Statistics and Probability Letters

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1 Statistics and Probability Letters Contents lists available at ScienceDirect Statistics and Probability Letters journal hoepage: A CLT for a one-diensional class cover proble Pengfei Xiang a, John C. Wieran b, a Wachovia Bank, USA b Departent of Applied Matheatics and Statistics, 302 Whitehead Hall, Johns Hopkins University, 3400 N. Charles Street, Baltiore, MD 228, USA a r t i c l e i n f o a b s t r a c t Article history: Received 4 January 2008 Received in revised for 28 July 2008 Accepted 3 July 2008 Available online 9 August 2008 The central liit theore is proved for the doination nuber of rando class cover catch digraphs generated by unifor data in one diension. The class cover proble is otivated by applications in statistical pattern classification Elsevier B.V. All rights reserved.. Introduction The probabilistic behavior of rando Class Cover Catch Digraphs CCCDs is of considerable recent interest, due to their use in pattern classification ethods involving proxiity regions. Priebe et al. 200 found the exact distribution of the doination nuber of CCCDs for the unifor distribution in one diension. Under the sae conditions, DeVinney and Wieran 2002 proved the SLLN for the doination nuber. Wieran and Xiang 2004 further extended the SLLN to a class of continuous distributions in one diension. Xiang 2006 calculated the variance of the doination nuber for the unifor distribution in one diension. Solutions to the CCCD proble can be used to build classifiers. More details and exaples of the application of CCCDs to classification are presented in Priebe et al. 2003, Marchette and Priebe 2003, Ceyhan and Priebe 2005, Eveland et al. 2005, Ceyhan and Priebe 2006, DeVinney and Priebe 2006, Ceyhan et al. 2006, and Ceyhan et al The class cover proble The class cover proble CCP is otivated by its applications in pattern classification. The study of the CCP was initiated by Cannon and Cowen 2000, and has been actively pursued recently, because its solution can be directly used to generate classifiers copetitive with traditional ethods. For a foral description of the CCP, consider a dissiilarity function d : Ω Ω R such that dα, β = dβ, α dα, α = 0 for α, β Ω. We suppose X = {X i : i =,..., n} and Y = {Y j : j =,..., } are two sets of i.i.d. rando variables with class-conditional distribution functions F X and F Y, respectively. We assue that each X i is independent of each Y j, and all X i X and all Y j Y are distinct with probability one. For each X i, we define its covering ball by BX i = {ω Ω : dω, X i < in j dy j, X i }. A class cover of X is a subset of covering balls whose union contains all X i X. Obviously, the set consisting of all covering balls is a class cover. However, we want to choose a class cover to represent class X that is as sall as possible, to ake the classifier less coplex while keeping ost of the relevant inforation. Therefore, the CCP we consider here is to find a iniu-cardinality class cover. Furtherore, we can convert the CCP to the graph theory proble of finding doinating sets. The class cover catch digraph CCCD induced by a CCP is the digraph D = V, A with the vertex set V = {X i : i =,..., n} and the arc set A such that there is an arc X i, X j if and only if X j BX i. It is easy to see that the CCP is actually equivalent to finding a iniucardinality doinating set of the corresponding CCCD. Corresponding author. Tel.: ; fax: E-ail address: wieran@jhu.edu J.C. Wieran /$ see front atter 2008 Elsevier B.V. All rights reserved. doi:0.06/j.spl

