Eeden, C. (Constance) van

Kraft, C.H.

English

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stichting mathematisch centrum ~ MC AFDELING MATHEMATISCHE STATISTIEK SW 58/78 JUNI (DEPARTMENT OF MATHEMATICAL STATISTICS) C.H. KRAFT & C. VAN EEDEN BOUNDS FOR THE MEAN AND STANDARD DEVIATION OF LINEAR COMBINATIONS OF ORDER STATISTICS Preprint 2e boerhaavestraat 49 amsterdam BlBL!OTHEEK ~j~J\THEl\:'i/\T!SCH C[f",JTnUl\1 -, l\iv";STEF\DAf',Jl--P-'Lln:te.d a.:t ;the. Ma.:thema.:ti..c.ai. Ce.n:tll.e., 49, 2e. BoeJLhaa.vetd11.a.a.:t, Amo;teJLdam. The. Ma.:thema.:ti..c.ai. Ce.n:tll.e., 6ou.nde.d ;the. 11-:th 06 Fe.bnuall.y 1946, ,u, a non-pno6.U inl>:tltu.tion cumlng a.:t :the. pnomo.t.ion o0 puJc.e. ma.:thema.:ti..C6 and -l:t6 appUc.a.:ti..on!>. I;t ,l6 .6pon6oned by :the. Nethvifand6 GoveJLnmen:t :thnough :the Nethvifand6 Onganiza.:ti..on 6on the Advanc.eme.n:t o0 PuJc.e. Rueall.c.h (Z.W.O). AMS(MOS) subject classification scheme (1970): Primary 62G30; Secondary 62G35 Bounds for the mean and standard deviation of linear combinations of . . I) * order statistics by Charles H. Kraft & Constance van Eeden Universite de Montreaz 2) SUMMARY In this paper bounds are given for E0 r!=l asos and cr0 (E!=l asOs), where O1, .•. ,0N are the order statistics of x1, ..• ,~, a random vector with density p0 (x 1-e 1, ... ,xN-0N) for some e = (e 1, .•. ,eN) and pO(x1, ... ,xN) a completely arbitrary density with respect to Lebesgue measure in~- If the . N X. are exchangeable under pO the bounds depend only on E0 E a O and i N s= 1 s s crO(Es=l asOs) plus quantities that can be computed without knowledge of pO. The bounds are attainable for some distributions and nearly attainable for some others. · KEY WORDS & PHRASES: Order statistics, linear combinations of order stat-istics, bounds for means and standard deviations. I) Research supported by the Ministere de l'Education du Quebec, Grant from F.C.A.C., by the National Research Council of Canada, Grants No. A3O38 and No. A3114 plus supplemental travel grants and by the Mathematical Centre, Amsterdam. 2) This paper was written while the authors were on leave at the Mathematical Centre, Amsterdam. * This report will be submitted for publication elswhere. 1. INTRODUCTION Let p0 (x) = p0 (x 1, ... ,xN) be a given density with respect to Lebesgue measure in~ and consider the family p8 (x) = p0 (x 1-e 1, .•. ,~-eN), 8 = (e 1, •.• ,8N) E ~- Let (X1, .•• ,~) be a random vector whose density is p8 for some e and let 0 1, .•. ,0N be the order statistics of x1, ..• ,~. In thi·s b d f E ~N O d (~N O) paper oun s are given or 8 us=I ass an 0 8 us=I ass • These bounds are of the form (I.I) and ( I • 2) a O ) s s N a O - E l a 0 s s O s=l s s where a= (a 1, ... ,aN). The bounds (I.I) require no other assumptions than those above; (1.2) requires a further condition; exchangeability under p0 of the Xi' for ex-ample, is sufficient (see section 3 for the exact condition and also for a N bound for o8 (t:s=l a O) which uses only the