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Stochastic orderings related to ranking in an nth order record process
AMS 1970 Subject Classifications:Primary 60-00, BOEOS, 60lG99;Secondary 60H99. &,v words; Stochastic Orderings; Record Process; Ranking; fnformation Retrieval.
Introduction As applied statisticians we were consulted about a process in the field of information retrieval, of which a simplified description without technical details is given here. Consider a document retrieval system with a library containing m documents. When the system is asked for the n library documents with the highest score for a certain (arbitrary). question, the scores Xi of document i for this question are successively evaluated, for i = 1, 2, . . . ) m, A question consists of a few key words and the scores.are based on criteria such as thieoccurrence of these key words in the title of the document, the document itself etc If Xi belongs to the n highest observations among XfbXzS. . . ,XL document i is called a record. When document i is a record the system has to carry out a number of operations. The main operation is to put the information belonging the record document in a so-called hit-file. This hit-fife can contain the information of n documents, Each time i (>n) a record occur:gthe information of the hit-fife document with the lowest score is replaced by the information of the reword document i, esired d~~uments~ end of the described pro$ess the hiter CIeariy, designers of ret~~ev~~syst 79
f=ords on economica! grounds. If in the retrieval system the scores of certain dwuments tend to be higher thaln those of others, one may expect that the number
of records will decrease by placin,gthe former documents in front of the ladter in the library fiie. A first question that arises in this context is: if we have two rrankings vi&it &i&m < ..OrXkrXFe*1,***9.Xm) and g = (XI, ... ,Xk+ I,& .**,X& n Xk and Xk + I has to hold suchthat&< Xk+1-irnblies~~@)~&~%@~~~ where and R(h) are the number of records using ranking a and b, respectively? (For convenience we wili refer in the following to the ranking of Xi, X2, *., , X8#Mead of to the ranking of d-ments I,5 .*. , tn.) In studying this question we found a of stochastic order relations between random variables, about lwhich we e. As is often the case in mathematics, the results will not be presented in F in the constructive way we found them. First the hierarchy of stochastic and their relationship with known orderings are studied (Section 2); and next in Section 3 the role that these orderings a,lay in ranking in an n-th order record process is discussed. From the proois in Section 3 the reader can set how the orderings arise. In the special case n = 1 the process described above is known as the record Since the early fifties many investigations of record processes have been one, mostly using the assumption that XI, X2@ .,. , Xm are inctepfendent and rrtically distributed random variables; a survey may be found in Westcott (1977). case n > 1 is investigated in De Kroon and Zijlstra (1977).
In this section we introduce and discuss a hierarchy of stochastic orderings for two random variables UI and Vz; a partial ordering on the set of permutations is also given.
The orderings given Mow may be illustrated by considering the simultaneous ~~~ba~~~ity distribution of’ Ipi and L$ in R2. Each orderi is characterized by the that the drivability mass of subsets of a specific for exceeds the probability mltletric image about the fine u1 = ~2. some subsets of R2 have been drawn, which are typical for the orderings
Fig. 1. Subsets of R 2, which are typical for the orderings of Definition 2.1.
U1< U2 iffP~v&LI1
(2.2)
U1c U2 iff Ip(U1e ws U2)~P(U2< ws U1) rl for all reals w.
(2.3)
U1sU2
iffP(v~U1cw,U1+z~U2
U1< U2 iffP(v~W1
(2.4) (2.5)
U1< 4% iffP(v~&~b/n)gP(vSUz~U1) $1 libr all reals v. UlS5U2 iff P(U1c w, W1s W2)3;P(U2< for all reals w,
w, LQ
Ul),
VI% W2 iffP(U1sU2)gP(Uz$Wl)
(2.7) (2.8)
From Definition 2. I or Fig. 1 it is clear that by starting with ~3..or s * more ref!ned series of orderin s may be defined. In the case of the ~-o~~~~~ T CllJF has been done by Yanagimoto and ~ib~ya (1 12
.
have ,&eady ‘beendescribed, see for instance Pfanza
) or Yanagimoto and Sibuya (1972). In order to m&e the reader more familiar we will explain briefly how the r-orderings are usuall!~defined, if U1 and Uz are independent r;mdom variables with differentiable disaribution function Gi(u) =t: “P(&< u) and density funactiong,(u), i= I, 2. 2.2. r3 (monotone likelihood tafio)
I[tis easily seen that under this assumption for Ul and
Uz
(2. I) is equivalent to W3
sr(Vlg2(X)hg?(V)(tI(x),
for ail u and x with VEX. However (2.9) means tha?.g:L@)/gl(u) is no&ecreasing, we have a monotone likelihood ratio. r2 (monotoncl&(1a)/G1(u))
Let G,(u) = %- G(u), i = 1,2. Because of the independence of UI and uivafent to
U2
(2.2) is
~C;‘~(w)-GI(v))C2(HY)~{G2~W)-Ci2(v:))~~lGW),
for all v and w with v s w.
