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Estimating Average Worth of theSelected Subset from Two-ParameterExponential PopulationsAditi Kar Gangopadhyay a & Somesh Kumar ba Department of Mathematics , Indian Institute of Technology ,Roorkee, Indiab Department of Mathematics , Indian Institute of Technology ,Kharagpur, IndiaPublished online: 02 Sep 2006.

To cite this article: Aditi Kar Gangopadhyay & Somesh Kumar (2005) Estimating Average Worth of theSelected Subset from Two-Parameter Exponential Populations, Communications in Statistics - Theoryand Methods, 34:12, 2257-2267, DOI: 10.1080/03610920500257220

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Communications in Statistics—Theory and Methods, 34: 2257–2267, 2005Copyright © Taylor & Francis, Inc.ISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920500257220

Ranking and SelectionMethodology

Estimating AverageWorth of the Selected Subsetfrom Two-Parameter Exponential Populations

ADITI KAR GANGOPADHYAY1 ANDSOMESH KUMAR2

1Department of Mathematics, Indian Institute of Technology,Roorkee, India2Department of Mathematics, Indian Institute of Technology,Kharagpur, India

Suppose independent random samples are available from k�k ≥ 2� exponentialpopulations

∏1� � � � �

∏k with a common location � and scale parameters �1� � � � � �k,

respectively. Let Xi and Yi denote the minimum and the mean, respectively, of theith sample, and further let X = min�X1� � � � � Xk� and Ti = Yi − X; i = 1� � � � � k.For selecting a nonempty subset of �

∏1� � � � �

∏k� containing the best population

(the one associated with max��1� � � � � �k�), we use the decision rule which selects∏i if Ti ≥ c max�T1� � � � � Tk�, i = 1� � � � � k. Here 0 < c ≤ 1 is chosen so that

the probability of including the best population in the selected subset is at leastP∗�1/k ≤ P∗ < 1�, a pre-assigned level. The problem is to estimate the averageworth W of the selected subset, the arithmetic average of means of selectedpopulations. In this article, we derive the uniformly minimum variance unbiasedestimator (UMVUE) of W . The bias and risk function of the UMVUE are comparednumerically with those of analogs of the best affine equivariant estimator (BAEE)and the maximum likelihood estimator (MLE).

Keywords Average worth; Best affine equivariant estimator; Exponentialpopulations; Subset selection; Uniformly minimum variance unbiased estimator.

Mathematics Subject Classification Primary 62F10; Secondary 62C15.

1. Introduction

The problem of estimation after selection arises in various industrial, agricultural,medical, and economic experiments. For example, a textile designer chooses thebest quality cloth from k available varieties for his usage. Naturally, he would beinterested in estimating the durability of the best cloth that he has selected. A doctor

Received May 5, 2003; Accepted May 27, 2005Address correspondence to Aditi Kar Gangopadhyay, Department of Mathematics,

Indian Institute of Technology, Roorkee 247667, India; E-mail: aditifma@iitr.ernet.in

2257

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2258 Gangopadhyay and Kumar

has experimented k types of drugs for a particular disease and selects the best one.He would like to have an estimate of the effectiveness of the selected drug forthe successful treatment of the disease. In the recent past, these types of problemshave received considerable attention from researchers. Dahiya (1974), Hsieh (1981),Cohen and Sackrowitz (1982), Hwang (1993), and Venter (1988) have proposedvarious estimators for estimating the mean of the selected normal population.The exponential, uniform, and gamma population models have been investigatedby Sackrowitz and Samuel-Cahn (1984), Vellaisamy et al. (1988), Vellaisamy andSharma (1988), Vellaisamy (1992b), and Kumar and Kar (2001), etc.

All of the abovementioned studies relate to the case when the selectionrule selects only one population. However, in various practical situations, one isinterested in not only the best, but also a subset of good populations (containingthe best). The problem of estimation after subset selection was initially formulatedand studied by Jayaratnam and Panchapakesan (1984) for two normal populations.They proposed three classes of estimators for the average worth of the selectedsubset and compared numerically their biases and mean squared errors.

