ESTIMATION OF THE SMALLER AND LARGER OF TWO PARETO SCALE PARAMETERS
JUNG Soo WOO AND CHANG SOO LEE
1. Introduction
Many authors have utilized a Pareto distribution because of its wide applicability in socio-economic, physical and biological phenomena.
The problem considered in this paper is estimation of the minimum and maximum of two unknown Pareto scale parameters. Similar problems were considered by Blumenthal and Cohen( 1968a, b, and c), who were estimating the large translation parameter of two symmetric(mainly normal) distribution. Kushary and Cohen(l989) considered a similar but somewhat different problem when it was known which populations correspond to each scale parameter. These authors found improved estimators of scale parameters, in a general setup, under the condition that the first sample corresponded to the smaller scale parameter and the second to the larger. Recently, Carpenter and Pal( 1992) have con- sidered the estimation of the smaller and larger of two exponentially unknown location parameters. Elfessi and Pal(1992) have considered the estimation of the smaller and larger of two uniform scale parame- ters.
Let Xi},' .. , X m , i = 1,2, be a pair of independent random samples from populations which are Pareto distributed with unknown scale parameters Ai, i = 1,2, and a common known shape parameter 0 as follows ; for i = 1,2,
(1.1) f(Xi;O,Ad = oAfx;<o+lJ, 0 < Ai < Xi, 0 < o.
Received November 18, 1994.
1991 AMS Subject Classification: 62F11.
Key words: MLE, minimax bias estimator, minimax risk estimator.
Define 8 1 = minimum(AI, A2) and 8 2 = maximum(A1' A2). A sufficient statistic for Ai is Yi = max: {XiI'··· ,Xin }, i = 1,2. Our goal is to estimate 81 and 8 2 based on Y1 and 12.
In the next section, we shall introduce several estimators of 8 1 and 8 2 , and compare them in terms of standardized bias and risk under square error loss function. The following definitions are used to clarify the various criteria used in this paper.
DEFINITION 1.1. Let 6 i be an estimator of 8 i , i = 1,2.
(a) The standardized biases-bias) of 6; is defined as
(1.2)
(b) The risk of 6 i under the square error loss is defined as
(1.3)
The reason why B(6 i ) and R(6 i ) are considered instead of the usual bias and mean square error is the fact that they are invariant under scale transformations.
2. Estimation of the smaller and larger of two scales
Define Zl = min{Y 1 , Y 2 } and Z2 = max:{Yi, Y2}. ML estimators of 81 and 8 2 are
Note that the reparametrization from (AI, A2) to (81,8 2 ) is not one-to- one. Zehna(1966) defined the induced likelihood function L 1 (8I,8 2 IYI, Y2) as
(2.2)
U = {(AllA2)181 = min(A}, A2) ,82 = max(A1, A2)}.
It is easy to see that uW) ,6~1») maximizes the induced likelihood function L 1 (8},8 2 IY},Y2), and hence (6P),6~1») is said to be an ex- tended maximum likelihood estimator (MLE) of (8},82). Another way to de:6.ne an MLE is to use the joint density of (Zl' Z2) as (see Dudewicz(1968)) the likelihood function in hope that (Z}, Z2) carries all of the information relevant for estimation of (81,82). The following Lemma gives the likelihood function L2(8}, 82 1z}, Z2) of (8I,82) based on (Z}'Z2).
LEMMA 2.1. The joint density of (Z}, Z2) is
(2.3) 8 1 :5 Zl :5 8 2 :5 Z2
8 1 :5 8 2 :5 Zl :5 Z2·
Proof. Note that the joint dJ. of (Zll Z2) is G(ZI, Z218I,82) =P{Zl :5 Z}'Z2 :5 Z2}
=P{Z2 :5 Z2} - P{Zl > Zl, Z2 :5 Z2}
=A 1 - A 2 ,
Now differentiating G(z}, Z2) with respect to Zi, i = 1,2, we can get the
joint density function of Zl and Z20
(2.4)
It is interesting to note that on the region 1 1 = {( 8t, ( 2 )101 ::; Z1 ::;
8 2 ::; Z2}, the likelihood function L1(01,02Izt,Z2)(= g(zt,z2101l82)) is maximized at (Zt,Z2) and max L1(81l02IzllZ2) = (na)2(z1·Z2)-1. On the other hand, on the region 1 2 = {(8t, ( 2 )18 1 ::; 8 2 ::; Z1 ::; Z2}
is maximized at (Z1,Z1) and max L2(81,82Izt,Z2) = 2(na?z;a-1.
z2na- 1. Hence an estimator maximizing L 2( 8b 821zb Z2) is
(
A(2) A(2»)_{(Zt,Z2), if (~)na<t
8 1 ,8 2 - Z
(ZllZ1), if (~)na ~ t.
