Concomitants of Multivariate Order Statistics With Application to Judgment Poststratification

Transcription

1 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 1 Concomitants of Multivaiate Ode Statistics With Application to Judgment Poststatification Xinlei WANG, Lynne STOKES, Johan LIM, and Min CHEN We genealize the definition of a concomitant of an ode statistic in the multivaiate case, develop geneal expessions fo its density, and establish elated popeties. We study the concomitant of a nomal andom vecto in detail and discuss methods fo calculating its moments. Futhemoe, we apply the theoy to develop new estimatos of the mean fom a judgment poststatified sample, whee poststata ae fomed 8 by ank classes of auxiliay vaiables. Ou estimatos ae shown to be moe efficient than existing ones and obust against violations of the 67 9 nomality assumption. They ae also well suited to applications equiing cost efficiency KEY WORDS: Best linea unbiased estimato; Gaussian quadatue; Least squaes estimato; Ranked set sampling; Selection diffeential INTRODUCTION 1 7 Let (X h, Y h H h=1 be H independent andom vectos fom a common bivaiate distibution. Denote by X (:H the th-odeed X-vaiate, 1 H. The concomitant of the th-ode statistic of X is defined to be the Y-vaiate paied with X (:H and is denoted by Y [:H]. Popeties of concomitants have been studied by many authos (e.g., Bhattachaya 197; Sen 1976, 1981; David, O Connell, and Yang 1977; Yang 1977; Goel and Hall 199; Nagaaja and David 199; David and Nagaaja (00, sec. 6.8 have povided an oveview. Applications of concomitants include thei use in estimating coelation (Banett, Geen, and Robinson 1976, in anking and selection (Yeo and David 198; David 199, and in anked set sampling (RSS (Stokes In this aticle we extend the definition of concomitants to the multivaiate case, develop geneal expessions fo thei distibutions, and establish elated popeties. That is, we study the distibution of an Y-vaiate associated with odeed components of an absolutely continuous X-vecto. Fo example, suppose that X h contains the scoes of the hth employee on two peemployment sceening measues and that Y h contains his o he scoe on a late job pefomance measue, fo a sample of H employees. Ou theoy would allow evaluation of the distibution of the job pefomance measue fo an employee anked, say, best on both sceening tests. It would also allow compaison of that distibution to the concomitant job pefomance measue fo an unsceened employee o to one scoing best on a single sceening measue, to evaluate ou selection pocedue. Ou theoy was motivated by an application of concomitants to judgment poststatification (JP S (MacEachen, Stasny, and Wolfe 00, a method closely elated to RSS. Both JP S and RSS, ae useful when the vaiable of inteest, Y, is expensive to measue but can be anked, at least appoximately, much moe cheaply. The anking is efeed to as judgment anking. Both RSS and JP S allow bette estimation of the mean of Y, whee the eduction in vaiance is povided by statification. A anked set sample can be thought of as a statified sample, in which judgment anks define the stata. A judgment poststatified sample can be thought of as a simple andom sample (SRS, in which judgment anks define the poststata. This makes JP S moe pactical than RSS fo some applications, whee the eseache may be amenable to beginning with an SRS with the option of using auxiliay data late but eluctant to beginning with a nonstandad design, such as a RSS (MacEachen et al. 00. A common method of judgment anking in RSS is though an accessible auxiliay vaiable X, making Y a concomitant. We intoduce a simila idea fo JP S in Section 5. As in conventional poststatification, we can use multiple auxiliay vaiables fo foming poststata. When the anks of these auxiliay vaiables jointly define poststata, we need the theoy and popeties of the concomitant of multivaiate ode statistics to develop and compute JP S estimatos of the mean and investigate thei popeties. The aticle is oganized as follows. In Section we intoduce concomitants of bivaiate X-vectos and pesent analytical esults. In Section we apply these esults to the nomal case and show how to compute means and vaiances of the concomitant. In Section we extend ou methods with staightfowad modifications to the highe-dimensional case. In Section 5 we fist eview methods of mean estimation that have been suggested fo JP S samples using anking infomation fom moe than one auxiliay vaiable, then popose new estimatos with attactive popeties that ae available when cetain distibutional assumptions about the data can be made. We epot esults of simulation and empiical studies compaing the estimatos. We conclude with a bief discussion in Section CONCOMITANT OF BIVARIATE ORDER STATISTIC 10.1 The Geneal Theoy Let (X h1, X h, Y h H h=1 be an iid andom sample fom a tivaiate distibution, whee the andom vaiables X 1 and X ae absolutely continuous. Denote the ode of X h1 among X 11,...,X H1 by R h:h and denote the ode of X h among X 1,...,X H by S h:h. We conside the andom vaiable Y h given the anks R h:h = and S h:h = s, called the concomi- Xinlei Wang is Assistant Pofesso ( swang@mail.smu.edu and tant of the th-ode statistic of X 5 1 and the sth-ode statistic Lynne Stokes is Pofesso ( slstokes@mail.smu.edu, Depatment of 11 5 Statistical Science, Southen Methodist Univesity, Dallas, TX Johan of X, and denoted by Y h[,s:h]. Fo simplicity, we ignoe the 11 Lim is Assistant Pofesso, Depatment of Applied Statistics, Yonsei Univesity, subscipts H and h and denote the concomitant as Y 55 [,s], its 11 Seoul, 10-79, Koea, on leave fom the Depatment of Statistics, Texas A&M 56 Univesity, College Station, TX??? ( johanlim@stat.tamu.edu. Min Chen is a Doctoal Candidate, McCombs School of Business, Univesity of 0 Ameican Statistical Association Texas, Austin, TX??? ( minchen@mail.utexas.edu. The authos thank Jounal of the Ameican Statistical Association Shin-Jae Lee at Seoul National Univesity fo poviding data on human teeth???? 0, Vol. 0, No. 00, Theoy and Methods size. DOI /