(2.10)
After adding G~w)&(w) to the left and right hand s:ideof (2.10) we get
G.4‘ w)/CQ w)s &(v)/C:~(v),
for all v anri IVwith v_s tlr.
(2.11)
is n?rans that &(u)/~&4) is non-decreasing, i.e. the derivative of (u) if non-negative and hence (2.12) So the ordering r2 is an ordering according to f’ailurerates. 2.4. f 1 (.um~i stochastic ordering) er the irldependence-assumption for U1 and (12(2.3) is equivalent to G~(~)G~(w)~G~(w~~,(w,,
for ail HA
(2.9 3)
Z(W)to both sides of (2.13) we okin Lz
i(w),
for all w.
(2. f4)
This is the usual definition of the well-known stochastic ordering of randant \ at-x [1959, Section 3.3). $9
docility all &xamplcsthat sltmwcertain implications not to hold,
s3--_r2
sIA
-80
Fig. 2. Implication scheme, general WC.
are collected in Appendix A. When studying the relationship of the order!ngs we distinguish between the genera8 case and the case with independent random variables. In the general case we have: Propositiorlr2.2, For the order relationsgiven in Definition 2.1 the im#cation scheme of Fig. 2 hole&s and only thisscheme is valid. Proof. The given implications can be shown directly by ta.king unions rend limits, The invafi.dity of other implications has been proved by Pfanzagl (1964) and Y’anagimoto and Sibuya (1972) for the r-orderings and the ordering SO; the invalidity of the remaining ones folio VCJfrom the counter examples in the appendix. Note that together with the given implications r2+3 implies s2+s3. The independent case differs from the general case cm two points, as the three 1’ollowing,propositions show. Proposithn 2.3. Let UI and U2be indepertdent randomvariables.If rhe probability distribut,-!onsof C/r and UZ are both discrete or both have a differenti!zble distribution @n&an Ut $ UZimplies Ut *$ U2.
Proof. 1lnthe independent case the Definition (2.2) is equivalent to C;!(rd)/C,(tl) non-decreasing, as we showed in (2,ll). Furthermore (2.5) can be written as f G&4) dGl(u)l: j Cl(u) dG&r), Iv* w) [WV w)
for all v and w with vs w. (2.15)
(i) If UI and U2have discrete d&ributions (2.15) implies ~2@4)[G1(u +) -- Cl(u)] ;z G1(u)[G204
“3r(addling -
2(u) to both si&s)
k5 Togather
+) - Gz(uN,
‘I*
with the continuity from the left of &(u)~/
C;@)/‘G 1(u) is non-decreasing.
44 (2.16) implies that
(ii) If GI and G2 are differentiable on (s,b) with derivatives gI and g2 we can deduce from (2.15)
s
dug
2(tl)gl(u)
j (3,&)g2(zi) rfu,, for all v, w E (s; I).
(2.1,s)
Iv**‘b
Iv.Kl
from measure theory that @:t7) is equivalent to dz(u)gl@) ig t(~)g~cu) for all u E (s, t), which implies thal &(u)/~~(u) is non-decreasing on (sb1). This completes the proof. c
know
. We wonder whether the proposition holds without restrictions to the tions. From the proof it is clear that the proposition is true if 911and & havce iate ‘mixed” distributions. But what hiappens when GI or G2 is not differentiable in a certain interval?