assumptions for (I.I)). s s With the condition of exchangeability of the Xi under p0 the functions f 2, f 3 and f 4 can be computed without knowledge of p0 . Examples will be given to show that the bounds are in some circum-stances exactly attainable and in others almost attainable. In what follows, section 2 contains the derivation of the bounds for the mean, section 3 the derivation of the bound for the standard deviation. The computation of the functions f 1, f 2, f 3 and f 4 when the Xi are exchange-able under Po is discussed in section 4. Distributions for which the bounds are attained as well as examples where they are almost attained are given in section 5; section 6 contains some comparisons between actual values and their bounds. 2 2. BOUNDS FOR THE EXPECTATION OF r!=t a.sos. It will be helpful to consider first the expectation of a single order statistic, 0 • s Straightforward computation gives E O = l f x. Pe(x)dx, 6 s . J -J T. s J where Tj = {xj 1 $ ••• $ XjN} and j = {j 1, ••• ,jN} is a permutation of {l, ••• ••• ,N}. Let y = x -6, r = 1, ••. ,N, then r r r Eeos = i f (y. + e. )po(y)dy J JS JS B. J with B • = { yJ· + 6 J. $ ••• J 1 1 $ yJ· + SJ· }. With A. = N N 1 {i 1, •• ~,iN} a permutation of {1, .•• ,N}, we have (2. 1) l I i,j A.B. l. J $Yi}, i = N where i and j range independently through the N! permutations of l, ... ,N. Concerning the sets A.B. we have the following lemma. l. J LEMMA 2.1. For each (i,j) there either exists k such that (i) k E { is , • , , , iN} and (ii) or A.B. is empty. l. J for aU y E A.B. l. J PROOF. Suppose there exists j such that A.B. f 0 and for which there is no l. J k with (i) and (ii). Then for every k E {i 8 , ••• ,iN} we have for some y E A.B., l. J which implies, because that for all k E {i , •.. ,iN} . s for some y E A.B .. l. J 3 This means that, for all k E {is•···,iN}, k is strictly to the right o~ js in the j-permutation, but this is a contradiction because {is•···,iN} con-tains N-s+l integers and the number of integers strictly to the right of Js is only N-s. In the same way the following lemma can be proved. LEMMA 2.2. For each (i,j) there either exists k such that (i) and (ii) or A.B. is empty. l. J for aU y E A.B. l. J If we now define the random variables whose values on each A. are l. (ls min { e . , . . . ' e . } (2. 2) 1. s 1.N 1, ... ,N, s = e = max { e . ' . . . ' e . } us 1.1 1.s then we have, on each A.B., l. J (2.3) y. + 8fs s y. + e. s y. +8 s = l, ••• ,N. l. JS JS l. us' s s Integrating with respect to Po and using ( 2. 1) gives 4 (2.4) $ E0o + E0e s us s = l, ••• ,N, where E0els = Ii P0 (Ai)els' Clearly, for a monotone nondecreasing function h(0) s Eoh ( 0 + e O ) :::; Eeh ( 0 ) $ Eoh ( 0 + e ) ' s ~s s s us s = l, •.. ,N, can be obtained as above. A similar result holds for a function h(O1, ••• ,0N), where his monotone in each variable; the upperbound and lowerbound for E8h(0 1, ••• ,0N) are determined by the direction of the monotonicity of h in each argument. This result will now be used to find bounds for the expecta-tion of linear combinations IN a O . First note that on A.B. we have, for s= l s s 1. J each s = l, .