Obviously, (8~2),8~2») i= (8P),8~1») with probability P{(Zt/Z2)na ~
1/2} > o. The reason these estimators can differ is straightforward, since the estimators axe chosen to maximize two different likelihood functions and these likelihood functions are not necessarily propor- tional. For further discussion on this phenomenon, see Pal and Berry (1991). We shall derive the s-bias and risk of 8~2) later.
In the following, we suggest various estimators of 8 1 and 8 2 ' and derive their biases and MSE expressions. In each case, the expressions depend on the parameter value only through 0 = 8t/8 2 ,O < 0 ::; l.
Before going into the details, we first show that both 8 1 and 8 2 can not be estimated unbiasedly as long as we look at Z1 and Z2 only.
From (2.3), it is easy to show that for i = 1,2, na ~ 2 , (2.5)
Therefore,
(2.6)
IB(9(l»)I-1 1 . -na 1 -'-1 _ sna-1 ' ( n a ---' 1)(2na - na 1) , I
IB(9(1»)! _I 1 ono: na I
2 - na - 1 + (na - 1 )(2na - 1) .
So, the usual estimators are biased. It is tempting to estimate 010na-1
and 02sna using suitable unbiased estimators and hence to obtain un-
biased estimators of 8 i , i = 1,2. But the following result shows that
it is impossible to obtain such unbiased estimators based on the Zi'S
alone.
THEOREM 2.2. If we assume (h < ()2 (i.e., Al =1= A2), then there does not exist any unbiased estimators (function of Zl and Z2 only) of 0i, i = 1,2.
Proof. From (2.5), note that ()l and ()2 are unbiasedly estimable if and only if ()16 ncr - 1 and ()28nct are unbiasedly estimable. So it is enough to show that ()18 no - 1 and ()28nat are not unbiasedly estimable and we will prove it by contradition.
Suppose there is a T1 = T1(Zl, Z2) such that E[Td = ()18 no - 1 , o < ()l < ()2. Then
(2.7)
The right hand side(RHS) of (2.7) is a function of ()2 only, whereas the first term on the left hand side(LHS) depends on 8 2 as well as ()l'
Therefore, (2.7) is true for all ()l and (h (0 < 8 1 < ()2 < (0), if and only if
(2.8) 1 9 2 i 1 Zl, Z2 zl ( ) -(no+l)d Zl = 0 ; . l.e., 9 1
l b i 1(Zl,Z2)Z;(nct+l)dz1 = 0, 'v'a,b; 0 < a < b < 00.
But (2.8) implies that J~2 i(Zl, Z2)Z;(no+l)dz 1 = 0 for any ()2 < Z2,
which implies the fact that LHS of (2.7) is zero, and it is yielding a contradiction since the RHS is nonzero. Similarly, one can prove that there does not exist any T2 = T2(ZI,Z2) such that E[T 2] = 8 2 8 ncr •
To improve over the standard estimators ep) and 8~1), we consider the class of scale equivariant estimators of ()i, i = 1,2, given as
C i ={9 i(c)=C,Zi; O<c}, i=1,2.
Note that,
(2.9) I c na -I-cv cna-1 na I
na - 1 (na - 1)(2na - 1) ,
I na 6 na na I
= l - c na _ 1 -c (na-l)(2na-l)·
Clearly, it is impossible to :find a Co such that IB(Di(eo»1 is minimun in the class Ci for all 6,0 < 6 :::; 1. Therefore , we adopt the minimax absolute s-bias approach i.e., choose c o (> 0) such that
(2.10)
The following Theorem gives the optimal estimators of () i in C i, i = 1,2, under the criterion (2.10).
T L 2(na-1){2na-1)· 1 2 T'h t' t
HEOREM 2.3. et Ci = na(4na+2i-5) ' 1 = ,. e es lIDa ors
Di(e;) = CiZi is the minimax bias estimator(according to (2.10)) for (Ji,
i = 1,2.