(2.10) and the well-klnownpropeaty that 01 < U2 iff jK(u)dG#Qr ~~(~~d&(u) for every non-increasing functioa K(U), u E fil (see Lehmann (1959,, 2.1 l))* we have [
C:(u)
dGl(u)l
( QI.kb
j
Gz(i4
dGzr[u)z
t--Q),WI
j
(%(u) dGz(u)
(-ah w
Prop~sltion 2.5. If WIand U2 are independent !randomvariabks the implication scheme of Fig. 3 holds and only this scheme is valid (with the restriction made in ~~o~~sit~on2.3).
r3 -
r2
1
s3 --S2
t
rl
--
-~llo
I
sllA
-SO
Fig. 3. implication scheme, iodrtpendent car;@.
directly b,y taking anions and limits or by ty of otJljerimplications has been proved by s 5hrll ones
M,
Z(,&cta,
J, R. M.
de Ktoan
/ Stochastic
orderings
ftorn
a tecotd ptmss
8s
It is llmportant to know whether the defined orderings are partial orderings, i.e. ase .refl.exive, anti-symmetric: {andtransitive, As is usual transitivity is the crucial property. * For independent random variables the r-orderings (and so $3) are partial orderings as the reader can easily verify (cf. Pfanzagl(l964)). The orderings sI, slA and SOare not transitive. This can be seen from Examples 5 and 6 in Appendix A, For $2 we have: Proposition 2.6, For independent random variables s2 is a partial ordering. Proof. The proof is trivial under the assumptions of Proposition 2,3, as ~2~2 in that case, The following proof holds without those restrictions. Let UI, & and UJ be independent random variables with UI s UZand UZ2 Us. Let gi= dGi/‘d(Cir+ Gz+ G3) be thleRadLn-Nikodym derivative of G:iwith respect to Gi + Gz;L’G3, i= 1,2,3; the derivatives exist because Gi is absolutel,y continuous with respect to Gt + G2+ GJ, Now (2.5) can be written as j Gz(u)gl(u) d(G + G2 + G3)(u) z Ivj w)r:du)gzCu) W% + G2+ G)(u), * Iv,w)
for all v and tt, with v< W; however, this is ecprivalent to C2(u)gl(u) z 131(u)g2(u),
for all u.
(2.18)
for all M.
(2.19)
In the same way we have C3(u)g2(u) g G2@)g3(u)b
From (2.18) and (2.19) it follows easily that &(u)g~(u) g It’;l(u)gJ(u)for all u, i.e. Ur < &I. Using (2.18) we see that <: is reflexive and anti-symmetric. s2
d!
2.7. AIJ ordering on the set perrmtations In Sefction 3 we will need an ordering on the set of permutations. Let A be set of all permutations of theem elements of the set C = {Xl, X2:,. . . pIL). If the elements of C’are totally ordered with respect to <, we are able to define an ordering on A. 3.
The ,permutatl;og RE A precedes
mz&blel to tmnqform a inrto k/ 4 number of s IcOrfectl~gan inversion; co tim! an inversion h2terchmge of Xe, und X+, for some k wit s clear that X&e
._,<-
it
~4 is a maximal element of a totatly
d A From this it foilows that if < is a partial ordering all maximal subwts of A hrcvethe same maximal element. ,2,3) with the usual orderfng e, then A has two maxilrnat
a1 problem concerning ranking in an mth order record ut the que&tion: if oncehas two rankings a = (Xi, . . . ,&, what relation e between xk iand m) and b=dXI,...,X&+ l,Xk, .*., X,&, to ho!d szlch that AK .Xk+ t implies; EIR(a)g ER(b)? This question is in the foIWing propositions for different values of k and n. From the E;tetitle.+orderings arise. omc additional notation. For each ranking c = (XCl,XC*.. . , &J we R,(c) = 1, if XC,is a record, =O, if&,isnot
arecord,
i=I,i: ,..., m.
more, let Y and Z be the Cn- l&h and nth order statistic from above xk- I, that have the distribution function I&z). Finaly we
o Z, we also call & a record. 1) is independent o&f(XI, ..n,xk-I),
!,...,&-1) iff&<4~k-+l. s2 ent of the rankin
Now we have
for every distribution j~unction ~(~~~2). From this and definition (2.5) the ~ro~o~it~~o~ fotkws.
Ri(a\)= Ri(b), iic:1::+ 1; SO ER(s) g ER(b) is equivalent to &??k +~(a)= 1) g P(Rp+ l(b) = .I). Z==~~in(;~~~, . . . ) xk - I) with d~$tribution function Lz(z) = L(oo,z). Therefore PrdS
for every distributl~u fulrction t&9. From this and definition (2.7) the proposition follsws.