•. ,N, (2.5) y. 1. s + /J.. . ' 1.J s say. Summing (2.5) with respect to s gives N (2.6) I s=l /J. .. 1.J s N = I s=l Hence we can write N (2. 7) I a (y. + e. s=l s J s JS -Ne. N ) = I s=l N = I s=l N asy· + I a /J. .. l.s s=l s l.J s N I -asyi + (a - ao)/J. .. + a0Ne, s=I s l.J s s where a.0 is an arbitrary constant. Since, by (2.3), /J. .. 1.s 1.n the interval 1.J s [8 0 ,e ], we have ~s us N N (2.8) I a (y. + e. ) 2". I asyi + I (as - a O) 8 ls + s=I s J s J s s=l s as>ao I -+ (a - ao)e + a0Ne s us a <a s 0 5 and N (2.9) I s=l + (a. - a ) e + s O us I -+ (as - ao) e .ts + a0Ne. as<ao Integrating with respect to Po gives the following lower and upper bounds N N (2. 10) Ee I a 0 2: Ea I a 0 + I (as - ao)Eoe .ts + s s s s s=l s=l as>ao I -+ (a - ao)Eoe + a0Ne s us as <ao and N N (2.11) Ee I a 0 :::;; Eo I a 0 + I (a - ao)Eoe + s s s s s us s=l s=l as>ao I (as - ao)Eoe.ts + -+ a 0Ne. as<ao The distance between these bounds is N ( 2. 1 2) I I a - a 0 I E0 < e - e o ) , s=l s us -l..S which is minimized by choosing a0 as the median of the as with respect to the probabilities (2.13) E0 ( e -e o ) us -l..S N s = l, ... ,N. I E0 (e -er1) s=l us ,{._s Sometimes a shorter bound can be found by maximizing (2.10) and minimizing (2.11) separately. Consider the upper bound (2.11) which can be written 6 N N fe I a. 0 - E I a. 0 s s=l s s 0 s=l s s N N s I a. E e - I (a. - a.o)Eo (e - el ) + a.0 [N8 - I E0e J s O us s us s us s=l a.s<a.o s=l N = I a. Eoe + g(a.o). s=l s us Because g(a.0 ) is continuous and has a nondecreasing derivative (except for a.0 = a.s' s = 1, ... ,N) which is nonpositive for small a.0 and nonnegative for large a.0 , g(a.0) either has a minimum or an interval of minima and a mini-mizing a.0 is an a.(so) which satisfies the two inequalities N I E0 (e - el ) I E e - -s NS us s 0 us a. ( s) <a. ( s O) s=l N I E0 (e - el ) I -::?: E e - NS, a. ( s >5a. ( s O ) us s s=l 0 us where a.(l) s •.. s a.(N) are the ordered values of a.s. In the same way an a.0 maximizing the lower bound for E8 IN a. 0 is s=l s s an a.(so) which satisfies the two inequalities N I E0 (e -el) - I Eo 8ls s NS -us s s=l a. (s) <a. (so) N I Eo(S - el ) - I Eo 8-ts· ::?: NS -a.(s)sa.(so) us s s=l An example where a shorter bound is obtained by maximizing (2.10) and min-imizing (2.11) separately is the following. Let N = 4, 0 I = 0 < 02 = e = 3 e4 = e and let the Xi be p0-exchangeable. Then, for a.4 < a. 1 < a.3 < a 2 , the median of the a.5 with respect to the probabilities (2.13) is a 3 . Using this 7 median for in the bounds for E EN a O - E EN a O gives aO 8 s= I s s O s= I s s lower bound upper bound 3 I and the length of this interval is 8[4 (a2 - a 1) + 4(a3 - a 4)]. The upper bound is minimized for a0 = a 1 and the upper bound then becomes The lower bound is maximized for a0 = a 3 which is the median. The differ-ence between