On Finite Mixture of two-Component Exponentiated ...

Journal of A pplied Sciences Res earch, 5(10): 1351-1369, 2009 © 2009, INSInet Publication

On Finite Mixture of two-Component Exponentiated Gamma Distribution 1

A.I. Shawky and 2R.A. Bakoban

1

King Abdulaziz University, Faculty of Sciences, Department of Statistics, P.O. Box 80203, Jeddah 21589, Saudi Arabia. 2 King Abdulaziz University, Faculty of Sciences, Branch Girls, Department of mathematics, P.O. Box 4269, Jeddah 21491, Saudi Arabia. Abs tr act: This article is cons idered with the problem of es timating the parameters , reliability and failure rate functions of the finite mixture of two components fro m e xp o nentiated gamma dis tributions . The maximum likelihood and Bayes methods of es timation are us ed. A n approximatio n fo rm due to Lindley (1980) is us ed for obtaining the Bayes es timates under the s quared e rro r los s and LINEX los s functions . Comparis ons are mad e b e t w e e n thes e es timators and the maximum likelihood ones us ing M onte Carlo s imulation s tudy. Key words : Finite mixture; Statis tical properties ; M aximum likelihood; Bayes ian analys is ; S q u a red error los s function; LINEX los s function; Lindley's approximation; Expone n t ia t ed gamma dis tribution. INTRODUCTION One of the important fa milies of dis tributions in lifetime tes ts is the exp onentiated gamma (EG) dis tribution with probability dens ity function ( p.d.f.)

(1.1) and the cumulative dis tribution function ( c.d.f.) is given by

(1.2) s ee Shawky and Bakoban [3 0 -3 4 ]. W hen the s hape parameter è = 1 in bo t h (1.1) and (1.2) g ive the p.d.f. and c.d.f. of gamma dis tribution with s hape parameter á = 2 and s cale p a ra meter â = 1, i.e., G(2,1). M ixtures of life dis tributions occur when two different c a us es of failure are pres ent, each with the s ame parametric form of life dis tributions . In recent years , the finite mixtures of life dis tributions have pro v e d t o be of cons iderable interes t both in terms of their methodological dev e lo p me n t a n d practical applications [2 2 ,1 9 ,2 3 ,2 1 ,7 ]. On characterizat ion of mixtures w ere s tudied by, Nas s ar and M ahmoud [2 5], Nas s ar [ 2 4 ] , Gharib [1 1 ] and Is mail and El Khodary [1 4 ]. M any authors Corresponding Author:

in teres ted with inferences on mixtures of exponential dis tributions among them Rider [2 8 ], Everitt and Hand [9 ], A l-Hus s aini [2], Bartos zewicz [4 ] a nd Jaheen [15 ]. Radhakris hna et al.[2 7 ] deriv e d moments and maximum likelihood es timators of the parameters of t w o component mixture generalized gamma dis tribution. A ls o, A hmad et al.[1] deriv e a pproximate Bayes es timation for mixtu re s of two W eibull dis tributions under type II cens oring. On finite mixture o f t wo component Go mpertz dis tribution cons idered by A lHus s aini et al . [ 3 ]. Several papers dis cus s ed normal mixtures , for example, Hos mer [1 3 ] and Holgers s on and Jorner [1 2 ]. M oreover, J ohn [1 6 ] briefly outlines the us e of t h e methods of moments and maximum likelihoo d in es timating the parameters of two component gamma mixtures . Further, a mixture o f two gamma dis tributions applied to rheumatoid arthritis was dis cus s ed by M a s u y a ma [ 2 0 ] . A ls o , Gh a rib [ 1 0 ] o b t ained two characterizations of a gamma mixture dis tribution. Furt h e r, a mixture of two invers e W eibull dis tribution was s tudied by Sult a n et al.[3 6 ]. A ls o, Els herpieny [8 ] es timat e d t h e p a ra meters of mixed generalized exponentionally dis tributions . A random va ria b le T is s aid to follow a finite mixture dis tribution with k componen t s , if the p.d.f. of T can be written in the form Titterington et al.[3 9 ]. (1.3)

where, for

is the

p.d.f.