88
hf. Ziil;srro 3.P.M
deKnwnl4Stochustk odetiagsjiumr
s
tetxm#pti
for every distribution function L&z) =L(QQ,z). From this imd Ikfititi~~ @.6j the proposition foIlows. Notice that Proposition 3.3 holds for n = k = 11if s 1 is repkt~ed by SO. Fin;aliy, Themam 3.4 gives conditions for C th a: are sufficient to indicate an optimal ranking.
Proof. The theorem is a direct consequence of the Propositiorzs 3.1,3.2,3.3 and the remark following DeZinition 2.7. otice that for n = 1 and sl (cf. Proposition 3.3) a theorem such\ as 3.4 cannot be ~~~~AMA . V‘ mm.unw=wu,h*n1~“111I~wv Y c1 is not a partial ordering.
Appendix A
‘Thisappendix contains counter examples, to which is referred in the text.
is example shows that WI2 U;!does not imply Ue2
U2
in the general case. In
three examples UI and Ur itre independent; therefore the counter y in the general and independent c2se.
Frum U1<:
the U2.
SIA
example we SW UI < Uz and UI s U2, si
but
not
Ut C. Qi, $2
UI 5 U2,
or
Note th@ .E’ZJI = 2, Z and E&= 2.096.
Example3 ---II
-PY
The example shows that Ut 5 I72does not imply 4716 11~and that UI 2 UZdoes
sum ---
0
4
3 2
0
1
0
8
I
@r/aand
t
SOare nut transitive)
I
M. J!~M&I, J. F. M. de Krooxlf S?mhu&c orderings from IPrecord p recess
91
We wish $0 thank Tsm Snijders of the University of Groningen fbr his interest in
Kroon, J.PM, de and Zijistra, M, (1977).A combinatorial problem in the field of information retrieval: ISA-R-SE’note 89, N.V. Philips, Eindhoven. Lehmann, E.L, !195$1).Testktg Staktical Hypotheses. Wiley, New York, Pfanzagl, J. (1964).On the topological structure of some ordered families of distributions. Am. MM. SW&t. 35,Y216- l%ka. Westcott, M. (1977). The random record model. P~Gc.Roy. sot. tondun A 356, 5:!9-547. Yana@moto, T . and M, Sibuya (1972). Stochastically larger component of a random vector. Ann. hf. Stat. Muth., 24, 259-269.
Introduction As applied statisticians we were consulted about a process in the field of information retrieval, of which a simplified description without technical details is given here. Consider a document retrieval system with a library containing m documents. When the system is asked for the n library documents with the highest score for a certain (arbitrary). question, the scores Xi of document i for this question are successively evaluated, for i = 1, 2, . . . ) m, A question consists of a few key words and the scores.are based on criteria such as thieoccurrence of these key words in the title of the document, the document itself etc If Xi belongs to the n highest observations among XfbXzS. . . ,XL document i is called a record. When document i is a record the system has to carry out a number of operations. The main operation is to put the information belonging the record document in a so-called hit-file. This hit-fife can contain the information of n documents, Each time i (>n) a record occur:gthe information of the hit-fife document with the lowest score is replaced by the information of the reword document i, esired d~~uments~ end of the described pro$ess the hiter CIeariy, designers of ret~~ev~~syst 79
f=ords on economica! grounds. If in the retrieval system the scores of certain dwuments tend to be higher thaln those of others, one may expect that the number
of records will decrease by placin,gthe former documents in front of the ladter in the library fiie. A first question that arises in this context is: if we have two rrankings vi&it &i&m < ..OrXkrXFe*1,***9.Xm) and g = (XI, ... ,Xk+ I,& .**,X& n Xk and Xk + I has to hold suchthat&< Xk+1-irnblies~~@)~&~%@~~~ where and R(h) are the number of records using ranking a and b, respectively? (For convenience we wili refer in the following to the ranking of Xi, X2, *., , X8#Mead of to the ranking of d-ments I,5 .*. , tn.) In studying this question we found a of stochastic order relations between random variables, about lwhich we e. As is often the case in mathematics, the results will not be presented in F in the constructive way we found them. First the hierarchy of stochastic and their relationship with known orderings are studied (Section 2); and next in Section 3 the role that these orderings a,lay in ranking in an n-th order record process is discussed. From the proois in Section 3 the reader can set how the orderings arise. In the special case n = 1 the process described above is known as the record Since the early fifties many investigations of record processes have been one, mostly using the assumption that XI, X2@ .,. , Xm are inctepfendent and rrtically distributed random variables; a survey may be found in Westcott (1977). case n > 1 is investigated in De Kroon and Zijlstra (1977).