these upper- and lowerbounds is 8[¾(a 2 - a 1) + ¼<a 1 - a 4 ) J and the difference between the two lengths is ¼s (a3 - a 1) > 0. I If x1, ... ,¾ are p0-exchangeable, then p0 (Ai) = N!' the distributions of 8 0 and 8 do not depend on p0 and are simple to compute. Some distri-,t_S US butions of 8ls and Sus are given in section 4. N 3. UPPERBOUND FOR THE STANDARD DEVIATION OF I s=l From (2.8) and (2.9) we have N I a (y. + 8. ) is s=l s J s JS N I I (a - ao) 8 + (3.1) :::; asyi + s us a O • s s I (as - ao) 8 .ts + s=l s as>ao as<ao N I I (as - ao) 8ls + I ~ a.Sy i + (a - ao) e + s=I s us s as>ao as<ao Then, if N N Y I ri <Y. + o . ) - [ () I (i o , s= I 5 J s J s s= I 5 5 -a 0Ne and -a0Ne. 8 N X = l s=I a. y. S 1 s N - Ea L a. o, s=I s s zl = I (a.s - a.a)(els -Eaels) + I a.s<a.a a.s>a.a z2 = I (a. - a.a)(e - Eae ) + I s us us a.s>a.a a.s<a.a N N A= a. Ne - E I a. 0 + Ea I a. 0 + a e s s s s s=I s=I + N N B = a.aNe - Ee I a. 0 + Ea I a. 0 + s s s s s=I s=I + (3. I) becomes (3.2) (a - a.a)(e - Eae ) , s' us us (a.s - a.a) (els -Eaels), I (a.s - a.a)Eaels + a.s>a.a I (a. - a.a>Eae , s us a.s<a.a I a.s>a.a (a. - a.a)Eae + s us I (a.s - a.a)Eae ls' a.s <ct.a To find an upper bound for cr(Y) N = cr(Es=I a.sos) consider parabolas z 1+z 2+A+B 2 k (X + 2 ) + l with k and l such that With a little algebra one sees that k > I and are necessary. Let Z +B-Z -A 2 l = ~( 2 I ) k-1 2 and k will be chosen after integration. Integrating with respect to pO gives (3.3) z 1+z 2+A+B 2 Z +B-Z -A 2 s kEO (X + 2 ) + ~ E ( 2 I ) k-1 0 2 for all k > I. 9 Minimizing the right hand side of (3.3) with respect to k > I one finds for the minimizing value of k t Z +B-Z -A 21..!_ E ( 2 I ) 2 k = I + 0 2 z 1+z2+A+B 2 EO(X + 2 ) Substitution of this value in (3.3) gives (3.4) N ae < I s=I a O ) s s z 1+z 2+A+B 2 -21 Z +B-Z -A ..!_ s {Eo(X + 2 ) } + {Eo( 2 2 I )2}2. This bound for a 0 (I:=I asOs) contains E0 I:=I asOs; also, it depends on the correlation between X and z 1 + z2 . Using (2.10) and (2.11) one obtains IA+BI --'-----'- = 2 N < ..!_ \ - 2 l s=I a 0 s s N + Eo I s=I a O + s s N +-2 I (a-a,o)Eo(e +0o)I s=I s us -l-S I a - a0 I E0 ( e - e o ) • S US -l-S 10 N Substituting this bound in (3.4) gives the following bound for cr 0 (rs=l asOs) N ae< l s=l (3.5) a O ) ~ s s 1 N a o + -2 I < a - aO) < a + a o ) ) + s s s=l s us ~s 1 N 2 2 + -4 ( I I a - a 0 I E0 ( e - e o ) ) } + s=l s us ~s 1 22 I a - a 0 I ( e - e o ) } J • s us ~s If X and z1 + z2 are uncorrelated under pO, for example if the Xi are pO-exchangeable, the bound (3.5) becomes (3.6) N cr 0 ( I s=l a O ) ~ s s 2 N I 2 N ~ foO< I a o) + -4 a0 < I <a -aO)(e +ao )) s= 1 s s s= 1 s us ~s I 1 N 2 2 + -4 ( I I a - ao I Eo ( e - e O ) ) } + s=l s us ~s I 1 N 2 2 + [-4 E0{ I I a - a0 I (e - e O ) } J s=l s us ~s 4. THE DISTRIBUTION OF (0 0 ,e ) WHEN THE X. ARE pO-EXCHANGEABLE. ~s us ----- i --Let 0j < ••• < eM be the ordered values of the e's, let, fork= I, •.