2 224 P. Xiang, J.C. Wieran / Statistics and Probability Letters Previous results The doination nuber of a CCCD is the cardinality of the CCCD s iniu doinating set. In the CCCD setting, we denote the doination nuber by Γ n, X, Y, or siply by Γ n,. DeVinney and Wieran 2002 proved the Strong Law of Large Nubers SLLN for the special case Ω = R, F X = F Y = U[0, ]: Theore.. If Ω = R, F X = F Y = U[0, ], and = rn, r 0,, then Γ n, li = gr n + n r2r + 3 3r + 4r + 3 a.s. Wieran and Xiang 2008 generalized this result by proving the SLLN for general distributions in one diension: Theore.2. If Ω = R, the density functions f X, f Y are continuous and bounded on [a, b], and /n r, r 0,, then Γ n, f X, f Y b li = g r fy u f X udu a.s., n n f X u where gr is the sae as in Theore.. a As a first step in proving the Central Liit Theore CLT for the doination nuber Γ n,, Xiang 2006 calculated the liiting variance for Γ n,. The calculation is very technical and lengthy, but the final result can be siply stated as follows, with an outline of the calculation given in Appendix A: Theore.3. If Ω = R, F X = F Y = U[0, ], and /n r, r 0,, then VarΓ n, vr = 536r r r r r r + 3 4r A central liit theore In this paper, we prove the CLT for the doination nuber generated by uniforly distributed data: Theore.4. If Ω = R, F X = F Y = U[0, ], and /n r, r 0,, then /2 Γn, E[Γ n, ] L N0, σ 2, where σ 2 = li Var[Γ n,] the exact liiting value is given in Theore.3. In our proof of the CLT, we extensively use the concept of negative association, so in the next sub-section we introduce soe of its basic properties and consequences..4. Negative association The concept of negatively associated NA rando variables was introduced and carefully studied by Joag-Dev and Proschan 983. Since then, liit theores for this type of rando variables have been well established. See Newan 984 and Taylor et al Definition.. Rando variables X i, i =,..., k, are said to be negatively associated NA if for every pair of disjoint subsets I, J of {,..., k}, and any increasing functions f I and f J the following covariance exists, Cov { f I X i, i I, f J X j, j J } 0, NA ay also refer to the rando vector X,..., X k. The following proposition is obvious fro the definition. Proposition.. Increasing functions defined on disjoint subsets of a set of NA rando variables are NA. Furtherore, several types of rando vectors were proven to be NA by Joag-Dev and Proschan 983, particularly: Proposition.2. A ultinoial rando vector is NA.

3 P. Xiang, J.C. Wieran / Statistics and Probability Letters Proposition.3. Let X,..., X k be k independent rando variables with PF 2 log-concave densities. Then, given k i= X i, the rando vector X,..., X k is NA. In soe situations, a dependence condition called negative dependence, which is weaker than negative association, is used. Definition.2. Rando variables X i, i =,..., k, are said to be negatively dependent ND if for any real nubers x i, i =,..., k, and PX i > x i, i =,..., k PX i x i, i =,..., k k PX i > x i, i= k PX i x i. i= Note. By choosing the functions in Definition. to be indicator functions of the events {X i > x i } and the events {X i x i }, it is readily apparent that negative association iplies negative dependence. Taylor et al proved that the SLLN holds for row-wise ND rando variable arrays. We apply one part of their theore: Theore.5. Let {X k, : k, } be row-wise ND rando variable arrays such that E[X k, ] = 0 for each k and. If X k, M for all k and for soe constant M <, then X /p k, k= a.s. 0, 0 < p < 2. Newan 984, Theore established the CLT for ND sequences. First, a distributional liit theore for row-wise ND rando variable arrays was proved: Theore.6. Suppose X k, and Y k,, k,, are triangular arrays such that for each and k, rando variable X k, is equidistributed with Y k,. Assue for each, the rando variables X k,, k =,..., are ND, but Y k,, k =,..., are independent. If in addition, CovX i,, X j, li i<j = 0, then /2 X k, converges in distribution if and only if distribution. /2 Y k, converges in distribution to the sae liit By applying the classical CLT for bounded i.i.d. rando variable arrays to the {Y k, } in the theore above, we conclude that: Theore.7. Let {X k, : k, } be identically distributed row-wise ND rando variable arrays such that E[X k, ] = 0 for each k and. If X k, M, and CovX k,, X l, then li k<l X /2 k, k= L N0, σ 2, where σ 2 Var[ k= = li X k,]. = 0, Reark. Since negative association iplies negative dependence, Theores.5.7 will all hold if the {X k,, } are row-wise NA. : k In Appendix B, we show that the {N j, : 0 j, } in the CCCD proble are row-wise NA.