A.I. Shawky, Permanent address: Fac. of Eng. at Shoubra, P.O. Box 1206, El M aadi 11728, Cairo, Egypt. 1351

J. App. Sci. Res., 5(10): 1351-1369, 2009 component and the mixing proportions,

s atis fy the j s uch that

conditions

and

The

and

Teic h e r [ 3 8 ] h a s

s tudie d t he clas s of gamma mixtures and proved its identifiable. W e now s how the identifiability of a mixture of k E G components in the following theorem.

corres ponding c.d.f. is given by Theorem 1: A finite mixture of k exponentia t ed gamma components is identifiable. (1.4) Proof: Teicher [ 3 7 ] s h o w ed that a finite mixture of exponential components is identifiable. If where

is the

c.d.f. component. The and

it

reliability function (RF) of the mixture is given by fo llo w s that (1.5)

. The trans formatio n is

one- to-one and onto , s o a finite mixture of EG(

),

j =1, 2, …, k components is identifiable. It follow s t h at where

is the

reliability component. ,

The hazard function (HF) of the mixture is given by a n d fo r a ll i, there exis ts s ome j

implies that s uch that where f(t) and R (t) are defined in (1.3) and (1.5), res pectively, i.e.

and

mixt u re o f k identifiable.

Therefore, a finite =1, 2, …, k , componen t s

is

Mi xture of k EG Components : By indexing the p.d.f. (1.1) and c.d.f. (1.2), j =1, 2, …, k then s ubs tituting in (1.3) and (1.4), the p.d.f. and c.d.f. of a finite mixture of (1.6) k EG(

),

j =1, 2, …, k components are given,

res pectively, by A mixture is identifiable if there exis t a one-to-o n e corres pondence between the mixing dis tribution and a res ulting mixture. Th a t is , t h e re is a unique characterization of the mixture. A clas s D of a mixture is s aid to be identifiable if and only if, for all f(t)å D the equality of the two repres entations [3 9 ]

(1.7) and

(1.8) implies that

and for all i there exis ts s ome

where, for j =1, 2, …, k ,

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and

J. App. Sci. Res., 5(10): 1351-1369, 2009

By obs erving that R(t) = 1-F(t) and

RF of a mixture of EG(

the

), j =1, 2, …, k components can obtained from (1.5) and (1.8) as

(1.9)

Dividing (1.7) by (1.9), we obtain the HF of a mixture of EG(

), j =1, 2, …, k components as

(1.10)

If k =2, the p.d.f., c.d.f., RF and HF of a mixture of EG(

), j =1, 2, components are then g iven, res pectively,

by

(1.11)

(1.12)

(1.13) and (1.14)

where f(t) and R(t) are given, res pectively, by (1.11) and (1.13). This article is cons idered w it h finite mixture of exponentiated gamma dis tributions . Statis tical properties for finite mixture of k exponentiated gamma components are derived in s ection 2. M aximum likelihood es timators of the two s h a pe parameters , reliability and failure rate functio n s of a mixture of two exponentiated gamma dis tributions are derived from comp le te and type II cens ored s amples in s ection 3. In s ection 4, Bayes es timators of the two s hape parameters , reliability and failure rate functions of a mixture of two expo n e ntiated gamma dis tributions are derived under the s quared error los s and LINEX los s functions . A ls o, M onte Carlo

1353

s imu lation concluding es timators cons idered

s tudy are made in s ection 5. Finally, remarks a bout comparis ons between thes e and the maximum likelihood ones are in s ection 6.