Proof First we consider estimator of (}1. Let
B (l) _ c na - 1 -cv cna-l na
I - na -1 (na -1)(2na -1)"
Then, for any estimator D 1 (e) = CZI, (c > 0),
{ B (l)
~
1 ,
IB{(}l(e»! ~ . . ' ( 1 )
-B 1 ,
if c>~ . - 2nar
if c < .lli!=.!. na Case(i). c ~ 2~:~1. In this case :6IB(D1(e»)1 < 0 and
~
na
max IB((}l(e»1 = c 1 - 1,
0<69 na-
which is increasing in c. Therefore, min max IB(D 1 (e»1 = 2 1 2.
O<e 0<6$1 na -
Case(ii). 0 < C ~ n~;l. In this case 16/B(81(c»)1 > 0 and
A
2no:
max IB(8 1 (c»)1 = 1 - c
2 l'
0<6$1 no: -
which is decreasing in c. Therefore, min max IB(6 1 (c»)1 = 2 1 1 o<c 0<6<1 no: - Case(iii). ~ no, < C < l.!!g=l 2no, . -
1
L e t t5 - n - {2 no: - 1 - (no,-1)(2na<-1)} CRQ ~ . Th en
where B~2) = _B~l).
So, max IB(8 1 (c»)1 = max {c no: 1 - 1,1 - c
2 2no: 1} ,
0<6<1 no: - na -
- A A
1
which implies that min max o<C 0<6<1 IB(8 1 (c»)1 = IB(8 1(c»)lc=Cl = 4 no: - 3' h 2 Ro,-1) 2Ro,-1) -
were Cl = no, 4no,-3 .
Combining the preceding three cases, we can get the required result.
The result for 8 2 is proved as the similar method.
The following Lemma provides the risks of 8 i (c), i = 1,2, which are easy to derive.
LEMMA 2.4. The risks of 6 i (c), i = 1,2, are given by
2cno: t5 Ro, - 1 _ c 2 no: t5 Ro,- 2 (no: - 1)(2na - 1) (no: - 1)(na - 2)
{
c2no: 2cno: }
+ - +1
no: - 2 no: - 1 ' (b) R[B ] - { c 2
no: _ 2cno: } t5 no,
2(c) - (no: - 1)(no: - 2) (no: - 1)(2na -1)
{
c2na 2cno: }
+ - +1 .
no: - 2 no: - 1
Again, there does not exist any Co > 0 such that Bi(co) has the uniformly smallest risk in the class C i , i = 1,2. A minimax approach can be taken to an optimal value of c so that 8 i (c) minimizes the maximum risk in the class Ci, i = 1,2.
T HEOREM 2 5 . . L et c i = 2nar+i-3' 2(na-2)' 1 = 1 2 ,. T'h en
Then 8 n ~ 1 and min max R[B 1 (c)] = 1 -
0<cO<6~1
Case(i). c ~
na(a - 2) (na-1p'
C ase (..) 11 • C _ > 2(na-1) 2na-1 . Th en,
Proof· For estimating 9 1 , note that 16R[B2(c)] ~ 0 if 8 n ::;: 8 ::;: 1 , and ::;: 0 if 0 < 8 ::;: 8 n , where 8 n = Ci2:::~;) .
2(na-1) 2na-1 •
where ' 1 MU) = na-2 c 2 na _ na-1 2cna + 1 and M(2) 1 = na-1 c 2 na _ .1£!!!!.. na-1 + l.
Also,
.!!-M(l) > Oi£ > nO! - 2 = ° d.!!-M(2) > 0'£ > nO! - 1 = 00
d c 1 - C - nO! - 1 - c ,an d c 1 - 1 C - 2 nO!- 1 - c . M oreover, nun c . M(l) 1 = M(l)1 1 c=c· an d ' Inln c M(2) 1 = M(2)\ 1 c=c··· Al . so, notethatM(l);> M(2)H C> 2(na-:~) == c . So
1 - 1 - 2na-1 00
{ M (2)
A I '
max R[9 1 (c)] = (1)
0<6<1 - M 1 , c> Coo'
. (2) ( 1 ) .