2 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. Jounal of the Ameican Statistical Association,???? 0 pobability distibution function (pdf as f [,s] (y, the ank andom vaiables as R and S, and the bivaiate ank distibution P[R h:h =, S h:h = s] as π s, wheneve no ambiguity exists. Theoem 1. Suppose that (X 1, X, Y follows a tivaiate 5 X 6 6 distibution with a joint pdf f (x 1, x, y. Letm(X 1, X and 65 v(x 1, X denote the conditional mean and vaiance of Y, = θ 8 E[Y X 1, X ] and va[y X 1, X ]. Then the distibution of the concomitant Y [,s] among the H iid andom vectos is given by 1 k θ 1 k θ s 1 k θ H s+1+k 67 k=l X { U C k f 1 [,s] (y = θ1 k θ 1 k θ s 1 k θ H s+1+k Futhe, fo the mean of Y [,s],wehave 71 1 k=l X 7 15 Y 7 f (x 1, x, y dx 1 dx } { U C k 18 θ 77 1 k θ 1 k θ s 1 k θ H s+1+k X 19 k=l X = E [ m ( ] X 1(,s, X (,s, 78 f (x 1, x dx 1 dx } 1, (1 whee U = min( 1, s 1 and L = max(0, + s H 1, X is the suppot of the distibution of the X-vecto, (H 1! C k = k!( 1 k!(s 1 k!(h s k!, 7 86 θ 1 (x 1, x = P(X 1 < x 1, X < x, θ (x 1, x = P(X 1 < x 1, X > x, θ (x 1, x = P(X 1 > x 1, X < x, and θ (x 1, x = P(X 1 > x 1, X > x. The mean and vaiance of Y [,s] can be expessed by µ [,s] = E [ m ( X 1(,s, X (,s ] ( and σ 10 [,s] = E[ v ( X 1(,s, X (,s ] + va [ m ( X1(,s, X (,s ], ( whee (X 1(,s, X (,s ae bivaiate ode statistics of (X 1, X. Poof. Fist, we can wite X f [,s] (y = f (y x 1, x,, sp(, s x 1, x f (x 1, x dx 1 dx π s 51 Example 1. Suppose that U 1, U, Y unifom(0, 1 and iid. 110 = X f (x 1, x, yp(, s x 1, x dx 1 dx, π s ( because f (y x 1, x,, s = f (y x 1, x. In the spiit of the wok of David et al. (1977, it can be shown that U yielding the numeato of (1. Similaly, we can show that its denominato, the bivaiate ank distibution, is π s = p(, s x 1, x f (x 1, x dx 1 dx U C k µ [,s] = = f (x 1, x dx 1 dx. (6 yf [,s] (y dy m(x 1, x f (,s (x 1, x dx 1 dx whee Y is the suppot of the distibution of Y, and f (,s (x 1, x is the joint pdf of (X 1(,s, X (,s, that is, f (x 1, x R =, S = s. Similaly, the vaiance of Y [,s] can be witten as (. Remak 1. If (X 1, X and Y ae independent [i.e., f (x 1, x, y = f (x 1, x f (y], then it immediately follows fom (1 that f [,s] (y = f (y. Remak. Suppose that thee exists a monotonic function ψ( such that X = ψ(x 1. In this case f (x 1, x = f (x 1 I(x = ψ(x 1 and f (x 1, x, y = f (x 1, yi(x = ψ(x 1, whee I( is the indicato function. Based on (1, it is easy to veify that both π s and f [,s] (y degeneate to the univaiate case. When ψ( is inceasing (o deceasing, if = s (o = H + 1 s, then π s = 1/H and f [,s] (y = f (y x 1 f ( (x 1 dx 1 = f [, ] (y, X 1 whee f [, ] (y is the distibution fo Y [, ], the concomitant of the th-ode statistic of X 1, and f ( (x 1 is the distibution fo X 1(,theth-ode statistic of X 1 ; othewise, π s = 0, and f [,s] (y does not exist. Conside the application of concomitants to anking and selection of employees. Remaks 1 and fomalize the intuitive notion that if the sceening tests ae unelated to the pefomance measue, then using them fo selection is of no benefit, and if the sceening tests ae identical, then the second one is of no maginal benefit. Let X i = (Y + U i / foi = 1,. We illustate Theoem 1 by deiving f [1,1] (y and f [1,] (y fo H =, whee we condition on the anks of X = (X 1, X. The theoem equies both joint f (x 1, x, y and maginal f (x 1, x densities. The fome can be detemined to be unifom ove the egion J :0 y 1 and y/ x 1, x (y + 1/, that is, p(, s x 1, x = C k θ1 k θ 1 k θ s 1 k θ H s+1+k, (5 k=l f (x 1, x, y = I J (x 1, x, y. (7

3 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. Wang et al.: Concomitants of Multivaiate Ode Statistics 8 67 Figue 1. The Sampling Space of f(x 1,x. of this is given in Section The maginal density is found by integating the joint density ove the appopiate egion to obtain 8x (aea A 8x 1 (aea D (x f (x 1, x = x (aea B (8 (x 1 x + 1 (aea E 8(1 x 1 (aea C 8(1 x (aea F. Aeas A F ae shown in Figue 1. To find f [1,1] (y, fist compute fom (6 π 11 = X θ f (x 1, x dx 1 dx, whee θ must be detemined sepaately fo each aea of Figue 1 (e.g., in aea A, θ = 1 x 1 x + x 1x x /. The esult is that π 11 = 1/. The numeato of (1 becomes 7 86 f (y, R = 1, S = 1 = θ dx 1 dx, 1 y/ y/ 90 which, afte some calculation, can be shown to be f [1,1] (y = 1 0 ( 5y 0y + 50y 15y (9 fo 0 y 1. Noting that π 1 = 1/ π 11 = 1/6, a simila calculation yields f [1,] (y = 1 10 (7 + 15y 0y + 15y. (10 Suppose that (X 1, X, Y, with joint distibution (7, denote scoes on two sceening tests and a pefomance measue fo an employee. The advantage in pefomance expected fom an employee who pefoms best (in this case the lowest value, as fo speed tests on one sceening test can be measued by the selection diffeential, which Nagaaja (198 defined as 10 η [1] = µ [1, ] µ y, (11 σ y whee µ y = E(Y, σy = va(y, and µ [1, ] = E(Y [1, ].Fom (9 and (10, we have 5 11 f [1, ] (y = π 11f [1,1] (y + π 1 f [1,] (y = 1 π 11 + π 1 (5 y y + y fo 0 y 1, yielding µ [1, ] = /60 and ( η [1] = 60 1 / Genealizing the selection diffeential (11 to the bivaiate concomitant, we compute ( 1 η [1,1] = 0 1 / fom (9. Compaing these two shows that the additional sceening test impoves selectivity by about 50%. It is sometimes staightfowad to calculate moments of the concomitant diectly fom the density (1, as in Example 1. In othe cases, calculation is easie using ( and (; an example. Simplifying Popeties Computing densities of concomitants using Theoem 1 is tedious. Howeve, symmety in the distibution of (X 1, X, Y can be exploited to educe the numbe of calculations equied to obtain densities and moments fo the set of all concomitants. In this section we pesent some useful esults fo that pupose. Fist, we make some obsevations about the ank distibution π s. Fo convenience, let H + 1 and s H + 1 s. Popety 1. A monotonically inceasing tansfomation on X 1 o X does not change π s. A monotonically deceasing tansfomation on X 1 leads to π s = π s, and a monotonically deceasing tansfomation on X leads to π s = π s, whee π s is the bivaiate ank distibution based on the tansfomed vaiables. 