2. S tatis tical Properties : 2.1 Moments and S ome Meas ures : The about the origin, EG(

moment

, of a mixture of

), j =1, 2, …, k , components with p.d.f. (1.7)

in the non-clos ed form is

J. App. Sci. Res., 5(10): 1351-1369, 2009

(2.1)

that is , for pos itive real value of

takes the clos ed form

(2.2)

where

(2.3)

where

W h e n k = 2, t h e

, of a mixture of two EG (

moment about the origin,

), j = 1, 2,

components with p.d.f. (1.11) is given by

The a b o v e c lo s e d form of EG(

allows us to derive the following forms of s tatis tical meas ures for the mixture of

), j =1, 2, …, k components :

1- Coefficient of variation:

(2.5)

2- Skewnes s :

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J. App. Sci. Res., 5(10): 1351-1369, 2009

(2.6)

3- Kurtos is :

(2.7)

where

where abbreviation is The

res ult

developed

above

can

, of the mixture of EG(

be

derived

from the

moment

generating

function (m.g .f.),

), j =1, 2, …, k , components as follows .

(2.8)

W hen

(2.8) gives the m.g.f. of EG dis tribution with p.d.f. given in (1.1).

The mean deviation,

, of the mixture of EG(

), j =1, 2, …, k , components ,

random variable X with p.d.f. (1.7) in the non-clos ed form is

(2.9) where which is defined in (2.2) with r =1, and F(x) is the c.d.f. (1.8) . that is , for pos itive real value of

M.D. takes the clos ed form

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J. App. Sci. Res., 5(10): 1351-1369, 2009

(2.10)

Median and Mode: The median of the mixture of EG( form.

), j =1, 2, …, k , components can not be foun d in e xp lic it

W e derive the median m as the numerical s olution of the following equation:

(2.11)

where

is a real number.

Next, to find the mode for the mixture of EG( to t, s o (1.7) gives

), j = 1, 2, …, k , components , we differentiate f(t) with res pect

(2.12)

Then by equating (2.12) with zero, we get mode(s ). W e obs erve that the mixt u re o f E G (

), j =1, 2, …, k ,

components , may be unimodal (s ee Fig. 1) or bimodal (s ee Fig. 2) with mode(s ) can be found numerically by s olving (2.12). Figure 1 s hows s ome dens ities of EG(

Fig. 1: Shapes of EG(

), j =1, 2, components and their unimodal mixtures .

), j =1, 2, components and their mixtures with (p

1356

).

J. App. Sci. Res., 5(10): 1351-1369, 2009

Fig. 2:

Shapes of EG(

), j =1, 2, components and their mixtures with (p

mixture, The - - - - curve for EG(

), The ….. curve for EG(

), The non-das hed curve for ).

3. Maximum Lik elihood Es timation: Suppos e a T y p e -II cens ored s ample is the time of the tes t of n given,

it e ms

where

component to fail. This s ample of failure t ime s a re o b t a ined and recorded from a life w h os e life times have the mixture of

=1, 2, components with p.d.f. a n d c .d .f.

res pectively, by (1.11) and (1.12). The likelihood function in this cas e [1 7 ] can be written as :

,

(3.1)

where f(t) and R(t) are given, res pectively, by (1.11) and (1.13). Figure 2 s hows s ome dens ities of EG(

), j =1, 2, components and their bimodal mixtures .

The natural logarithm of the likelihood function (3.1) is given by

(3.2)

A s s uming that the parameters ,

and

are unknown, the likelihood equations are given, for j=1, 2, by

(3.3)

From (1.11) and (1.13), res pectively, we have

(3.4)

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J. App. Sci. Res., 5(10): 1351-1369, 2009 and

(3.5)

where

(3.6) and where, for j=1, 2, (3.7)

of

Subs tituting (3.4) and (3.5) in (3.3), we obtain

is a prio r d e n s ity function

and W e choos e the random variable

to

follow gamma dis tribution with s hape parameter , i.e., G(

(3.8)

a n d s cale parameter function is

,1). Its dens it y

(3.9)