AnO!( a - 2)
Smce M 1 lc=c.. > M 1 Ic=c·, mm max R[9 1(c)] = 1 - ( 1 )2 '
o<c O<6~1 nO! -
and this is attained at c = co. Combining cases (i) and (ii), we get the
required result.
To estimate (}2, note that l.sR[8 2 (c)] ~ 0 if c ~ cl, and ~ 0 if c < cl.
Hence,
. • . • 4(na)2(na - 2)
mm max R[(}l(c)] = mm max R[Ol(C)] = 1 - ( )(2 )2
C~cl 0<.s~1 c<c 1 0<69 na - 1 na - 1
and hence this is attained at c = cl.
We now show that the minimax(absolute s-bias and risk) estimators obtained in the last two Theorems are uniformly better than the MLE's in terms of absolute s-bias and risk. From (2.10), Lemmas 2.1 and 2.4, Theorems 2.2 and 2.5, we can obtain the following result.
THEOREM 2.6.
(a) (b)
if 1 ~ 8 ~ A-I if 0 < 8 < >. -1 ,
if 1 ~ 8 ~ >.-1
if 0 < 8 < >.-1,
(a) (b)
Now we focus our attention on the problem of estimating (}2 only because the 8~2) (a modified MLE) developed in (2.4) is nonsmooth and quite different from the other estimators. The following lemma gives the first and second moments of iW).
LEMMA 2.7. Let A = 2n\,. Then E['W)] ={ Al + A 3
A 2 +A 3
E[8~2)]2 ={ B l + B 3
B 2 +B 3
where, Al = --!!.2- ncr- l (}28-l + o;:;r-'--;'o;;,-;;--'-~
_ ncr >.ncr+3ncr+ 1 () 8 ncr 2 ncr-l) 2ncr-l) 2 ,
A 2 = n~~l ()2 + 2(ncr-l)(2ncr-l) [na + 4~cr - Ana -1]82 8ncr ,
A - ncr[>.ncr+ncr-l] () 8 ncr 3 - (ncr-l)(2ncr-l) 2 ,
B = --!!.2-(}2 + ncr[>.2ncr-ncr+2] ()28-(ncr-2) _ ncr[>.2 ncr +3ncr-2] ()28 ncr
1 ncr-2 1 4(ncr-l)(ncr-2) 2 4(na-l)(ncr-2) 2 ,
B - 2 - ncr-2 ncr 0 2 2 + 4(ncr-l)(ncr-2) ncr [ + no 6 + );2 - 4ncr >.2 no ](}28 ncr 2 ,
and B = ncr[>.2ncr+ncr-2]O 8 ncr 3 2(ncr-l)(ncr-2) 2 .
Proof. From Lemma 2.1, it can be shown.
Next, we shall consider Bayes estimators of fh(i = 1,2) under square error loss and a noninformative prior.
Consider the noninformative prior 7r(8t, 8 2 ) ex (8 1 .8 2 ), 0 < 8 1 :::;
8 2 < 00.
According to the Bayes theorem, the joint posterior distribution and the marginal posterior distribution of 8 1 and 8 2 , respectively, are ob- tained by
(2.11)
8 { 2na(zl . z2)-na8~nOl-\ 0 < 8 2 S Zl g( 21zt, Z2) = ( ) -nallna-1 1I <
na z2 u 2 ' zl < U2 _ Z2·
Therefore, the Bayes estimators of 8 1 and 8 2 under square error loss and a noninformative prior are
eBN [na ()i8na na ] Z · 2
i = na + 1 + -1 (na + 1)( 2na + 1) i, Z = 1, ,
where 8 = Zl/Z2 is an estimator of 6.
From (2.11), we can obtain the s-bias and risk of erN, i = 1,2. -
LEMMA 2.8. The s-bias olOflN, i = 1,2, are (a)
IB(OBN)/ = I 1 _ (na)2 6 nOl- 1
... .1 . (na'-'-.1)(na+l)2(na-'-1)(2na'-1)
+ (na)2 onOlI
2(na + 1)(2na + 1) ,
(b)
IB(oBN)1 _I 1 + (na)2 6 nOl
2 - (na -l)(na + 1) 2(na -1)(2na -1)
_ (na)2 onOl+1/
2(na + 1)(2na + 1) ,
And the risk of9f N ,i = 1,2, are (c)
A