4 226 P. Xiang, J.C. Wieran / Statistics and Probability Letters Proof of the CLT 2.. Basic idea For j =, 2, 3,...,, let Y j denote the jth order statistic of Y,..., Y, and define Y 0 = 0 and Y + =. Define α j, as the iniu nuber of covering balls required to cover the N j, X-points contained in [Y j, Y j+ ]. It should be noted that Γ n, = α j,; thus, the original CCP is decoposed into + sub-ccps of finding the doination nuber α j, in the interval [Y j, Y j+ ]. Since α j,, j = 0,...,, are dependent, the classical CLT theore cannot be directly applied. However, given N j,, j = 0,...,, the doination nubers α j,, 0 =,...,, are conditionally independent. Therefore, we can calculate the conditional characteristic function f for α j, on the σ -field generated by N j,, j =,...,. Using the Taylor expansion, a lengthy calculation shows that f can be expressed in ters of E[α j, N j, ] and Var[α j, N j, ]. We prove that these two sequences of rando variables are both negatively associated. Thus, applying liit theores for row-wise NA arrays, we conclude that f converges to a constant alost surely. By the doinated convergence theore, the unconditional characteristic function E[f ] goes to the sae constant, hence the result follows by the convergence theore for characteristic functions Detailed proof and Denote F = σ N 0,,..., N,, the σ -field generated by N j,, j = 0,...,. Let Z j, = /2 αj, E[α j, ] it f t = E e Z j, F. By Lea A.3, the Z j,, j = 0,..., are conditionally independent given F, so f t = [ ] E e itz j, F. Again by Lea A.3, we know that Z j, only depends on N j, given F, so f t = [ ] E e itz j, N j,. Using the Taylor expansion e iz = + iz 2 z2 + Az, where Az z 3 6, the conditional characteristic function of Z j, can be written as where [ ] [ ] E e itz t 2 j, F = + ite Zj, [ N j, 2 E Z 2 ] j, N j, + r, j, = [ [ ] ] tzj, j, E 3 AtZ j, N j, E 6 N j,. r Therefore, by substituting the forula for E [ e itz j, F ] into the expression for f t, we obtain log f t = [ ] log E e itz j, F j, = log + ite[z j, N j, ] t2 [ 2 E Z 2 ] j, N j, + r. Again, by the Taylor expansion log + δ = δ δ2 δ 3 + rδ, where rδ for δ <, 2 24 j,

5 P. Xiang, J.C. Wieran / Statistics and Probability Letters log f t can be further expanded as log f t = [ ] t 2 ite Z j, [ N j, 2 E Z 2 ] j, N j, + r ite[z j, N j, ] t2 [ 2 2 E Z 2 ] 2 j, N j, + r j, + r 2 j,, 2 where r 2 j, [ ] ite Z 24 j, [ N j, t 2 E Z 2 ] 2 j, N j, + r j, 3. [ Recall that r tzj, j, is bounded by E 3 N 6 j, ]. Since α j, is bounded by 2 as shown in Priebe et al. 200, Z j, = αj, E[α /2 j, ] 4. Thus r j, t C 3/2 3 3/2 where C = C t 32 3 t 3. We now proceed to consider the quadratic ter in Eq. 2. By the sae reasoning used to derive the bound on r, we conclude that E[Zj, N j, ] 4, /2 [ E Z 2 ] j, N j, 6. Using Inequalities 3 5, we ay write the quadratic ter in Eq. 2 as j, j, 4 5 t 2 2 E[Z j, N j, ] 2 + r 3 j,, 6 where r 3 j, C /2 It reains to check r 2. Based on the bound for r 2 given in Forula 2, and Inequalities 3 5, when is sufficiently large, r 2 j, j, 24 ite[z j, N j, ] t2 [ 2 E Z 2 ] j, N j, + r j, j, 3 C /2 Now we put all the pieces together. By substituting Forula 6 into Eq. 2, we have log f t = ite[z j, N j, ] t2 [ 2 E Z 2 ] j, N t 2 j, + 2 E[Z j, N j, ] 2 + r + j, r2 + j, r3 j, so f t = e = it E[Z j, N j, ] t2 2 it E[Z j, N j, ] t2 2 Var[Z j, N j, ] + Var[Z j, N j, ] r j, +r2 j, +r3 j,, r + j, r2 + j, r3 j,, 9 and taking expectations yields that the characteristic function E[f t] equals it E[Z j, N j, ] t2 Var[Z 2 j, N j, ] r j, E +r2 j, +r3 j, e. 0 Note that by Lea B.2, the rando variable array { {N j, } is NA. Because E[α j, N j, ] E[α j, ] is an increasing function of N j,, by Proposition., the rando variable array E[α j, N j, ] E[α j, ] } is also NA. Hence, by the CLT for row-wise NA arrays Theore.7, we get E[Z j, N j, ] = /2 E[αj, N j, ] E[α j, ] L N0, σ 2,