9 (y+1/ (y+1/ Popety. If the joint pdf f (x 1, x of X 1 and X is symmet- 88 ic [i.e., f (x 1, x = f (x, x 1 ], then π s = π s. Popety. If f (x 1, x = f ( x 1, x, then π s = π s. Popety 1 is obvious fom obseving that the ank of any obsevation is invaiant to a monotonically inceasing tansfomation. The othe two popeties have been poven by David et al. (1977. Example (Example 1 continued. Popeties 1 can be used to calculate π 1 and π. Obseve fom (8 that f (x 1, x = f (x, x 1 ; thus π 1 = π 1 = 1/6 fom Popety. Define Z i = X i 1/ foi = 1,. Accoding to Popety 1, (Z 1, Z has the same ank distibution as (X 1, X. Because the joint density of Z 1 and Z satisfies g(z 1, z = g( z 1, z, Popety yields π = π 11 = 1/. Next, we obseve some popeties of the concomitant distibution that follow diectly fom Theoem 1. Coollay 1. Suppose that thee exist monotonic functions ψ 1 ( and ψ ( such that Z 1 = ψ 1 (X 1, Z = ψ (X, and the joint pdf of Z 1, Z, and Y satisfies g(z 1, z, y = g(z, z 1, y. Then (a if both ψ 1 (inceasing and ψ o both ψ 1 (deceasing and ψ, then f [,s] (y = f [s,] (y, and (b if ψ 1 and ψ o ψ 1 and ψ, then f [,s] (y = f [ s, ] (y. Coollay. Suppose that thee exist monotonic functions ψ 1 ( and ψ ( such that Z 1 = ψ 1 (X 1, Z = ψ (X. Then (a if the joint pdf of Z 1, Z, and Y satisfies g(z 1, z, y = g( z 1, z, y, then f [,s] (y = f [, s] (y, and (b if g(z 1, z, µ y + d = g( z 1, z,µ y d, then f [,s] (µ y + d = f [, s] (µ y d

4 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. Jounal of the Ameican Statistical Association,???? 0 Example (Example 1 continued. Fom (7, f (x 1, x, y = f (x, x 1, y. Thus Coollay 1 yields f [,1] (y = f [1,] (y. Because the joint density of Z 1, Z, and Y satisfies g(z 1, z, 1/ + d = g( z 1, z, 1/ d, Coollay yields f [,] (1/ + d = f [1,1] (1/ d. Lety = 1/ + d; then f [,] (y = f [1,1] (1 y = ( + 15y + 0y + 10y 15y /0. σ 6 In the following theoem, we establish popeties of the mean µ [,s] and vaiance σ of a concomitant, whee the distibution of Y is not equied to be symmetic [,s] 68 Theoem. Suppose that thee exist monotonic functions 1 71 ψ 1 ( and ψ ( such that (a Z 1 = ψ 1 (X 1, Z = ψ (X To calculate µ [,s] and σ 1 [,s] using (1 and (15, values of 7 and thei joint pdf is symmetic about 0, that is, g(z 1, z = g( z 1, z, and (b E(Y Z 1 = z 1, Z = z is a linea function of z 1 and z. Then the mean of the concomitant of bivaiate ode statistics of (X 1,X satisfies µ [,s] + µ [, s] = µ y (1 fo {1,...,H} and s {1,...,H}. Futhemoe, if va(y z 1, z = va(y z 1, z, then the vaiance of the concomitant satisfies σ [,s] = σ [, s]. (1 R = z 1 p(, s z 1, z φ(z 1, z dz 1 dz π 8 s Fo the poof see Appendix A. Note that in Theoem, if H is odd, then µ [(H+1/,(H+1/] = = u 7 1 θ 86 µ y. π k=l R 1 k θ 1 k θ s 1 k θ H s+1+k Example. Conside a type of egession setup: Y = β 0 + β 1 X 1 + β X + ɛ whee ɛ is independent of X 1 and X and follows a distibution with mean 0. Also, assume that X 1 and X can be linealy tansfomed so that thei joint pdf afte tansfomation is symmetic about 0. Then (1 and (1 hold. Moe esults about µ [,s] and σ 5 [,s] can be obtained easily distibution and θ i θ i ( µ 1, ρ 1 µ 1 + (1 ρ1 µ, 9 fom Coollaies 1 and. Fo example, (1 and (1 follow diectly fom pat (b of Coollay.. THE NORMAL CASE Hee we discuss the special case of the concomitant of the ode statistics of a bivaiate nomal andom vecto. Let (X 1, X, Y be tivaiate nomal with means µ 1, µ, and µ y ; vaiances σ1, σ, and σ y ; and coelations ρ 1, ρ 1y, and ρ y. M 10 C k {ω i ω j t i θ1 k Popeties of the nomal distibution allow the conditional mean π θ 1 k θ s 1 k θ H s+1+k }, 10 k=l j=1 i=1 and vaiance of Y given X = (x 1, x to be witten as m(x 1, x = µ y + (τ 1 z 1 + τ z σ y and v(x 1, x = (1 τ 1 ρ 1y τ ρ y σ y, whee τ 1 = (ρ 1y ρ y ρ 1 /(1 ρ1, τ = (ρ y ρ 1y ρ 1 /(1 ρ1, and z j = (x j µ j /σ j, j = 1,. Fom (, [ µ [,s] = µ y + σ y τ1 E ( Z 1(,s + τ E ( ] Z (,s = µ y + σ y [ τ1 E ( Z 1(,s + τ E ( Z 1(s, ], (1 whee (Z 1, Z T has the standad bivaiate nomal distibution with coelation ρ 1 and (Z 1(,s, Z (,s ae the bivaiate ode statistics of (Z 1, Z with joint density g (,s (z 1, z. The second line follows because E(Z (,s = E(Z 1(s,.Fom(, [,s] = [ τ1 va( Z 1(,s + τ va ( Z 1(s, + τ 1 τ cov ( Z 1(,s, Z (,s + 1 τ 1 ρ 1y τ ρ y ] σ y. (15 Noting that τ 1 and τ ae the standadized egession coefficients, (1 and (15 suggest that the concomitant and elated ode statistics etain the same lineaity as in multiple egession. the means, vaiances, and covaiances of the bivaiate standad nomal ode statistics ae needed. Tables of these moments fo H =,, and ae available in an ealie wok (Wang and Stokes 005. The method that we used to obtain these tables is biefly outlined hee. To calculate the mean, one must evaluate E ( Z 1(,s = z 1 g (,s (z 1, z dz 1 dz R ( U C k exp[ (u 1 + u ] du 1 du / π s, (16 whee φ(, is the joint pdf of the standad bivaiate nomal fo i = 1,...,. The second line follows fom (5 afte the change of vaiables z 1 = u 1 and z = ρ 1 u 1 + (1 ρ1 u. Simila expessions can be witten fo π s, E(Z1(,s and E(Z 1(,sZ (,s. These wee all evaluated numeically using Gaussian quadatue. Fo example, the numeato of (16 was appoximated by U M whee θ l θ l ( t i, ρ 1 t i + (1 ρ1 t j fo l = 1,...,, t i is the ith zeo of the Hemite polynomial H M (t, and ω i is the ith weight facto. (Tables of t i and ω i ae available fo M = 1 0 in Salze, Zucke, and Capuano 195. Due to the symmety of the nomal density, the popeties in Section. can be used to educe the numbe of numeical evaluations needed to obtain µ [,s] and σ[,s]. Popety 1 shows (by standadizing X 1 and X that π s is elated only to ρ 1 and so can be calculated based on Z 1 and Z.Fom Popeties and, π s = π s = π s. Theoem shows that E(Z 1(,s + E(Z 1(, s = 0, and so one need only calculate elements in the uppe tiangula matix of [E(Z 1(,s ] H H.The numbe of covaiance (vaiance calculations needed is educed by obseving that cov(z 1(,s, Z (,s = cov(z 1(s,, Z (s, and