Bas ed on the above cons iderations , the prior dens ity

where, for j=1, 2, and i=1, …, r,

and

a re given by (3.6) and

function of

is given by

(3.7), res pectively. The s o lution of the two nonlinear likelihood equations (3.8) yields the maximum likelihood es timate (M LE)

of

(4.1)

The M LE’s of R(t) and H(t) are given, res pectively, by It is well known that the pos terior dens ity function (1.13) and (1.14) a ft e r re p la c in g

and

by their of

corres ponding M LE’s ,

given the obs ervations , denoted by

and is given by

4. Bayes Es timation: Let

and

be independent (4.2)

random variables . The joint prior dens ity of the random vector

is thus given by It then follo w s t h a t the Bayes es timator function of t h e parameter

1358

of a

is given by the ratio

J. App. Sci. Res., 5(10): 1351-1369, 2009

w h e re and

is given by (3.1), is the region in the

the pos terior dens ity

by (4.1) plane on which

is pos itive.

(4.3) The ra t io of the integrals (4.3) may thus be approximated by us ing a form due to Lindley [1 8 ] which reduces , in the cas e of two parameters , to the form

(4.4)

where

matrix (3.1).

For

element in the

w h e re

is as given by

For

where

and

is given in (4.1). Finally,

and

Now, we apply Lindley's form (4.4), we firs t obtain the elements

and

which can be obtained as

(4.5)

where

(4.6)

(4.7)

and for

(4.8)

(4.9)

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J. App. Sci. Res., 5(10): 1351-1369, 2009

and

(4.10)

(4.11)

(4.12) Fo r and

t h e functions

and

are as given by (3.6) and (3.7),

and

by (3.9)

is given by

(4.13)

Furthermore,

(4.14)

(4.15)

(4.16)

(4.17)

where, for

and

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J. App. Sci. Res., 5(10): 1351-1369, 2009

For

and

(4.18)

(4.19)

(4.20) and

res pect to the pos terior d e n s it y of â. By a res ult of Zellner[4 1 ] , t h e (unique) Bayes es timator of â, denoted (4.21) by

under t h e LINEX los s is the value

which

In Bay es ian es timation, we cons ider two types of los s functions . The firs t is the s quare d error los s function (quad ra tic los s ) which is clas s ified as a s ymmetric function and ass ociates equal importanc e to the los s es for overes timation and underes timation of equal magnitude. The s e c o nd is the LINEX (linearexponential) los s function which is as ymmetric, was introduced by Varian [4 0 ]. Thes e los s functions were widely us ed by s everal authors ; among of them Rojo [2 9 ], Bas u and Ebrahimi [5 ], Pandey [2 6 ], and Soliman [3 5 ]. The quadratic lo s s for Bayes es timate of a parameter â, s ay, is the pos terior mean as s uming that

minimizes (4.23), is given by

exis ts , denoted by be expres s ed as

es timated u s ing Lindley's approximation from (4.4), and their es timates are obtained as follows :

The LIN EX los s function may

(4.22) where

(4.24) provided that the expectation

exis ts

and is finite [6]. 4 .1 Es timation under S quared Error Los s Function: 4.1.1 Bayes Es timation of th e Vector of Parameters : The two parameters ,

and

can be approximately

i.The Bayes es timate of the parameter

The s ign and ma gnitude of the

Set

in (4.4). Then

s hape paramet e r c reflects the direction and degree of as ymmetry res pectively. (If

the overes timation

is more s erious than underes timation , and vice-vers a). For c clos ed to zero, the LINEX los s is approximately s quared error los s and therefore almos t s ymmetric. The pos terior expectation of the LINEX los s function Equation (4.22) is

(4.23) where

denoting pos terior expectation

for

and

with

Subs tituting the above functions and (4.14-17) in (4.4) yields the Baye s e s timate under s quared error los s 1361

J. App. Sci. Res., 5(10): 1351-1369, 2009 function,

of

ii. The Bayes es timate of the parameter Set

in (4.4). Then

and Subs tituting the above functions and (4.14-17) in (4.4) y ields the Bayes es timate under s quared error lo s s for function,

of

in (4.4), where R(t) is given as in (1.13). Then, for

4.1.2 The Bayes Es ti mate of th e R F: S e t

(4.25)

and

where

(4.26)

and

are as given, res pectively, by (1.1), (1.2) and (3.7).