Jayaratnam and Panchapakesan (1986) considered the problem of estimatingthe average worth of the selected subset from two exponential populations withunknown and different scale parameters and location parameter zero. They showedthat the natural estimator is positively biased and adjusted this natural estimatorto get an unbiased estimator when the scale parameters are same. However, asone of the scale parameters approaches zero, the bias of the natural estimatordecreases to zero whereas the unbiased estimator obtained for the common scaleparameter case has considerable negative bias. They also observed that the naturalestimator and the unbiased estimator are not comparable in terms of the meansquared error (MSE). This problem was further investigated by Vellaisamy andSharma (1990) who obtained the UMVUE. They also showed that the naturalestimator is inadmissible with respect to the squared error and the scale invariantloss functions and obtained an improved estimator. The numerical performances ofthese estimators were also compared.

Misra (1994) generalized the results of Vellaisamy and Sharma to k gammapopulations. The problem of estimating the average worth of the selected subsetfrom exponential populations for the location parameter case has been investigatedby Misra et al. (1998), who obtained the UMVUE. Anand et al. (1998) consideredthe case of uniform populations. They derived the UMVUE and also proved ageneral inadmissibility result. The natural estimator and the UMVUE were alsoshown to be inadmissibile. Simultaneous estimation of parameters of populations inthe selected subset has been studied by Vellaisamy (1992a, 1996).

In the present article, we consider the problem of estimating the average worthof the selected subset from k�≥2� exponential populations with a common locationbut different scale parameters. In Sec. 2, we introduce the notation and derive theUMVUE. In Sec. 3, numerical comparisons of the biases and the mean squarederror risk functions of the UMVUE, an analog of the best affine equivariantestimator and an analog of the maximum likelihood estimator, are made.

2. The Notation and Derivation of the UMVUE

Suppose independent random samples �Xi1� � � � � Xin�, �n ≥ 2�, i = 1� � � � � k areavailable from k exponential populations

∏1� � � � �

∏k with

∏i having the associated

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Estimating Average Worth of the Selected Subset 2259

probability density

fi�x� =1�i

e−�x−��/�i � x > �� �i > 0� (2.1)

Let Xi and Yi denote the minimum and the mean of the ith sample, respectively.Further, define X = min�X1� � � � � Xk� and Ti = Yi − X, i = 1� � � � � k. It can be provedthat �X� T1� � � � � Tk� is a complete and sufficient statistic (see Ghosh and Razmpour,1984, for a proof for the case k = 2). Also, X and T = �T1� � � � � Tk� are independentlydistributed with respective probability densities

fX�x� = n

( k∑i=1

�−1i

)exp

{−n�x − ��

( k∑i=1

�−1i

)}� x > �� �i > 0� (2.2)

and

fT �t� = nkn−1�n− 1�( k∑

i=1

�−1i

)−1( k∏i=1

�−ni

)�n�−k

( k∏i=1

tn−1i

)( k∑i=1

t−1i

)

× exp{−n

k∑i=1

ti�−1i

}� ti > 0� �i > 0� i = 1� � � � � k� (2.3)

(see Kumar and Kar, 2001). We are interested in selecting a nonempty subset S of�∏

1� � � � �∏

k� such that S contains the best population. Here the best population isthe one associated with max��1� � � � � �k�. A correct selection (CS) is defined as theevent in which S contains the best population. Following the general formulation ofsubset selection procedures by Gupta and Panchapakesan (1972), Misra et al. (2005)proposed the procedure R given by

R: Select the population∏

i in S if Ti ≥ cmax�T1� � � � � Tk�, where c ∈ �0� 1 is chosenso that

inf�P��CS �R� = P∗�

� = ��� �1� � � � � �k� and P∗ is the preassigned minimum probability of correctselection �1/k ≤ P∗ < 1�.

Jayaratnam and Panchapakesan (1984, 1986) and Vellaisamy and Sharma(1990) defined the average worth of the selected subset as the mean of thepopulations in S. Thus, in our case, the average worth W is given by

W =∑

i∈S��+ �i�

�S� = �+∑

i∈S �i

�S� (2.4)

where �S� denotes the cardinality of S.To derive the UMVUE of W , we first prove a lemma on the lines of the (U,V),

a method of Robbins (1987).