5 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 5 Wang et al.: Concomitants of Multivaiate Ode Statistics 5 cov(z k(,s, Z l(,s = cov(z k(, s, Z l(, s fo any k, l = 1,, the latte of which is an intemediate esult in the poof of Theoem. Example 5. Suppose that (X 1, X, Y, the joint distibution of which is tivaiate nomal, denote scoes on two sceening tests and a pefomance measue fo an employee. The advantage in pefomance expected fom an employee who pefoms best (in this case the lowest value, as fo speed tests on both sceenes among H =,, o competitos can be measued by the selection diffeential. It can be witten, using (1, as 8 67 η [1,1] = µ [1,1] µ y = ρ 1y + ρ y E ( Z 1(1,1. ( ρ 1 One might expect that if the fist sceening test wee pefect (ρ 1y = 1, then the second one would povide no advantage in the selection pocess. This is incoect. To see why, note fist that ρ y = ρ 1 in this case. Then, fom (17, η [1,1] = E(Z 1(1,1 and η [1] = E(Z (1, whee Z (1 is the fist-ode statistic of a standad nomal andom vaiable. Figue (b shows the atio η [1,1] /η [1] as a function of ρ 1 fo H =,,. The atio is >1, but the advantage is geatest when the second sceene has a coelation of aound.70 with the pefomance measue. Its advantage diminishes to 0 as ρ 1 (= ρ y inceases to 1. An intuitive explanation fo this is that even pefect anking info σ mation does not povide complete infomation about the mean. 1 y 7 This expession shows that the selection diffeential inceases in magnitude as sceening tests gow moe effective (i.e., lage values of ρ 1y and ρ y and as the numbe of competitos inceases [because E(Z 1(1,1 is an inceasing function of H]. One would also expect less advantage as sceening tests become moe simila (i.e., ρ 1 inceases, but this is not clea fom (17 because E(Z 1(1,1 inceases in ρ 1. Figue (a displays η [1,1] fo H =,, and as functions of ρ 1 fo two modeately effective sceening tests (ρ 1y = ρ y =.5. It confims that the second test is less useful fo selection as it becomes moe simila to the fist. The second anking vaiable, to the extent that its infomation diffes fom that of the fist, can still impove estimation. Note also that the fine ae the pefect anke s poststata (lage H, the less additional infomation emains fo the second anke to povide.. EXTENSION TO THE MULTIVARIATE CASE In the pevious sections we investigated the concomitant of bivaiate ode statistics. We now seek analytic expessions fo the geneal case, the concomitant of multivaiate ode statistics whee the numbe of X vaiables. Let (X h, Y h H h=1 be an iid andom sample fom a multivaiate distibution with a joint pdf f (x, y, whee X T h continuous vecto of length m. Denote the ode of X hi among X 1i,...,X Hi by R hi and the ank vecto associated with X h by R T h = (R hi m i=1.givenafixedh, we conside the concomitant of multivaiate ode statistics of X h, that is, the andom vaiable Y h given R h. To obtain its density, ( can be genealized as 6 (a is an absolutely X f (x, yp( x dx 9 f [] (y = π, (18 whee π = X p( xf (x dx.onlyp( x is needed, which can be computed by ecusion. To illustate the idea, we descibe the method fo deiving p( x fo m = fom that fo m =. Following in David et al. (1977, we epesent the ways in which the compound event R h1 = 1 and R h = given X h1 = x 1 (b 1 and X h = x can occu in the following table: 100 X h < x X h > x 10 X h 1 < x 1 k 1 1 k X h 1 > x 1 1 k H k H H H Then p( 1, x 1, x, can be obtained fom the multinomial distibution, with the fou outcomes defined by the cells of the table. We split each of the fou cells futhe by a thid vaiable: 5 11 X h 5 < x X h > x 11 X h 55 < x X h > x X h < x X h > x X h 1 < x 1 l 0 k l 0 l k l Figue. An Example fo the Nomal Case. (a Selection diffeential X 57 h 116 fo pais of modeately efficient (ρ 1y = ρ 1 > x 1 l 1 1 H 1 + H 1 y =.5 sceening tests. (b Ratio of selection diffeential fo one and two sceenes when one test is 117 k l (l 0 + l 1 + l + (k + l 0 + l 1 + l 58 1 H H 1 59 pefect. ( H = ; H = ; H =. 118

6 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 6 6 Jounal of the Ameican Statistical Association,???? 0 Afte splitting, label the eight cells 1,...,8 and denote the numbe of obsevations in the jth cell by t j,1 j 8(i.e.,t 1 = l 0, t = k l 0, t = l 1,...,t 8 = H k + l 0 + l 1 + l. Thus p( 1,, x 1, x, x = { } 8 [θ j (x 1, x, x ] t j 8 C 67 k,l0,l 1,l, units. This ank infomation is used to classify the n measued 9 k,l 68 0,l 1,l A j=1 units into H poststata. MacEachen et al. (00 poposed as whee A is an intege set {k, l 0, l 1, l k 0; l 0 0; l 1 0; l 0; t j 0, fo 1 j 8} and C k,l0,l 1,l (H 1!/ { 8 j=1 t j!}. Define θ j (x 1, x, x as the coesponding pobability in the jth cell, that is, θ 1 (x 1, x, x P(X 1 < x 1, X < x, X < x, θ (x 1, x, x P(X 1 < x 1, X < x, X > x, and so on. Similaly, p( x, x m+1 can be deived fom p( x by patitioning each of the m cells into two subcells based on the value of x m+1 and then applying the multinomial distibution with m+1 possible outcomes. Fom (18, we can obtain an analytic expession fo f [] (y when the length of x > by ecusion. With tivial modifications, the popeties discussed in Section can be genealized to m >. Conside the nomal case, whee (( (X, Y T µx N h=1, 7 µ y 197. MacEachen et al. (00 showed that the asymptotic 86 ( ρx ρ diag(σ x,σ y xy ρ T diag(σ x,σ y. (19 xy 1 The mean and vaiance of the concomitants can be expessed as genealizations of (1 and (15 as µ [] = µ y + σ y ρ T xy ρ 1 x E( Z ( (0 and whee Z ( is a vecto of multivaiate ode statistics of Z that follows the standad multivaiate nomal distibution with the coelation matix ρ x. Calculation of the moments of the standad nomal multivaiate ode statistic involves p( z, foexample, E ( Z 1( = R m z 1 p( zφ(z dz, π 10 which can be evaluated numeically using Gaussian quadatue, simila to the method used in the bivaiate case. 5. APPLICATION TO JUDGMENT POSTSTRATIFICATION 5 ˆµ (m M 111 Hee we apply the theoy of the pevious sections to suggest = 1 H ni=1, ( ˆp ih 5 h=1 11 new estimatos of the mean fom J PS samples and to examine thei popeties Backgound MacEachen et al. (00 intoduced JP S sampling as an altenative to RSS fo estimating the mean of Y, which is expensive to quantify but elatively cheap to