Subs tituting (4.25), (4.26) and (4.14-17) in (4.4) yields the Bayes es timate under s quared error los s function, of in (4.4), where H(t) is giv e n a s in (1.14). T h e n , fo r

4 .1 .3 The Bayes Es timate of the HF: Set

(4.27)

(4.28)

where

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J. App. Sci. Res., 5(10): 1351-1369, 2009

,

w h e re (3.7). For

and

are as given, res pectively, by (1.1), (1.2), (3.6), (4.21) a n d

and

(4.29)

where

Subs tituting (4.27-29) and (4.14-17) in (4.4) yields the Bayes es timate un d e r s q u a re d e rro r los s function, H(t)

of

4.2 Es timation Under LINEX Los s Function: On the bas is of the LINEX los s function (4.24), t h e Ba yes es timate of a function

of the unknown parameters

and

is given by

(4.30)

where 1363

J. App. Sci. Res., 5(10): 1351-1369, 2009

(4.31)

Sup p o s e

s o we can apply Lindely's approximation cited previous ly as it was us ed to evaluate

(4.4). So we obtain the following: 4.2.1 Bayes Es timation of the Vector of Parameters : The two parameters ,

and

c a n b e a p p ro ximately

es timated us ing Lindley's approximation from (4.4), and their es timates are obtained as follows : i. The Bayes es timate of the parameter Set

in (4.4). Then

for

Subs tituting the above functions and (4.14-17) in (4.4) then into (4.30) yields the Baye s es timate under LINEX los s function,

of

ii. The Bayes es timate of the parameter Set

in (4.4). Then

for

Subs tituting the above functions and (4.14-17) in (4.4) then into (4.30) yields the Bayes es timate und e r LINEX los s function,

of in (4.4), where R(t) is given as in (1.13). Then, for

4.2.2 The Bayes Es ti mate of th e R F: S e t

(4.32)

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J. App. Sci. Res., 5(10): 1351-1369, 2009

for

(4.33)

and

(4.34)

where

and

are as given, res pectively, by (1.1), (1.2) and (3.7).

Subs tituting (4.32-34) and (4.14-17) in (4.4) then into (4.30) y ie ld s t he Bayes es timate under LINEX los s function, of R(t)

4.2.3 The Bayes Es timate of the HF: Set

in (4.4), where H(t) is given as in (1.14). Then,

for

where

is defined in (4.27), then

(4.35)

where (4.36)

for

(4.38)

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J. App. Sci. Res., 5(10): 1351-1369, 2009

w h e re

and

are given, res pectively, by (1.1), (1.2), (3.6), (4.21),

(4.36) and (3.7). Subs tituting (4.35), (4.37), (4.38) and (4.14-17) in (4.4) then into (4.30) y ie ld s the Bayes es timate under LINEX los s function,

of H(t)

5. S imulation S tudy: W e o b tained, in the above Sections , Bayes ian and non-Bayes ian es timates of the vector parameters

relia b ilit y , R ( t ) and failure rate, H(t) functions of a mixture of two EG(

), j =1, 2, components .