• , M, Nk be the number o{ ej ~ ek and let NO = 0. If the Xi are pO-exchange-able, then PO(Ai) = N! and a somewhat lengthy but straightforward combina-torial argument gives, for the joint distribution of 0 0 and e , ~s us ( 4. I) where A= (Nk-Nj-1)! (s-1-Nj-l)!(Nk-s)! (Nk-Nj)! (s-1-Nj)!(Nk-s)! B = 0 (Nk-I-Nj-1)! C = (s-1-Nj-l)!(Nk-l-s)! 0 (Nk 1-N.)! - J (s-1-Nj)!(Nk-l-s)! D = 0 A - B - C + D 0 otherwise, if s ~ N. + I J if not if s :,; Nk-1 if not if N. + I :,; s :,; Nk-1 J if not. I I From (4.1), or by direct computation, one obtains for the marginal distri-butions of els and eus (4. 2) = (N-Nk-1\ \ N-s+I) ( N-Nk \ \N-s+I} 12 and (4.3) ( N) = (Nk) _ (Nk-1\. P (8 =a') I s O us k \s s / If the e's are all .unequal then Nk = k and the joint and marginal distribu-tions of 8 0 and a become ,l..S US ( k-j-1) (k-j) (k-s) . 1 + 1 • s-J- ; ,s-J if J :$; s :$; k (4.4) 0 otherwise, (N-k ffi (4.5) PO(als a') N-s N-s+1 = = = k N s \N-s+1 and (k-1\ {N-s+1\ (4.6) P0 (a a') =~= s N-k+1 = us k N-s+1 N S/ \N-k+1 Two simple limiting distributions for els' when the S's are all unequal, are the following. If i ➔ A, 0 <A< 1, then (4.7) k = 1,2, •••• If s = N- a, a fixed, then (4.8) X Po<e.e.s :$; e[Nx]) ➔ (a+ 1) I (1 - t)adt 0 Similar results hold for the distribution of 8 us O<x<I. 13 If M = 2, i.e. the two sample problem, then the joint distribution of (80 ,e ) is .(..S us (4.9) e us = e i, 8us = e') = 2 I -e = e') us 2 = k+I~ \N-s+I) ~ ~N-~+I~ \.s} \.N-s+I) where n = N 1 and m = N - N 1 • The marginal distributions are (4.10) and P0 (e us (4.11) P0 (e us = e') = I I -( m N-s+I N N-s+I =~ \N-s+I) (n \ = 8 I) I = t!j = 8 I) = 2 {n \ t~) Note that, for the two sample case, one of the distributions of els and eus is degenerate, so els and eus are independent. 14 5. ATTAINABILITY In this section distributions are given for which the bounds are attained, as well as examples where they are almost attained. EXAMPLE a. If p0 is concentrated on a single Ai' say i = {1, ... ,N}, and the order of the e's is the same as i, here e 1 ~ ... ~ 8N, then els= eus = eis' here 8 , with probability I for every sand the inequalities (2.10) and s (2.11) for the mean and the inequalities (3.5) and (3.6) for the standard deviation become equalities. I EXAMPLE b. If the as are all equal, say equal to N, and a.0 is the common value then the inequalities (2.10), (2.11), (3.5) and (3.6) are equalities. EXAMPLE c. The inequalities (2.10), (2.11), (3.5) and (3.6) are continuous as the e's approach equality and hence can be made arbitrary close to equality for 8 - 8 . sufficiently small. max min EXAMPLE d. 8 - 8 . does not have to be close to zero to get approximate max min equality. For Nodd let x1, •.. ,~+2 be independent. For large N there is practically no difference between the distribution of the median of x1, ... , ~+Z from U(O,l) and the median of x1, .•. ,~ from U(O,I). Hence, for 8 > I, there is, for large N, practically no difference between the distribution of the median of x1, ... ,~+2 from U(O,I) and the median of x1, •.. ,~+2 for x 1, ..