

(39)

∆45 = ∆54 = −n∑i=1

1

(Ui + Vi)2

{(Ui + Vi)

(log(θ)ψ(η)Ui

)−(ψ(η)Ui + log(θxβi )Vi

)Ui log(θ)

}(40)

and

∆55 =nθδ(log(θ))2

(1 + θδ)2−

n∑i=1

1

(Ui + Vi)2

{(log(θ)

)2

Ui(Ui + Vi)−(

log(θ))2

U2i

}, (41)

where

Ui = θα+δΓ(η)xαβ−1i ,

Vi = θηΓ(α)xηβ−1i ,

ψ(a) =Γ′(a)

Γ(a)



and

Γ(n)(a) =

∞∫0

ta−1(

log(t))ne−tdt

is the nth order derivative of gamma function.

The estimated variance-covariance matrix of the parameters θ, α, β, η and δ can be calcu-lated by

V ar(θ) Cov(θ, α) Cov(θ, β) Cov(θ, η) Cov(θ, δ)Cov(α, θ) V ar(α) Cov(α, β) Cov(α, η) Cov(α, δ)Cov(β, θ) Cov(β, α) V ar(β) Cov(β, η) Cov(β, δ)Cov(η, θ) Cov(η, α) Cov(η, β) V ar(η) Cov(η, δ)Cov(δ, θ) Cov(δ, α) Cov(δ, β) Cov(δ, η) V ar(δ)

=

∆11 ∆12 ∆13 ∆14 ∆15

∆21 ∆22 ∆23 ∆24 ∆25

∆31 ∆32 ∆33 ∆34 ∆35

∆41 ∆42 ∆43 ∆44 ∆45

∆51 ∆52 ∆53 ∆54 ∆55

−1

The diagonal elements V ar(θ), V ar(α), V ar(β), V ar(η) and V ar(δ) are the asymptoticvariances of the estimators of θ, α, β, η and δ respectively. The approximate 100(1 − ϕ)%

two-sided confidence intervals for θ, α, β, η and δ are θ ± Zϕ2

√V ar(θ), α ± Zϕ

2

√V ar(α),

β ± Zϕ2

√V ar(β), η ± Zϕ

2

√V ar(η) and δ ± Zϕ

2

√V ar(δ) respectively, where Zϕ

2is the

upper ϕ2th

percentile of a standard normal distribution.

* Here we use the likelihood ratio (LR) test to compare our model with its sub models fora given data. For testing η = δ = 1, the LR statistic is given by

ω = 2(

log l(θ, α, β, η, δ)− log l(θ, α, β, 1, 1)),

where θ, α, β, η and δ are the unrestricted estimates and θ, α and β are the restrictedestimates. The LR test rejects the null hypothesis if ω > χ2

d, where χ2d denote the upper

100d% point of the χ2 distribution with 2 degrees of freedom.

8. Application

In this section, we demonstrate the usefulness of the GSLMD by fitting a real data set.The data set was given by Bjerkedal (1960). The data are the survival times of guinea pigsinjected with different doses of tubercle bacilli. We used the data set obtained under theregimen 6.6. Guinea pigs are known to have high susceptibility to human tuberculosis. Evenan infection initiated with a few virulent tubercle bacilli will lead to progressive disease anddeath. The data set consists of 72 observations and are listed below:

Survival times of 72 guinea pigs under regimen 6.60.12, 0.15, 0.22, 0.24, 0.24, 0.32, 0.32, 0.33, 0.34, 0.38, 0.38,0.43, 0.44, 0.48, 0.52, 0.53, 0.54,0.54, 0.55, 0.56, 0.57, 0.58, 0.58, 0.59, 0.60, 0.60, 0.60, 0.60, 0.61, 0.62, 0.63, 0.65, 0.65, 0.67,



0.68, 0.70, 0.70, 0.72, 0.73, 0.75, 76, 0.76, 0.81, 0.83, 0.84, 0.85, 0.87, 0.91, 0.95, 0.96, 0.98,0.99, 1.09, 1.10, 1.21, 1.27, 1.29, 1.31, 1.43, 1.46, 1.46, 1.75, 1.75, 2.11, 2.33, 2.58, 2.58, 2.63,2.97, 3.41, 3.41, 3.76.