In this section we introduce and discuss a hierarchy of stochastic orderings for two random variables UI and Vz; a partial ordering on the set of permutations is also given.
The orderings given Mow may be illustrated by considering the simultaneous ~~~ba~~~ity distribution of’ Ipi and L$ in R2. Each orderi is characterized by the that the drivability mass of subsets of a specific for exceeds the probability mltletric image about the fine u1 = ~2. some subsets of R2 have been drawn, which are typical for the orderings
Fig. 1. Subsets of R 2, which are typical for the orderings of Definition 2.1.
U1< U2 iffP~v&LI1
(2.2)
U1c U2 iff Ip(U1e ws U2)~P(U2< ws U1) rl for all reals w.
(2.3)
U1sU2
iffP(v~U1cw,U1+z~U2
U1< U2 iffP(v~W1
(2.4) (2.5)
U1< 4% iffP(v~&~b/n)gP(vSUz~U1) $1 libr all reals v. UlS5U2 iff P(U1c w, W1s W2)3;P(U2< for all reals w,
w, LQ
Ul),
VI% W2 iffP(U1sU2)gP(Uz$Wl)
(2.7) (2.8)
From Definition 2. I or Fig. 1 it is clear that by starting with ~3..or s * more ref!ned series of orderin s may be defined. In the case of the ~-o~~~~~ T CllJF has been done by Yanagimoto and ~ib~ya (1 12
.
have ,&eady ‘beendescribed, see for instance Pfanza
) or Yanagimoto and Sibuya (1972). In order to m&e the reader more familiar we will explain briefly how the r-orderings are usuall!~defined, if U1 and Uz are independent r;mdom variables with differentiable disaribution function Gi(u) =t: “P(&< u) and density funactiong,(u), i= I, 2. 2.2. r3 (monotone likelihood tafio)
I[tis easily seen that under this assumption for Ul and
Uz
(2. I) is equivalent to W3
sr(Vlg2(X)hg?(V)(tI(x),
for ail u and x with VEX. However (2.9) means tha?.g:L@)/gl(u) is no&ecreasing, we have a monotone likelihood ratio. r2 (monotoncl&(1a)/G1(u))
Let G,(u) = %- G(u), i = 1,2. Because of the independence of UI and uivafent to
U2
(2.2) is
~C;‘~(w)-GI(v))C2(HY)~{G2~W)-Ci2(v:))~~lGW),
for all v and w with v s w.
(2.10)
After adding G~w)&(w) to the left and right hand s:ideof (2.10) we get
G.4‘ w)/CQ w)s &(v)/C:~(v),
for all v anri IVwith v_s tlr.
(2.11)
is n?rans that &(u)/~&4) is non-decreasing, i.e. the derivative of (u) if non-negative and hence (2.12) So the ordering r2 is an ordering according to f’ailurerates. 2.4. f 1 (.um~i stochastic ordering) er the irldependence-assumption for U1 and (12(2.3) is equivalent to G~(~)G~(w)~G~(w~~,(w,,
for ail HA
(2.9 3)
Z(W)to both sides of (2.13) we okin Lz
i(w),
for all w.
(2. f4)
This is the usual definition of the well-known stochastic ordering of randant \ at-x [1959, Section 3.3). $9
docility all &xamplcsthat sltmwcertain implications not to hold,
s3--_r2
sIA
-80
Fig. 2. Implication scheme, general WC.
are collected in Appendix A. When studying the relationship of the order!ngs we distinguish between the genera8 case and the case with independent random variables. In the general case we have: Propositiorlr2.2, For the order relationsgiven in Definition 2.1 the im#cation scheme of Fig. 2 hole&s and only thisscheme is valid. Proof. The given implications can be shown directly by ta.king unions rend limits, The invafi.dity of other implications has been proved by Pfanzagl (1964) and Y’anagimoto and Sibuya (1972) for the r-orderings and the ordering SO; the invalidity of the remaining ones folio VCJfrom the counter examples in the appendix. Note that together with the given implications r2+3 implies s2+s3. The independent case differs from the general case cm two points, as the three 1’ollowing,propositions show. Proposithn 2.3. Let UI and U2be indepertdent randomvariables.If rhe probability distribut,-!onsof C/r and UZ are both discrete or both have a differenti!zble distribution @n&an Ut $ UZimplies Ut *$ U2.