W e adopted the s quared error los s and LINEX los s functions . The M LE’s are als o obtained. In order to as s es s the s tatis tica l p erformances of thes e es timates , a s imula tion s tudy is conducted. The e s t ima t es relative to LINEX los s and computations are carried out for cens oring percentages of 80% and 100% (complete s ample cas e), for eac h are c o mp uted, us ing (4.4) after cons idering the s amp le s ize. The mean s quare errors (M SE’s ) us ing appropriate changes acc o rd ing to Subs ections (4.2.1), generated random s amples of different s izes are (4.2.2) and (4.2.3). computed for each es timator. The random s amples are 5. The above (2-4) s teps are repeated 1000 times and the generated as follows : bias es and the mean s quare errors are computed for different s ample s izes n and cens oring s izes r. 1. F o r g iven values of the prior parameters and The computational (our) res ults were computed by us ing M athematica 4.0. In a ll a b ove cas es the prior g e nerate a random values for and from parameters chos en as =2 and =0.5 which yield the gamma dis tribution G( ,1), j=1, 2. the generated values of = 2.01307 and 2. U s in g and , obtained in s tep (1), generate =0.56372 (as the true values ). The true values o f R (t) ra ndom s amples of different s izes : n =15, 25 a n d 50 and H(t) when are computed to be from a mixt u re o f t w o E G( ), j =1, 2, components R(0.5)= 0.86723 and H(0.5) = 0.312305. The bias (firs t as given by (1.11). The computations are carried out for entries ) and M SE’s (s econd entries ) are dis played in s uch s ample s izes and cens ored s amples of s izes : 12, 20 Tables 1-4. T he computations are achieved under and 40, res pectively. complete and cens ored s amples . Tables 1, 2, 3 and 4 con t a ins es timated bias es and 3. The M LE’s of the vector M SE’s of M LE’s , Bayes es timators under quadratic and parame t e rs

are obtained by iteratively

s olving (3.8). T h e e s timators

LINEX los s functions o f res pectively.

and 6. Concluding Remark s In this paper we have pres ented the Bayes ian and maximum likelihood es timates of the vector parameters

of the functions R(t) and H(t) are the n computed at

reliability, R(t) and failure rate, H(t) function s o f

s ome values 4. The Bayes es timates relativ e to s quared error los s , and

and H(t)

are computed, us ing (4.4) after

t h e lifetimes follow a mixture of two =1, 2, components . The es timation are conducted on the bas is of complete and type-II cens ored s amples . Bayes es timators , under s quared error los s and LINEX los s functions , are derived in approximate forms by us ing

cons idering the appropriate changes according to Subs ections (4.1.1), (4.1.2) and (4.1.3). A ls o, the Bayes

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J. App. Sci. Res., 5(10): 1351-1369, 2009 Lindley’s method. The M LE’s are als o obtained. Our obs ervations about the res ults are s tated in the following points : 1. Table 1 s hows that the Bayes es timates u n der the quadratic los s function a re the bes t es timates as compared with the bias es of es timates under LINEX los s function or M LE’s . This is T ru e for both complete and cens o re d s amples . It is immediate to note that M SE’s decreas e as s ample s ize increas es . On the other hand the Bayes es timates under the LINEX los s function have the s malles t es timated M SE’s as compared with the es timates und er quadratic los s function or M LE’s . This is True for both complete and cens ored s amples .

2.

Table 2 s hows that the Bayes es t imates under quadratic los s function have the s malles t es timated M SE’s as compared with the Bayes es timates under LINEX los s func t io n or M LE’s . This is true only for complete s amples . On the other hand the Bayes es timates under the quadratic los s function have the bes t bias as compared with t h e es timates under LINEX los s function or M LE’s for cens ored s amples . A ls o, we note that M SE’s us ually decreas e as a complete s ample s ize increas es . Further, Bayes es timates under LINEX los s function have the s malles t M SE’s . This is True fo r s mall and moderate cens ored s amples .