We fit the density functions of the generalized Stacy- Lindley mixture Distribution(GSLMD), Lindley distribution (LD1) ((Lindley (1958)), two-parameter Lindley distri-bution (LD2) (Shanker et al. (2013)), generalized Lindley distribution (GLD1)(Zakerzadehand Dolati (2009)) and a new generalized Lindley distribution (GLD2) (Abouammoh et al.(2015)). The pdfs of the LD1, LD2, GLD1 and GLD2 distributions are respectively given

asf1(x; θ) = θ2

1+θ (1 + x)e−θx; x > 0, θ > 0,

f2(x;α, θ) = θ2

θ+α (1 + αx)e−θx; x > 0, θ > 0, α > −θ,

f3(x;α, θ, γ) = θ2(θx)α−1(α+γx)(γ+θ)Γ(α+1) e−θx; x > 0, α, θ, γ > 0

and

f4(x;α, θ) = θαxα−2

(θ+1)Γ(α) (x+ α− 1)e−θx;x > 0, θ ≥ 0, α ≥ 1.

Maximum likelihood estimates and moment estimates of the parameters of the distributionsand their χ2 values are given in Table 1. They indicate that GSLMD fits the data setbetter than the other distributions. Figure 2 shows the plots of the fitted densities. Red linerepresents GSLMD and all other lines represent LD1, LD2, GLD1 and GLD1.



0.25 0.6 0.85 1 1.25 1.5 1.75 2.5 3 40

0.2

0.4

0.6

0.8

1

x

f(x)

Fig. 2. Fitted probability density function for the real data set.



Tab

le1.

Com

par

ison

offi

tofGSLMD

usi

ng

diff

eren

tm

eth

od

sof

esti

mati

on

of

72

gu

inea

pig

su

nd

erre

gim

en6.6

Count

Observed

Exp

ecte

dfr

equency

by

meth

od

of

mom

ents

Exp

ecte

dfr

equency

by

ML

ELD

1LD

2GLD

1GLD

2GSLMD

LD

1LD

2GLD

1GLD

2GSLMD

0-0

.25

514

14

15

13

713

12

15

12

70.2

5-0

.623

16

17

17

17

16

18

20

19

19

18

0.6

-0.8

518

99

910

11

98

99

12

0.8

5-1

.06

55

55

65

65

55

1.0

-1.2

53

77

67

87

57

77

1.2

5-1

.50

65

55

57

59

77

71.5

-1.7

52

44

44

54

32

35

1.7

5-2

.52

77

77

86

44

57

2.5

-34

22

22

23

32

32

3-4

33

22

22

22

22

2T

ota

l72

72

72

72

72

72

72

72

72

72

72

df

64

34

26

43

42

Est

imate

dvalu

es

θ=

1.4

2α

=1.2

1α

=0.9

α=

2.1

α=

2.6

θ=

1.5

8α

=1.3

42

α=

1.2

31

α=

2.3

14

α=

2.8

3

of

para

mete

rsθ=

1.5

1γ

=1.5

5θ=

1.5

1θ=

2θ=

1.4

91

γ=

1.4

32

θ=

1.7

4θ=

2.3

7

θ=

1.5

β=

1θ=

1.6

9β

=1.4

2η=

1.5

η=

1.3

δ=

0.6

δ=

0.9

2

χ2

25.7

88

20.6

56

20.7

51

17.1

93

13.9

06

22.3

98

18.9

17

19.2

37

16.8

62

9.7

53



9. Conclusions

A mixture of two Stacy gamma distributions called Generalized Stacy-Lindley mixture dis-tribution (GSLMD) is proposed. Properties of GSLMD including moments, conditionalmoments, vitality function, inequality measures and uncertainty measures are derived. Maxi-mum likelihood estimation and moment estimation techniques are used to estimate the modelparameters. In addition to this, the observed information matrix and asymptotic confidenceinterval are also included. Finally, the GSLMD model is fitted to a real data set to illustratethe usefulness of the distribution and it is concluded that our model performs well comparedto the existing Lindley models.cc

Acknowledgement. The authors are highly thankful for some of the helpful comments ofthe referee.

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