Lemma 2.1. Let T = �T1� � � � � Tk� be a random vector with the density function asgiven in (2.3) and Ui�t�, t = �t1� � � � � tk�, i = 1� � � � � k be k real valued functions

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2260 Gangopadhyay and Kumar

for which

(i) E�Ui�T� exists for all � and i = 1� � � � � k;(ii) the integrals

Ii�t� =∫ ti

0Ui�t1� � � � � ti−1� x� ti+1� � � � � tk�

x

k∑(l=1l �=i

)

k∏(

j=1j �=l�i

) tj+

k∏(j=1j �=i

) tj xn−2 dx�

i = 1� � � � � k

exist and are finite for all ti > 0, i = 1� � � � � k.

Then the functions V1�t�� � � � � Vk�t� defined as

Vi�t� =n

tn−1i

∫ ti

0

[lix

n−1 + �1− li�tixn−2

]Ui�t1� � � � � ti−1� x� ti+1� � � � � tk�dx�

i = 1� � � � � k� (2.5)

where li = 1− t−1i /

∑kj=1 t

−1j , i = 1� � � � � k satisfy E�Vi�T� = �iE�Ui�T� for all � and

i = 1� � � � � k.

Proof. The proof follows using an integration by parts with respect to ti in theexpression for EUi�T�. One of the functions for integration by parts is taken ase−nti/�i .

We first obtain the UMVUE for V =∑

i∈S �i�S� . In order to apply Lemma 2.1,

we define functions Ui�t� by

Ui�t� =I�ti� ct�k��∑kj=1 I�tj� ct�k��

� (2.6)

where

I�a� b� = 1� if a ≥ b�

= 0� if a < b�

t = �t1� t2� � � � � tk� and t�k� = max�t1� � � � � tk�.Thus V = ∑k

i=1 �iUi�T� and if we obtain functions Vi�t�, i = 1� � � � � k using (2.5),then

∑ki=1 Vi�T� will be unbiased for E��V�.

For evaluating functions Vi�t� we follow an approach similar to the one usedby Anand et al. (1998) of partitioning the space of t-values into

(k�k+1�

2 − 1)sets.

Let T�1� ≤ · · · ≤ T�k� be ordered values of T1� � � � � Tk and T�0� = 0. Define sets�r�s by

�r�s ={T � T�r� < cT�k−1� ≤ T�r+1�� T�s� < cT�k� ≤ T�s+1��

for r = 0� 1� � � � � k− 2 and s = r� r + 1� � � � � k− 1� (2.7)

Then sets �r�s form a partition of R+k = ��t1� � � � � tk� � ti > 0� i = 1� � � � � k�. Let T ∗

denote the vector of ordered Ti’s, that is, T∗ = (

T�1�� � � � � T�k�

).

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Estimating Average Worth of the Selected Subset 2261

It is easy to observe then that

k∑i=1

Vi�T� =k∑

i=1

Vi�T∗��

Now we define Li = 1− T−1�i� /

∑kj=1 T

−1�j� , i = 1� � � � � k. Then

Vi�T∗� = n

Tn−1�i�

∫ T�i�

0

[xLi + �1− Li�T�i�

]xn−2Ui�T�1�� � � � � T�i−1�� x� T�i+1�� � � � � T�k��dx�

for i = 1� � � � � k� (2.8)

For i = 1� � � � � k− 1, we have

Vi�T∗� = n

Tn−1�i�

∫ T�i�

0

[xLi + �1− Li�T�i�

]xn−2

I�x� cT�k��∑j �=i I�T�j�� cT�k��+ I�x� cT�k��

dx�

It can be seen that

Vi�T∗� = 0� if T�i� < cT�k�

= n

Tn−1�i� �S�

[1n

(Tn�i� − cnTn

�k�

)Li +

T�i�

n− 1

(Tn−1�i� − cn−1Tn−1

�k�

)�1− Li�

]�

if T�i� ≥ cT�k��

Therefore, for T ∈ �r�s, r = 0� 1� � � � � k− 2, s = r� � � � � k− 2,

Vi�T∗� = T�i�

k− s

[n− Li

n− 1− cnLi

(T�k�

T�i�

)n

− ncn−1

n− 1�1− Li�

(T�k�

T�i�

)n−1]�

i = s + 1� � � � � k− 1�

= 0� for i = 0� 1� � � � � s (2.9)

and for T ∈ �r�k−1, r = 0� 1� � � � � k− 2

Vi�T∗� = 0� i = 0� 1� � � � � k− 1� (2.10)