• ,~ from U(O,I) and ~+I' ~+ 2 from U(8,8+1) and the inequalities will be almost equalities. 6. SOME COMPARISONS BETWEEN ACTUAL VALUES AND THEIR BOUNDS Let x1, x2 , x3 be independent with x1 and x2 double exponential with mean O and x3 double exponential with mean 8 ~ 0. Tables 6.1, 6.2 and 6.3 below give, for several values of 8, (a= a(Xi) = 12) the values of E8os and G 8 (0s), s = 1,2,3, together with the bounds (2.10), (2.11) and (3.6). For all three tables and all values of 8 the value of a.0 used for the bounds is a.0 = 0. This value of a.0 maximizes the lower bound (2.10), minimizes the upper bound (2.11) and is the median of the a. with respect to the s weights (2.13). 15 Table 6.1 Comparison of E8o1 and 0 8 (0 1) with their bounds E001 08(01) 8 lower value upper value upper bound bound bound 0 - 1 • 125 - 1 • 125 - I • 1 25 I . I 895 I. 1895 .040 - 1 • 1 25 -1. 1065 - 1 • 1061 1 • 189 5 1. 2060 . 1 0 -1 . 1 25 -1 .0803 -1 .0779 1.1896 1.2310 .4 0 - I . 125 -.9732 -.9364 1 • 1 904 1. 3640 0 -1 • 1 25 -.8488 -.6536 1 • 19 32 1. 6659 Table 6.2 Comparison of E8o2 and 0 8 (02) with their bounds E8o2 0 8 <02) 8 lower value upper value upper bound bound bound 0 0 0 0 .7993 .7993 .040 0 .0189 .0377 .7995 .8227 . 1 0 0 .0471 .0943 .8003 .8591 .4 0 0 • 1838 . 3771 .8154 1.0629 0 0 • 4111 .9428 .8852 1 • 5634 16 Table 6.3 Comparison of E8o3 and a 8 (o3) with their bounds Eeo3 cre(03) e lower value upper value upper bound bound bound 0 1 . 1 25 I • 125 I .125 1. 1895 1.1895 .04a 1 • 1439 I . 1443 1. 1816 I. 1895 1 • 2 I 28 • I a 1.1721 I.1747 1 . 2664 I. 1896 I. 2486 .4 a 1 . 3 I 36 I . 3551 1. 6907 I . I 9 20 I • 4427 a I . 5964 1. 8520 2.5392 I • 2138 I . 8996 As a second example, let x1, x2 , x3 , x4 be independent with x1 and x2 U(0,I) and x3 and x4 U(e,e+I), 0 ~ e ~ I. Tables 6.4 and 6.5 below give, for sev-1 °1+04 eral values of e, (a= a(X.) = ill) the values of E8o1, a 8 (o 1), E8 2 01+04 i and a 8 ( 2 ). In both tables and for all values of 8 the value of a0 used for the bounds in a0 = 0; this value maximizes the lower bound (2.10), mini-mizes the upper bound (2.11) and is the median of the a with respect to s the weights (2.13). Table 6.4 Comparison of E8o1 and a 8 (o 1) with their bounds E8o 1 ae(0l) e lower value upper value upper bound bound bound 0 . 2 . 2 .2 . 1633 . 1633 .04a .2 .2057 .2058 . 1634 . 1674 • I a .2 .2139 .2144 . 1638 . I 738 .4 a .2 .2494 • 2577 • 1700 .2091 a . 2 . 2967 .3443 • 1937 .2046 e 0 .04o . 1 a .4 a a Table 6.5 01+04 (01+04) Comparison of E6 2 and 0 6 2 with their bounds lower bound .5 .5029 .5072 .5289 .5722 Ee 2 ae 2 01+04 value upper value bound .5 .5 .1291 .5058 .5087 .1291 .5144 .5217 . 1294 .5577 .5866 . 1327 .6443 .7165 .1451 Departement de Mathematiques et de Statistique Universite de Montreal Montreal, Quebec, Canada. 17 upper bound .1291 . 1329 . 1387 . 1706 .2468 

Bounds for the mean and standard deviation of linear combinations of order statistics : (preprint)

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