ank by judgment. To obtain a JP S sample, one fist daws an SRS of n units fom a population and ecods the value of Y fo each, denoted by y i,1 i n. Fo each measued unit i, an additional sample of size H 1 is chosen at andom, and the ode of y i among the H units is assessed by some inexpensive (and likely impefect anking method not equiing measuement of the H 1 an estimato of µ y, ˆµ y = 1 H H ni=1 y i I ih ni=1, ( I ih whee I ih = 1ify i is assigned into the statum of ank h and I ih = 0 othewise. RSS diffes fom JP S sampling in that in the fome, judgment anking of the goup of H sample units occus fist, and then a specified ank is designated fo measuement fom the goup. Ranking of goups of size H continues until some specified numbe of units of each judgment ank ae quantified. The unweighted mean of such a sample can be shown to be unbiased fo µ y and to have smalle vaiance than an SRS of an equal numbe of measued obsevations (Dell and Clutte elative efficiency of ˆµ y to this RSS estimato is 1. Although RSS is descibed as using subjective judgment in anking, applications have often used the ank of an easily obseved auxiliay vaiable as a poxy fo the ank of Y. But what if infomation about the ank of moe than one auxiliay vaiable is available? It is difficult to use this infomation in RSS, because any paticula vecto of anks cannot be guaanteed to occu, and so pespecifying sample sizes fom stata defined by 7 σ 96 [] = σ y + σ [ ( y ρt xy ρ 1 x cov Z( ρ 1 x I ] joint anks is infeasible unless a multiple-laye design of RSS ρ xy, (1 is used (Chen and Shen 00. In contast, JP S uses ank in- fomation only fo estimation, not fo sample design. Ou goal is to use such ank infomation along with the measued y i to estimate µ y. An example of such data was discussed by Chen (00 fo estimating mean age of a population of fish. Aging a fish is expensive, but the ank of a fish s length and width, which ae coelated with age, among a goup of H fish can be easily obtained. MacEachen et al. (00 also cited the ability to use moe than one anke as an advantage fo JP S ove RSS. In the case whee assessments of anks ae available fom m ankes (o auxiliay vaiables, they poposed as an estimato of µ y, H ni=1 y i ˆp ih whee ˆp ih is the popotion of ankes classifying y i as having ank h; that is, they poate the measued value among the poststata eceiving any votes fom a anke. This estimato equies no distibutional assumptions fo its justification. When m = 1, ˆµ (m degeneates to ( M

7 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 7 Wang et al.: Concomitants of Multivaiate Ode Statistics 7 5. New Estimatos of Mean In this section we popose seveal new estimatos of µ y based on data fom a JP S sample, whee poststata ae defined on anks of m auxiliay vaiables. Ou poposed estimatos ae de- signed to take advantage of knowledge of the distibution of the concomitant. Hee we estict attention to the most tactable yet impotant case, the multivaiate nomal distibution. We fist assume that σ y, ρ xy, and ρ x in (19 ae known, and then ex µ y 68 amine the pefomance of the estimatos in the pactical case i=1 in which they ae estimated. Methods fo extension to the nonnomal case (with some mild distibutional assumptions ae 1 x 71 discussed in Section JP S again begins with selection of a andom sample of ˆµ (m n 1 (Ȳ[] δ []. 7 n units on which Y is measued. In addition, m elated and easily anked chaacteistics X ae available on each unit. Fo each i, an additional H 1 units ae andomly selected, and the anks of the components of X i among its H compaison units ae detemined. The vecto of anks is denoted by R i = (R 0 i1,...,r im. Thus thee ae H m poststata jointly gouped by 79 the anks R = (R i n i=1. Let PS denote the poststatum in which R i =, and π, n, and Ȳ [] denote the pobability, numbe, and sample mean of obsevations falling in PS.Letµ [] and σ denote the mean and vaiance within the poststatum, that is, [] = va(y i R i =. Define δ [] as the diffeence between µ [] and µ y, that is, δ [] µ [] µ y.finally, let I i be the indicato vaiable such that I i = 1ifR i = 81 [] 8 5 µ [] = E(Y i R i =, σ, which minimizes the sum of the weighted squaed and I i = 0 othewise. min µ y σ 87 9 We conside a class of linea JP S estimatos of the fom i=1 [] 88 ˆµ (m = w (n ( Ȳ [] c, ( whee the summation is ove all H m ˆµ (m [] (Ȳ[] ealizations of the ank vec- n /σ[] δ [], (8 9 to R i, n is the andom vecto containing the counts of Y in the H m poststata, w ( is a weight associated with PS that can be whee in the nomal case σ[] is given by (1. It is easy to show a function of n, and c is a constant associated with PS that can that ˆµ (m 6 is unbiased and that 95 be used fo bias coection. This class, denoted by E, contains familia membes as well as useful new ones. The SRS estimato Ȳ is in E, with w = n /n and c = 0. It obviously makes use σ, (9 8 va ( ˆµ (m n 97 9 [] 98 of neithe auxiliay no distibutional infomation. A vesion of the classical poststatified estimato that does use distibutional knowledge is ˆµ (m = π Ȳ[] E.Theπ s can be calculated fo nomal data. A nonpaametic vaiant of ˆµ (m that is also a membe of the class is ˆµ (m = ˆπ (nȳ [], whee ˆπ ( is an S 101 S 10 vs 10 5 estimate of the cell pobability π based on n. The cell poba- denoted by ˆµ (m 10 BLU, minimizes the vaiance of a subclass of E, bilities can be estimated by the anking pocedue (Deming and Stephan 190, because the maginal pobability fo each auxil- 8 iay vaiable ank is known to be 1/H because of chaacteistics 107 ˆµ (m BLU has the fom of ode statistics. The estimato of MacEachen et al. (00 is also a membe of E, because ( can be ewitten as ˆµ (m ˆµ (m BLU = (Ȳ[] 5 M a (nȳ [], [σ [] E(1/n ] 1 δ [], 111 = a (n = 1 H H b i n, i=1 b BLU in σ[] E(1/n 115 whee b i is the count of ank i in the ow vecto. Nowwe examine seveal new estimatos in this class suggested by commonly used estimation methods, each of which makes use of the distibutional knowledge though the stuctue of the moments of the concomitant of multivaiate ode statistics. We fist conside the odinay least squaes estimato