Now we evaluate Vk�T∗�. For this, we break the integral in two parts as

Vk�T∗� = n

Tn−1�k�

∫ T�k�

0

�xLk + �1− Lk�T�k��xn−2I�x� T�k−1��∑k−1

j=1 I�T�j�� cmax�x� T�k−1���+ I�x� T�k−1��dx

= n

Tn−1�k�

[ ∫ T�k−1�

cT�k−1�

�xLk + �1− Lk�T�k��xn−2

∑k−1j=1 I�T�j�� cT�k−1��+ 1

dx

+∫ T�k�

T�k−1�

�xLk + �1− Lk�T�k��xn−2

∑k−1j=1 I�T�j�� cx�+ 1

dx

]� (2.11)

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2262 Gangopadhyay and Kumar

Call the two integrals inside the square brackets in (2.11) as A and B. Then, it iseasy to see that for T ∈ �r�s and r = 0� 1� � � � � k− 2, s = r� r + 1� � � � � k− 1,

n

Tn−1�k�

A = T�k�

k− r

[�1− cn�

(T�k−1�

T�k�

)n

Lk +n�1− Lk�

n− 1�1− cn−1�

(T�k−1�

T�k�

)n−1]� (2.12)

For the evaluation of B, we consider further partitioning of the interval�T�k−1�� T�k� =

⋃k−1m=0 Pm, where

Pm = �T�k−1�� T�k� ∩[T�m�

c�T�m+1�

c

]�

and T�0� = 0. Then B = ∑k−1m=0 Bm, where

Bm =∫Pm

�xLk + �1− Lk�T�k��xn−2

∑k−1j=1 I�T�j�� cx�+ 1

dx� m = 0� � � � � k− 1� (2.13)

The expressions for Bm’s can be obtained after some algebraic calculations and weuse them to get

n

Tn−1�k�

B = T�k�

k− r

[n− Lk

n− 1−

(T�k−1�

T�k�

)n

Lk −n�1− Lk�

n− 1

(T�k−1�

T�k�

)n−1]�

if T ∈ �r�r � r = 0� 1� � � � � k− 2�

= T�k�

k− r

[{(T�r+1�

cT�k�

)n

−(T�k−1�

T�k�

)n}Lk

+ n�1− Lk�

n− 1

{(T�r+1�

cT�k�

)n−1

−(T�k−1�

T�k�

)n−1}]

+s−1∑

m=r+1

T�k�

k−m

[{(T�m+1�

T�k�

)n

−(T�m�

T�k�

)n}Lk

cn

+ n�1− Lk�

�n− 1�cn−1

{(T�m+1�

T�k�

)n−1

−(T�m�

T�k�

)n−1}]

+ T�k�

k− s

[n− Lk

n− 1−

(T�s�

T�k�

)nLk

cn− n�1− Lk�

�n− 1�cn−1

(T�s�

T�k�

)n−1]�

if T ∈ �r�s� r = 0� 1� � � � � k− 2� s = r + 1� � � � � k− 1�

Combining the expressions for A and B in (2.11), we get the value of Vk�T∗�. Using

it further in combination with Eqs. (2.9) and (2.10) then yields

k∑i=1

Vi�T∗� =

k−1∑i=r+1

T�i�

k− r

[n− Li

n− 1− cnLi

(T�k�

T�i�

)n

− ncn−1

n− 1�1− Li�

(T�k�

T�i�

)n−1]

+ T�k�

k− r

[n− Lk

n− 1− cnLk

(T�k−1�

T�k�

)n

− ncn−1

n− 1�1− Lk�

(T�k−1�

T�k�

)n−1]�

if T ∈ �r�r � r = 0� 1� � � � � k− 2�

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Estimating Average Worth of the Selected Subset 2263

=k−1∑i=s+1

T�i�

k− s

[n− Li

n− 1− cnLi

(T�k�

T�i�

)n

− ncn−1

n− 1�1− Li�

(T�k�

T�i�

)n−1]

+T�k�

[n− Lk

�n− 1��k− s�− cnLk

k− r

(T�k−1�

T�k�

)n

− ncn−1�1− Lk�

�n− 1��k− r�

(T�k−1�

T�k�

)n−1

−s∑

m=r+1

1�k−m��k−m+ 1�

{Lk

cn

(T�m�

T�k�

)n

+ n�1− Lk�

�n− 1�cn−1

(T�m�

T�k�

)n−1}]�

if T ∈ �r�s� r = 0� 1� � � � � k− 2� s = r + 1� � � � � k− 1�

Furthermore, the UMVUE of � is X − 1�n−1�

∑kj=1 T

−1j

(see Ghosh and Razmpour,

1984). Since∑k

i=1 Vi�T∗� is a function of the complete and sufficient statistic �X� T�,

we have proved the following theorem.