Proof. 1lnthe independent case the Definition (2.2) is equivalent to C;!(rd)/C,(tl) non-decreasing, as we showed in (2,ll). Furthermore (2.5) can be written as f G&4) dGl(u)l: j Cl(u) dG&r), Iv* w) [WV w)
for all v and w with vs w. (2.15)
(i) If UI and U2have discrete d&ributions (2.15) implies ~2@4)[G1(u +) -- Cl(u)] ;z G1(u)[G204
“3r(addling -
2(u) to both si&s)
k5 Togather
+) - Gz(uN,
‘I*
with the continuity from the left of &(u)~/
C;@)/‘G 1(u) is non-decreasing.
44 (2.16) implies that
(ii) If GI and G2 are differentiable on (s,b) with derivatives gI and g2 we can deduce from (2.15)
s
dug
2(tl)gl(u)
j (3,&)g2(zi) rfu,, for all v, w E (s; I).
(2.1,s)
Iv**‘b
Iv.Kl
from measure theory that @:t7) is equivalent to dz(u)gl@) ig t(~)g~cu) for all u E (s, t), which implies thal &(u)/~~(u) is non-decreasing on (sb1). This completes the proof. c
know
. We wonder whether the proposition holds without restrictions to the tions. From the proof it is clear that the proposition is true if 911and & havce iate ‘mixed” distributions. But what hiappens when GI or G2 is not differentiable in a certain interval?
(2.10) and the well-klnownpropeaty that 01 < U2 iff jK(u)dG#Qr ~~(~~d&(u) for every non-increasing functioa K(U), u E fil (see Lehmann (1959,, 2.1 l))* we have [
C:(u)
dGl(u)l
( QI.kb
j
Gz(i4
dGzr[u)z
t--Q),WI
j
(%(u) dGz(u)
(-ah w
Prop~sltion 2.5. If WIand U2 are independent !randomvariabks the implication scheme of Fig. 3 holds and only this scheme is valid (with the restriction made in ~~o~~sit~on2.3).
r3 -
r2
1
s3 --S2
t
rl
--
-~llo
I
sllA
-SO
Fig. 3. implication scheme, iodrtpendent car;@.
directly b,y taking anions and limits or by ty of otJljerimplications has been proved by s 5hrll ones
M,
Z(,&cta,
J, R. M.
de Ktoan
/ Stochastic
orderings
ftorn
a tecotd ptmss
8s
It is llmportant to know whether the defined orderings are partial orderings, i.e. ase .refl.exive, anti-symmetric: {andtransitive, As is usual transitivity is the crucial property. * For independent random variables the r-orderings (and so $3) are partial orderings as the reader can easily verify (cf. Pfanzagl(l964)). The orderings sI, slA and SOare not transitive. This can be seen from Examples 5 and 6 in Appendix A, For $2 we have: Proposition 2.6, For independent random variables s2 is a partial ordering. Proof. The proof is trivial under the assumptions of Proposition 2,3, as ~2~2 in that case, The following proof holds without those restrictions. Let UI, & and UJ be independent random variables with UI s UZand UZ2 Us. Let gi= dGi/‘d(Cir+ Gz+ G3) be thleRadLn-Nikodym derivative of G:iwith respect to Gi + Gz;L’G3, i= 1,2,3; the derivatives exist because Gi is absolutel,y continuous with respect to Gt + G2+ GJ, Now (2.5) can be written as j Gz(u)gl(u) d(G + G2 + G3)(u) z Ivj w)r:du)gzCu) W% + G2+ G)(u), * Iv,w)
for all v and tt, with v< W; however, this is ecprivalent to C2(u)gl(u) z 131(u)g2(u),
for all u.
(2.18)
for all M.
(2.19)
In the same way we have C3(u)g2(u) g G2@)g3(u)b
From (2.18) and (2.19) it follows easily that &(u)g~(u) g It’;l(u)gJ(u)for all u, i.e. Ur < &I. Using (2.18) we see that <: is reflexive and anti-symmetric. s2
d!