of µ y, denoted by ˆµ (m and defined as the estimato minimizing the sum of squaed distances fom each y i to the mean of its poststatum, namely n [ min I i yi ( ]. µ y + δ [] (5 Unde the nomality assumption, we have fom (0 that δ [] = σ y ρ T xy ρ 1E(Z (. Solving (5 yields = 58, ˆµ(m BLU 117 n Because E( ˆµ (m n = µ y and va( ˆµ (m n = n σ[] /n,we have E( ˆµ (m = µ y and va ( ˆµ (m [ ( (m = va E ˆµ n] + E [ va ( ˆµ (m n] = 1 π σ[] n. (6 Next, conside the weighed least squaes estimato, denoted by ˆµ (m distances to the poststata means, namely n [ ] yi (µ y + δ [] I i. (7 Solving (7 yields = n /σ = E [( 1 ] whee the n is multinomial with paametes n and π fo all H m possible. In addition, we might natually think of the best linea unbiased estimato, whose weights ae constant (i.e., not functions of andom vaiables. In ou JP S setting, this estimato, the unbiased estimatos of the fom w (Ȳ [] δ [], whee the weights w s ae esticted to be constant and sum to 1. with va ( ˆµ (m [σ[] E(1/n ] 1 = [ 1 ] 1. Now we poceed to compae the thee unbiased estimatos, ˆµ (m, and ˆµ(m, which ae all in E. Fist, ˆµ(m has the smallest vaiance among the thee, as we state in

8 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 8 8 Jounal of the Ameican Statistical Association,???? 0 Theoem. Second, ˆµ (m is easie to compute, especially when the numbe of poststata H m is lage, because it does good appoximation to the vaiance of ˆµ (m 61 not equie the vaiances σ[]. Finally, ˆµ(m and ˆµ(m BLU have when n is easonably lage. It also woks well fo small n if the diffeence simila expessions, because ˆµ (m 5 BLU can be obtained by eplac- between the uppe and lowe bound is small, which occus fo 6 ing 1/n by E(1/n in (8; they also have the same asymptotic vaiance, (n π /σ[] 1. Howeve, ˆµ (m is not quite 7 BLU 66 8 satisfactoy. It is not applicable when the sample size n is small and ˆµ(m, adding an exta anking vaiable impoves 67 compaed with the numbe of stata H m. In this case thee ae many empty cells with inestimable means, which will cause touble, because ˆµ (m assigns a pespecified nonzeo weight BLU 70 1 to each cell. In contast, ˆµ (m 71 and ˆµ(m ae both data-adaptive (X, X m+1, and ˆµ (m ( ˆµ(m is the coesponding estimato using the fist m anking vaiables X only. Then ˆµ (m by assigning nonzeo weights to nonempty cells only. Even if no empty cell occus, the pefomance of ˆµ (m is moe BLU is vey sensitive efficient than ˆµ (m to n and is much wose than that of ˆµ (m fo lage n, and ˆµ(m+1 is moe efficient 16 and ˆµ(m,asweshow than ˆµ (m fo any n (see App. B fo the poof. 75 in ou simulation. The following theoem establishes an optimal popety 19 fo ˆµ (m. gain can be expected. We investigated this fo the special case 78 of adding a second auxiliay vaiable to the fist. The asymp- Theoem. ˆµ (m 1 has the lowest mean squaed eo (MSE totic elative efficiency (ARE, ARE = lim n + [va( ˆµ (1 / 80 among the class of estimatos of the fom (. Poof. It is obvious that ˆµ (m s weights w (n = (n /σ[] / and eq. (6.8.b of David and Nagaaja (00, chap. 6.8 n /σ[] minimize va( ˆµ (m n = w (nσ [] /n. Because E( ˆµ (m n = µ y,wehaveva[e( ˆµ (m n] =0. Thus This theoem assues us that ˆµ (m is the most efficient 1 among the estimatos discussed, not limited to ˆµ (m 90, ˆµ(m, ˆµ ( ove ˆµ(1, in which we obseved the tightness of the two and ˆµ (m bounds in (0. As a esult, the values of the elative efficiency 91 BLU. Howeve, its vaiance (9 is not expessed in a wee vitually identical to those of the ARE fo the weighted 9 closed fom and so is had to compute. In the following coollay, we povide an uppe bound by compaing it with ˆµ (m also a lowe bound by consideing its asymptotic vaiance. 5 and 9 Coollay. The lowe and uppe bounds fo the vaiance of ˆµ (m The lowe bound (i.e., the asymptotic vaiance povides a nomal data in many cases. Finally, the following theoem justifies ou intuition that fo both ˆµ (m estimation efficiency, at least in an asymptotic sense. Theoem. Suppose that ˆµ (m+1 ( ˆµ (m+1 is the weighted (odinay least squaes estimato with anking vaiables Although Theoem establishes that the addition of anking vaiables is helpful, a pactical question is just how much va( ˆµ ( ], was computed fom the lowe bound in (0, (15, fo ρ 1 =.5,.5,.75; H =,, ; and a ange of values of ρ 1y and ρ y. Figue shows the esults fo two equally effective ankes (i.e., ρ 1y = ρ y and the thee values each of ρ 1 and H va( ˆµ (m E[va( ˆµ (m n] E[va( ˆµ (m 86 n] =va( ˆµ(m We see that the gain fom the second anke can be substantial; it inceases as eithe the anking quality o the numbe of 8 and MSE( ˆµ (m MSE( ˆµ (m 87, because ˆµ(m is unbiased. 9 anking classes inceases and deceases as the two ankes be- 88 come moe simila. We also computed the elative efficiency fo one, so ae not shown in Figue. 5. Simulation We have demonstated that unde the nomality assumption with known σ y, ρ xy, and ρ x, the weighted least squaes esti- ae 9 mato ˆµ (m is the most efficient among the class E including 98 0 ( π /σ[] va ( ˆµ (m 1 membes Ȳ, ˆµ (m 100 n π σ[] n. (0 M, ˆµ(m S, ˆµ (m vs, ˆµ(m, ˆµ(m, and ˆµ(m BLU. In pactice, howeve, these paametes will not be known, and the dis- 101 tibutional assumptions may not hold exactly. Thus we designed 10 5 (a (b (c Figue. Asymptotic Efficiency of ˆµ ( Ove ˆµ (1 fo Pais of Equally Effective Rankes: (a H =, ρ 1y = ρ y ;(bh=, ρ 1y = ρ y ;(ch=, ρ 1y = ρ y (, ρ 1 =.5;, ρ 1 =.5;, ρ 1 =