Theorem 2.1. The UMVUE for W is given by

�U�X� T� = X − 1

�n− 1�∑k

j=1 T−1�j�

+k−1∑

i=r+1

T�i�

k− r

[n− Li

n− 1− cnLi

(T�k�

T�i�

)n

− ncn−1

n− 1�1− Li�

(T�k�

T�i�

)n−1]

+ T�k�

k− r

[n− Lk

n− 1− cnLk

(T�k−1�

T�k�

)n

− ncn−1

n− 1�1− Lk�

(T�k−1�

T�k�

)n−1]�

if T ∈ �r�r � r = 0� 1� � � � � k− 2

= X − 1

�n− 1�∑k

j=1 T−1�j�

+k−1∑i=s+1

T�i�

k− s

[n− Li

n− 1− cnLi

(T�k�

T�i�

)n

− ncn−1

n− 1�1− Li�

(T�k�

T�i�

)n−1]

+T�k�

[n− Lk

�n− 1��k− s�− cnLk

k− r

(T�k−1�

T�k�

)n

− ncn−1�1− Lk�

�n− 1��k− r�

(T�k−1�

T�k�

)n−1

−s∑

m=r+1

1�k−m��k−m+ 1�

{Lk

cn

(T�m�

T�k�

)n

+ n�1− Lk�

�n− 1�cn−1

(T�m�

T�k�

)n−1}]�

if T ∈ �r�s� r = 0� 1� � � � � k− 2� s = r + 1� � � � � k− 1�

Remark 2.1. If we take c = 1 in the subset selection rule R, then the populationcorresponding to the largest Ti is selected and the estimation problem reduces tothat of estimating the mean of the selected exponential population. Kumar andKar (2001) considered quantile estimation of the selected exponential population.For b = 1, the quantile reduces to the mean. Thus, if we take c = 1, �U�X� T� is thesame as estimator �∗�X� T� of Kumar and Kar (2001) for b = 1.

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2264 Gangopadhyay and Kumar

3. Numerical Comparisons

In this section, we propose two new estimators of W and compare numerically theirbias and the mean squared error risk with that of the UMVUE. The results arepresented for the case k = 2. The form of the UMVUE �U for the case of k = 2 isgiven by,

�U = X − T�1�T�2�

�n− 1��T�1� + T�2��

+ T�1�

2

[n− L1

n− 1− cnL1

(T�2�

T�1�

)n

− ncn−1

n− 1�1− L1�

(T�2�

T�1�

)n−1]

+ T�2�

2

[n− L2

n− 1− cnL2

(T�1�

T�2�

)n

− ncn−1

n− 1�1− L2�

(T�1�

T�2�

)n−1]� if T ∈ �0�0�

= X − T�1�T�2�

�n− 1��T�1� + T�2��+ T�2�

[n− L2

n− 1− cnL2

2

(T�1�

T�2�

)nncn−1�1− L2�

2�n− 1�

(T�1�

T�2�

)n−1

− 12

{L2

cn

(T�1�

T�2�

)n

+ n�1− L2�

�n− 1�cn−1

(T�1�

T�2�

)n−1}]� if T ∈ �0�1�

Based on the ith sample, the best affine equivariant estimator of �+ �i isXi + n−1

nZi, (see, for example, Sharma and Kumar, 1994) where Zi = Yi − Xi. We

can use an analogue of this estimator for estimating W . Thus, for k = 2 we define

d1�X� Z� =X1 + X2

2+ n− 1

n

Z1 + Z2

2� if cT1 < T2 <

T1

c�

= XJ +n− 1n

ZJ� otherwise�

where TJ = max�T1� T2�, ZJ and XJ are Zi and Xi, respectively, from the populationcorresponding to TJ .