Table 1: Estimated biases ( first entries ) and MSE’s ( second entries ) of various estimators of

n

r

15

12

15

15

25

20

25

25

50

40

50

50

, c=2.5 -1.0711 1.59839 -1.08531 1.45857 -1.15158 1.59089 -1.11603 1.55293 -1.16001 1.54395 -1.17164 1.53987

-0.577146 0.752019 -0.491996 0.594549 -0.576757 0.759242 -0.418845 0.536734 -0.411279 0.50017 -0.197093 0.158257

Table 2: Estimated biases ( first entries ) and MSE’s ( second entries ) of various estimators of

n

r

15

12

15

15

25

20

25

25

50

40

50

50

for different sample sizes.

-0.776452 0.160452 -0.760799 0.579267 -0.727412 0.20037 -0.622922 0.410337 -0.564431 0.077494 -0.418115 0.20275 for different sample sizes.

, c=2.5 0.53883 0.503419 0.564431 0.533643 0.580301 0.542394 0.517879 0.468563 0.580396 0.500746 0.567017 0.499627

0.008351 0.043882 0.039072 0.029057 -0.000837 0.041848 0.026582 0.010961 -0.004068 0.028116 0.016411 0.003708

-0.10603 0.031005 -0.064143 0.029164 -0.069954 0.014245 -0.017597 0.015448 -0.06293 0.062669 -0.011661 0.00643

Table 3: Estimated biases ( first entries ) and MSE’s ( second entries ) of various estimators of R(t) for different sample sizes. n

r

15

12

15

15

25

20

25

25

50

40

50

50

, c=5 -0.135564 5.20895 -0.094805 1.9792 -0.119424 4.47587 -0.078305 1.35839 -0.001362 0.003671 -0.018272 0.15388

0.048574 0.048122 0.061839 0.044025 0.040625 0.03999 0.044069 0.019383 0.031492 0.02917 0.028585 0.004704

1367

0.007284 0.004098 0.048557 0.002935 -0.005528 0.003919 0.034665 0.00087 0.017442 0.003474 0.024907 0.002698

J. App. Sci. Res., 5(10): 1351-1369, 2009 Table 4: Estimated biases ( first entries ) and MSE’s ( second entries ) of various estimators of H(t) for different sample sizes. n

r

15

12

15

15

25

20

25

25

50

40

50

50

3.

4.

,c=5 0.056278 0.678701 0.085643 1.12149 0.094125 0.177431 0.090051 0.273923 0.069277 0.044136 0.112697 0.929472

-0.017486 0.059764 0.028995 0.05686 -0.033006 0.068893 0.027403 0.034304 -0.016413 0.041539 0.017439 0.006246

Table 3 s hows that the M LE’s have the bes t bias as compared with the Bayes e s t imates under LINEX los s function or quadratic los s functio n fo r large s amples for both complete and cens ored cas es . For s mall and moderate s amples Bayes es timates under the LINEX los s function have the bes t bias as compared with the e s t imates under LINEX los s function or M LE’s . This is True for both complete and cens ored s amples . It is immediate to note that M SE’s us ually d e creas e as s ample s ize increas es . On the other hand the Bayes es timates under the LINEX los s function have the s malles t es timated M SE’s as compared with the es timates under q u a dratic los s function or M LE’s . This is True for both complete and cens ored s amples . T a ble 4 s hows that the Bayes es timates under t h e LINEX los s function us ually have the s malles t es timated M SE’s as compared with t h e es timates under quadratic los s function or M LE’s . This is True for both complete and cens ored s amples . It is imme diate to note that M SE’s us ually decreas e a s s ample s ize increas es . On the other hand the Bayes es timates under the quadratic los s function us ually have the bes t bias as compa red with the es timates under LINEX los s function or M LE’s .

From the previous obs ervations , the es timation from a finite mixture of two EG componen t s data is pos s ible and flexible us ing Bayes approach, es pecially us ing as ymmetric lo s s function s uch as LINEX function, which is the mos t appro priate for all parameters as s hown from this article. W e als o, recommend to us e Bayes es timates u n der quadratic los s function for and H(t)

2.

3.

4.

5.

6.

7. 8.

9. 10.

11.

12.

13.

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