9 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 9 Wang et al.: Concomitants of Multivaiate Ode Statistics 9 1 Table 1. Compaing Efficiency of the JP S Estimatos With Estimated Paametes 60 H = ; sample size H = ; sample size H = 10; sample size 61 (ρ 1y, ρ y, ρ 1 Mean estimato (.9,.9,.65 ˆµ ˆµ ˆµ M ˆµ BLU (.8,.8,.5 ˆµ ˆµ ˆµ M ˆµ BLU (.8,.5,.5 ˆµ ˆµ ˆµ M ˆµ BLU (.5,.5,.5 ˆµ ˆµ ˆµ M ˆµ BLU NOTE: Note that due to the empty-cell poblems, ˆµBLU is not applicable when n is small compaed with H. 77 a simulation study fo two puposes: (1 to compae the pefomance of the estimatos when σ y, ρ xy, and ρ x ae unknown and must be estimated fom the data and ( to test thei obustness when the nomality assumption is violated. In ou peliminay simulations, we found that the estimato of MacEachen et al. (00 pefomed consistently best among the thee sampling 7 estimatos ˆµ (m M, ˆµ(m S, and ˆµ (m vs. Hence we included only ˆµ(m M, ae epoted in Table. Compaing the simulated values in Table 1 to these, we obseve that estimating these paametes has ˆµ(m 86 8 ˆµ (m 87,, and ˆµ(m BLU in the full study pesented hee, along a negligible effect when n 50. Fo smalle sample sizes, both with Ȳ as a benchmak estimato. In ou fist expeiment, we simulated JP S samples fom the standad multivaiate nomal distibution fo Y and two auxiliay vaiables (X 1, X fo fou sets of (ρ 1y,ρ y,ρ 1 : (.9,.9,.65, (.8,.8,.5, (.8,.5,.5, and (.5,.5,.5. WesetH to be,, o 10 and n to be 10, 0, 50, o 100. Because m is fixed at, we omit the supescipts in the discussion that follows. When calculating ˆµ, ˆµ, and ˆµ BLU fom each sample, we substituted estimates fo σ y and the ρ s, computed using standad methods. Table 1 epots the simulated elative efficiency of the fou JP S estimatos to the SRS estimato Ȳ fo each setting. Hee efficiency is defined as the atio of the vaiance of Ȳ to MSE of each JP S estimato, whee the MSE is estimated fom 0,000 eplicates. The esults in Table 1 show that the two least squaes estimatos outpefom the othe two estimatos in all cases, even though they use estimates of σ y and the ρ s. The pefomance of ˆµ is at most only slightly bette than that of ˆµ.Both estimatos pefom well fo even small n. The impovement fom the two paametic estimatos ove the nonpaametic estimato, ˆµ M, is consideable, especially when n is small and H is lage, as long as the anking vaiables ae effective. In contast, ˆµ BLU pefoms pooly oveall; it is sensitive to sample size and is not applicable when empty cells occu. Hence we do not conside ˆµ BLU futhe hee. To moe closely examine the effect of estimation of the unknown coelations and vaiance, we compute the asymptotic efficiency fo ˆµ ove Ȳ using the lowe bound in (0 and efficiency fo ˆµ ove Ȳ using (6. Thei theoetical values ˆµ and ˆµ lose some efficiency by doing so, although they still pefom bette than ˆµ M. In the second expeiment, we examine the pefomance of the JP S estimatos when the nomality assumption is violated. We conside thee cases: (1 (log X 1, log X, Y ae geneated fom the standad nomal distibutions with the fou sets of coelations as befoe, ( (log X 1, log X, log Y ae geneated fom the same distibutions as in case (1, and ( (X 1, X, Y follows the multivaiate unifom distibution descibed in Example 1. Hee we set H =, geneate 0,000 JP S samples fo each setting, and calculate ˆµ, ˆµ, and ˆµ M fom each. The fome two estimatos ae computed as if (X 1, X, Y wee multivaiate nomal, but using estimated values fo σ y and ρ s. Table epots the simulated efficiency. Seveal obsevations can be made fom Table. When the anking vaiables violate the nomality assumption but Y is still nomal, ˆµ and ˆµ pefom compaably and have efficiencies simila to those in the nomal case. When both X and Y ae lognomal, ˆµ outpefoms ˆµ, wheeas the situation is evesed when (X 1, X, Y have a multivaiate unifom distibution. As expected, the least squaes estimatos ae less efficient than in the nomal case when Y is no longe nomally distibuted. Supisingly, ˆµ M does not pefom as well as ˆµ Table. Theoetical Values of (asymptotic Relative Efficiency of ˆµ and ˆµ Theoetical (.9,.9,.65 (.8,.8,.5 (.8,.5,.5 (.5,.5,.5 value H = H= H= 10 H = H= H= 10 H = H= H= 10 H = H= H= 10 ARE( ˆµ 58, Ȳ RE( ˆµ, Ȳ

10 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p Jounal of the Ameican Statistical Association,???? 0 1 Table. Compaing Efficiency of the JP S Estimatos With Estimated Paametes (H = only When the Nomality Assumption Is Violated 60 (log X 1,logX,Y MVN; (log X 1,logX,logY MVN; Multivaiate unifom; 61 Mean sample size sample size sample size 6 estimato (ρ 1y, ρ y, ρ ˆµ (.9,.9, ˆµ ˆµ M ˆµ (.8,.8, ˆµ ˆµ M ˆµ (.8,.5, ˆµ ˆµ M ˆµ (.5,.5, ˆµ ˆµ M in any of the cases consideed o as well as ˆµ except when Y is heavy-tailed. This leads to ou belief that even with modeate deviation fom nomality, ˆµ and ˆµ still may achieve bette pefomance than ˆµ M, especially fo small n. 5. An Empiical Study: Human Teeth Width This section uses a eal dataset to compae the JP S estimatos of the mean. To examine thei pefomance in both infinite and finite population settings, samples wee selected with and without eplacement fom a small population containing measuements on teeth widths fo 95 subjects in a health suvey conducted in Seoul, Koea (Lee, Lee, Lim, Ahn, and Kim 006. All teeth wee measued by digital Venie calipes, a pocess that equies a -week taining peiod to maste. Hee ou goal is to estimate the mean width of teeth in the back of the mouth, using the middle teeth as anking vaiables. A pactical 7 86 and paamete estimates computed fom thei data wee taken as the tue population paametes. These included ρ 16 =.50, ρ 6 =.50, ρ 1 =.576, µ L6 = 10.95, and σl6 =.19. Standad diagnostic checking, pefomed on (U 1, U, L 6 though the maco %MULTNORM in SAS, did not eveal any goss violation of nomality. In this simulation, we set H = 5 and sample sizes n = 10, 15,...,55. To obtain a JP S sample of size n with eplacement, we epeated the following pocedue n times: A set of 5 subjects was andomly selected fom all 95 subjects, and bivaiate anking was done based on U 1 and U within the set; then 1 of the 5 subjects was andomly selected to ente the sample. In contast, to obtain a JP S sample of size n without eplacement, the set of five subjects selected on each daw wee excluded fom the dataset, so they wee not available fo the next selection. Fo each JP S sample of size n, we calcu- justification fo doing this is that a tooth close to the cente is lated ˆµ (, ˆµ(, and ˆµ( M, with poststata detemined by anks 9 5 much easie to ode than a tooth fathe back in the month. of U 1 and U, and ˆµ (1, ˆµ(1, and ˆµ(1 M, with poststata dete- 9 The widths of the fist uppe incisos, U 1 (the fist tooth fom the cente, and the fist uppe canines, U (the thid tooth fom the cente, wee used as anking vaiables. Selection of JP S samples was simulated, and estimates of the mean width of the fist lowe molas, L 6 (the sixth tooth fom the cente, wee calculated. The 95 subjects wee teated as the tue population, mined by U only. All least squaes estimatos wee computed using σ y and the ρ s estimated fom the sample. Figue shows values of the simulated elative efficiency of the six JP S estimatos to Ȳ fo each sample size n. Hee the MSE is estimated fom 100,000 eplicates. The figue shows that whethe sampling is with o without eplacement, the least (a (b Figue. An Empiical Study of Human Teeth Width. (a Sampling with eplacement. (b Sampling without eplacement. (, 1;, 1; ,M1;, ;, ;,M. 118