For the component problem, the MLE for �+ �i is Xi + Zi. Now, Xi + Zi = Yi.But we know that Yi = X + Ti. Therefore, another estimator for W can be con-structed by taking an analog of this estimator X + Ti (see, Kumar and Kar, 2001).

Table 1n = 4, P∗ = 0�95, c = 0�2334

�1 �2 B1 B2 R1 R2 RU

0.5 1.0 −0�090132 0.073662 0.104448 0.153389 0.1254511.0 1.0 −0�134586 0.064754 0.140636 0.183174 0.1516061.0 2.0 −0�180264 0.147324 0.417791 0.613554 0.5018041.0 3.0 −0�239207 0.257338 0.987354 1.539369 1.3366361.5 2.0 −0�227843 0.127122 0.464463 0.627702 0.4948511.5 2.5 −0�246884 0.173025 0.666430 0.944596 0.7571802.0 3.0 −0�314727 0.200606 0.993464 1.374690 1.092166

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Estimating Average Worth of the Selected Subset 2265

Table 2n = 10, P∗ = 0�95, c = 0�451020

�1 �2 B1 B2 R1 R2 RU

0.5 1.0 −0�016436 0.066758 0.051460 0.068955 0.0702411.0 1.0 −0�044682 0.049769 0.061461 0.074288 0.0644681.0 2.0 −0�032871 0.133517 0.205840 0.275820 0.2809661.0 3.0 −0�128409 0.132673 0.557082 0.687781 0.8176561.5 2.0 −0�064366 0.105494 0.207053 0.259736 0.2284561.5 2.5 −0�049325 0.157377 0.311697 0.410992 0.3846552.0 3.0 −0�072991 0.176886 0.451063 0.582618 0.532821

This may be defined as

d2�X� T� =X1 + X2

2+ T1 + T2

2� if cT1 < T2 <

T1

c�

= XJ + TJ � otherwise�

Misra et al. (2005) showed that the value of c is given by

∫ �

0Gn−1

(x

c

)gn�x�dx +

∫ �

0Gn

(x

c

)gn−1�x�dx = 2P∗�

where gn�·� and Gn�·� denote the probability density function and the cumulativedistribution function of gamma random variable having scale parameter 1 andshape parameter n.

The values of c for different values of n and P∗ are tabulated by Misra et al.(2005). In Tables 1–3, we present the bias and MSE risk values of estimators �U � d1,and d2 for k = 2. For values of c computed in Misra et al. (2005), the bias andrisk values are calculated using simulations based on 3,000 samples of sizes n =4� 10� 20 and for selected values of �1 and �2. These values do not depend on �.From the tables, we observe that the absolute bias of d2 is smaller than that ofd1 for small values of n. But when the difference between �1 and �2 increases, theabsolute bias of d1 becomes smaller than that of d2. However, as n increases the

Table 3n = 20, P∗ = 0�95, c = 0�582517

�1 �2 B1 B2 R1 R2 R3

0.5 1.0 −0�006222 0.039540 0.031400 0.036781 0.0468611.0 1.0 −0�018765 0.030194 0.031923 0.035719 0.0341551.0 2.0 −0�012444 0.079080 0.125599 0.147125 0.1874451.0 3.0 −0�122663 0.019152 0.382951 0.407300 0.4483071.5 2.0 −0�007208 0.082566 0.114419 0.134686 0.1353411.5 2.5 0�009801 0.122096 0.181176 0.219287 0.2636372.0 3.0 0�008754 0.142533 0.257255 0.309843 0.340907

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2266 Gangopadhyay and Kumar

performance of d1 becomes better than that of d2 with respect to absolute bias.With respect to the MSE, d1 always behaves better than the other two estimators.The MSE of the UMVUE is always smaller than that of d2. Similar trends havebeen observed for P∗ = 0�85� 0�90, n = 4� 10� 20� 30, and some other combinationsof values of ��1� �2�. We have also carried out computations for values of k = 3� 4� 5and different configurations of values of P∗� n� �1, and �2. The bias and the riskvalues are seen to show behavior similar to the case k = 2. We omit the tablesfor these cases. We have used the following notation in the tables: B1 = Bias of d1,B2 = Bias of d2, R1 = Risk of d1, R2 = Risk of d2, RU = Risk of �U .

Acknowledgments

The authors are thankful to the referees for some useful suggestions whichconsiderably improved the presentation of the article.

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