2.7. AIJ ordering on the set perrmtations In Sefction 3 we will need an ordering on the set of permutations. Let A be set of all permutations of theem elements of the set C = {Xl, X2:,. . . pIL). If the elements of C’are totally ordered with respect to <, we are able to define an ordering on A. 3.
The ,permutatl;og RE A precedes
mz&blel to tmnqform a inrto k/ 4 number of s IcOrfectl~gan inversion; co tim! an inversion h2terchmge of Xe, und X+, for some k wit s clear that X&e
._,<-
it
~4 is a maximal element of a totatly
d A From this it foilows that if < is a partial ordering all maximal subwts of A hrcvethe same maximal element. ,2,3) with the usual orderfng e, then A has two maxilrnat
a1 problem concerning ranking in an mth order record ut the que&tion: if oncehas two rankings a = (Xi, . . . ,&, what relation e between xk iand m) and b=dXI,...,X&+ l,Xk, .*., X,&, to ho!d szlch that AK .Xk+ t implies; EIR(a)g ER(b)? This question is in the foIWing propositions for different values of k and n. From the E;tetitle.+orderings arise. omc additional notation. For each ranking c = (XCl,XC*.. . , &J we R,(c) = 1, if XC,is a record, =O, if&,isnot
arecord,
i=I,i: ,..., m.
more, let Y and Z be the Cn- l&h and nth order statistic from above xk- I, that have the distribution function I&z). Finaly we
o Z, we also call & a record. 1) is independent o&f(XI, ..n,xk-I),
!,...,&-1) iff&<4~k-+l. s2 ent of the rankin
Now we have
for every distribution j~unction ~(~~~2). From this and definition (2.5) the ~ro~o~it~~o~ fotkws.
Ri(a\)= Ri(b), iic:1::+ 1; SO ER(s) g ER(b) is equivalent to &??k +~(a)= 1) g P(Rp+ l(b) = .I). Z==~~in(;~~~, . . . ) xk - I) with d~$tribution function Lz(z) = L(oo,z). Therefore PrdS
for every distributl~u fulrction t&9. From this and definition (2.7) the proposition follsws.
88
hf. Ziil;srro 3.P.M
deKnwnl4Stochustk odetiagsjiumr
s
tetxm#pti
for every distribution function L&z) =L(QQ,z). From this imd Ikfititi~~ @.6j the proposition foIlows. Notice that Proposition 3.3 holds for n = k = 11if s 1 is repkt~ed by SO. Fin;aliy, Themam 3.4 gives conditions for C th a: are sufficient to indicate an optimal ranking.
Proof. The theorem is a direct consequence of the Propositiorzs 3.1,3.2,3.3 and the remark following DeZinition 2.7. otice that for n = 1 and sl (cf. Proposition 3.3) a theorem such\ as 3.4 cannot be ~~~~AMA . V‘ mm.unw=wu,h*n1~“111I~wv Y c1 is not a partial ordering.
Appendix A
‘Thisappendix contains counter examples, to which is referred in the text.
is example shows that WI2 U;!does not imply Ue2
U2
in the general case. In
three examples UI and Ur itre independent; therefore the counter y in the general and independent c2se.
Frum U1<:
the U2.
SIA
example we SW UI < Uz and UI s U2, si
but
not
Ut C. Qi, $2
UI 5 U2,
or
Note th@ .E’ZJI = 2, Z and E&= 2.096.
Example3 ---II
-PY
The example shows that Ut 5 I72does not imply 4716 11~and that UI 2 UZdoes
sum ---
0
4
3 2
0
1
0
8
I
@r/aand
t
SOare nut transitive)
I
M. J!~M&I, J. F. M. de Krooxlf S?mhu&c orderings from IPrecord p recess
91
We wish $0 thank Tsm Snijders of the University of Groningen fbr his interest in
Kroon, J.PM, de and Zijistra, M, (1977).A combinatorial problem in the field of information retrieval: ISA-R-SE’note 89, N.V. Philips, Eindhoven. Lehmann, E.L, !195$1).Testktg Staktical Hypotheses. Wiley, New York, Pfanzagl, J. (1964).On the topological structure of some ordered families of distributions. Am. MM. SW&t. 35,Y216- l%ka. Westcott, M. (1977). The random record model. P~Gc.Roy. sot. tondun A 356, 5:!9-547. Yana@moto, T . and M, Sibuya (1972). Stochastically larger component of a random vector. Ann. hf. Stat. Muth., 24, 259-269.