11 JASA jasa v.00/1/09 Pn:0/06/006; 16:6 F:jasatm05511.tex; (R p. 11 Wang et al.: Concomitants of Multivaiate Ode Statistics 11 squaes estimatos have almost identical pefomance and with the two anking vaiables they have the best pefomance of all. In addition, the esults demonstate that the benefit fom using a second anking vaiable when sampling without eplacement is lage than that fom sampling with eplacement samples. Thus it may be safe to use the least squaes estimatos fo small populations DISCUSSION We have defined concomitants of multivaiate ode statistics and povided analytical expessions fo thei densities and moments. We have also illustated the use of the theoy by povid Fo notational claity, let Y 1 [,s];z explicitly denote the concomitant ing new estimatos that use anking infomation fom moe than 7 of the th-ode statistics of Z 1 1 and the sth-ode statistics of Z with one auxiliay vaiable fo impoving estimation of the mean. 7 mean µ [,s];z and vaiance σ 15 [,s];z We note that the poposed least squaes estimatos do not s denote the bivaiate ank 7 distibution of Z 1 and Z,andlet(Z 1(,s, Z (,s be the bivaiate 16 equie nomality. They ae available when cetain distibutional 75 ode statistics of (Z 1, Z with joint density g (,s (z 1, z. Because 17 assumptions about the data can be made: Develop- ψ 1 and ψ ae monotonic, to show (1 and (1, it is equivalent to ment of ˆµ equies that δ [] not be a function of µ y (say, show fo {1,...,H} and s {1,...,H}, δ 78 [] µ y fo each poststatum, and ˆµ equies that both 0 µ [,s];z + µ [, s];z = µ y (A.1 79 δ [] µ y and σ[] µ y. Suppose that fo each anking vaiable X i (1 i m, thee exists a monotonic function g i ( and such that Z 81 i = g i (X i and Z = (Z 1,...,Z m has a joint distibu- σ[,s];z = σ [, s];z. (A. 8 tion f (z; with the paamete set µ y (e.g., a special case is that each X i is fom a location-scale distibution family. Let Unde the conditions that E(Y z 1, z is a linea function of z 1 and z, 5 m(z; 8 m E(Y µ y Z = z, which is a function of z and a set and g(z 1, z = g( z 1, z, we can obtain 6 of distibutional paametes 85 m ;letv(z; v va(y Z = z, 7 µ [,s];z = µ y + β 1 E ( Z 1(,s + β E ( Z (,s (A. 86 which is a function of z and a set of distibutional paametes v. Then a sufficient condition fo δ [] µ y is m µ y, 9 and a sufficient condition fo σ 88 [] µ and y is v µ y, which follow diectly fom Theoem 1 and its highe-dimensional gene σ[,s];z = va(y z 1, z g (,s (z 1, z dz 1 dz 1 alization. These sufficient conditions may be milde than those Z 90 assumptions in most egession setups, because they do not equie lineaity o any othe functional fom fo the conditional β1 va( Z 1(,s + β1 β cov ( Z 1(,s, Z (,s + β va( Z (,s, (A. 9 expectations. In fact, the esult in Theoem (i.e., the optimality of ˆµ whee β 1 and β ae constants and Z is the suppot of the distibution 6 among linea estimatos is athe geneal. Unde the assumptions discussed ealie, this esult can be diectly extended to sit of (Z 1, Z. 7 Now conside E(Z 1(,s, which can be expessed by 8 uations in which the sampling space can be patitioned to stata 97 9 though eithe sampling design o poststatification. Howeve, 98 0 obtaining the bias-coection tem δ [] and the vaiance σ 99 [] fo E ( Z 1(,s = z 1 g (,s (z 1, z dz 1 dz. Z 1 each statum is nontivial. In ou JP S applications, these can be 100 Define z 1 = z 1 and z = z.then deived though the theoetical developments in Sections, 101 which geatly facilitate computation of the most efficient linea 10 E ( Z 1(,s = z 1 g (,s(z 1, z dz 1 dz. (A.5 Z 10 estimato. Thee ae othe examples in the liteatue in which a single Because g(z 1, z = g(z 1 6 anked vaiable is used to impove estimation of some paamete. The methods that we have developed hee can be used in 106, z, q s = q s,sothat g (,s (z 1, z = q(, s z 1, z g(z 1, z z 8. (A.6 q s 107 those applications as well. Fo example, the appoach of Banett et al. (1976 fo obtaining linea estimates of coelation coef- Fom (5, 50 ficients can be diectly adapted when infomation is available q(, s z 1, z = q(, s z 1, z, (A fom two o moe anking vaiables, using the moment expessions (0 and (1. by noting that θ 5 Othe useful applications will equie additional theoetical 1 (z 1, z = θ (z 1, z, θ (z 1, z = θ (z 1, z, θ (z 1, 11 z = θ (z 5 1 development. Fo example, the popeties of concomitants of, z,andθ (z 1, z = θ 1 (z 1, z. Inseting (A.7 in (A.6 11 and (A.8 in (A.5 yields 55 exteme ode statistics have been a topic of study fo thei g (,s (z 1, z = g (, s (z 1, z (A use in anking and selection (Yeo and David 198; Anold and Beye 005. The notion of exteme ode statistics of a vecto of anking vaiables can be defined in vaious ways, with the best way undoubtedly depending on its pupose. Finally, we note that we have assumed that the numbe of anking classes is the same fo all anking vaiables. Thee ae applications in which a genealization to the case in which one anking vaiable allows H classes, wheeas anothe that allows H classes may be needed. Fo example, conside the employee selection poblem in which not evey candidate had the complete battey of sceening tests. In that case, it would be useful to have a way to examine popeties of the concomitant of multivaiate ode statistics, some of which ae patially anked. APPENDIX A: PROOF OF THEOREM and E ( ( Z 1(,s + E Z1(, s = 0. (A.9 118