6 228 P. Xiang, J.C. Wieran / Statistics and Probability Letters where { σ 2 = li Var E[α, N, ] + Covα,, α 2, }. Therefore, by the convergence theore of characteristic functions, we obtain it E[Z j, N j, ] E e e t2 2 σ 2. { Siilarly, we can prove that the rando variable array Var[α j, N j, ] } is also NA, hence by the SLLN theore for row-wise NA arrays Theore.5, we have Var[α j, N j, ] a.s Var[Z j, N j, ] = σ 2 2, where σ 22 li E [ Var[α, N j, ] ]. Thus, t2 2 e Var[Z j, N j, ] a.s e t2 2 σ 2 2. Fro the bounds we obtained in 3, 7 and 8, r j, +r2 j, +r3 j, e e 0 =. The two convergence results iediately above produce the following: t2 2 e Var[Z j, N j, ] Therefore, by Eq. 0, it E[f t] E e it E e t2 = E e 2 E[Z j, N j, ] r j, +r2 j, +r3 j, E[Z j, N j, ] e Var[Z j, N j, ] t2 2 σ 2 2 t2 Var[Z 2 j, N j, ] a.s e t2 2 σ r j, +r2 j, +r3 j, by the doinated convergence theore. Cobining this with Forula, we obtain where li E[f t] = li it E[Z j, N j, ] e r j, +r2 j, +r3 j, e t2 2 σ 2 2 e t2 2 σ t2 2 σ 2 2 = e t2 2 σ 2 t2 2 σ 2 2 = e t2 2 σ 2, σ 2 σ σ 2 [ = li Var E[α, N, ] ] + Covα,, α 2, + [ li E Var[α, N j, ] ] Var[Γ n, ] = li. Recalling the definitions of f t and Z j, given at the beginning of the proof, we finally obtain [ E e it /2 Γ n, E[Γ n, ] ] e t2 2 σ 2. Thus, the result follows by the convergence theore for characteristic functions.

7 P. Xiang, J.C. Wieran / Statistics and Probability Letters Further research In this paper, we have established the CLT for the doination nuber of CCCDs in the case of the unifor distribution in one diension. Further research directions consist of extending the CLT to ore general distributions in one diension, and finally obtaining a siilar result in higher diensions. As any applications of CCCDs arise in higher diensions, proving the CLT in this situation would significantly benefit evaluation of CCCD-classifiers. Acknowledgents The authors gratefully acknowledge financial support for this research fro Johns Hopkins University through the Acheson J. Duncan Fund for the Advanceent of Research in Statistics. They also thank Carey Priebe for encourageent and helpful discussions regarding research on class cover catch digraphs and related statistical classification ethods. Appendix A. Calculation of the variance of the doination nuber In this appendix, we outline the calculation of the exact and liiting variance of Γ n,. Recall that Y 0, Y +, Y j denotes the jth order statistic of Y,..., Y, and the rando variable α j, is the iniu nuber of covering balls needed to cover the N j, X-points located between Y j and Y j+. The rando variables α j,, j = 0,, are referred to as the external coponents, and α j,, j =,..., as the internal coponents. A.. Expressions in ters of expectations First, copute the conditional oents of α j,, using the conditional distribution forulas given in the following theore by Priebe et al. 200: Theore A.. If F X = F Y = U[0, ], then for j {0,,..., }, if N j, = 0, then α j, = 0; for j {0, }, if N j, > 0, then α j, = ; for j {, 2,..., }, if N j, = n j, > 0, then Pα j, = N j, = n j, = Pα j, = 2 N j, = n j, = n j, Based on the above forulas, straightforward calculations yield 0 N j, = 0, j = 0,..., N Eα j, N j, = j, > 0, j = 0, N j, > 0, j =,...,, 9 4 N j, and 0 N j, = 0, j = 0,..., 0 N Varα j, N j, = j, > 0, j = 0, N 4 N j, > 0, j =,...,. j, 8 4 2N j, To deterine the arginal and joint distributions of {N j, }, we use the following two leas: A. A.2 Lea A.. Given L j, Y j+ Y j = l j,, j = 0,,...,, the rando vector {N j, } is ultinoially distributed with paraeters {n, l j, : l j, = }. Lea A.2. The density function of L j, is f l j, = l j,, and the joint density function of L j, and L j2, is f l j,, l j2, = l j, l j2, 2.

8 230 P. Xiang, J.C. Wieran / Statistics and Probability Letters Two consequences of these two leas are used repeatedly in our calculations: PN j, = 0 = + n PN j, = 0, N j2, = 0 = + n + n. Using the identity Varα j, = E[Varα j, N j, ]+Var[Eα j, N j, ], we can calculate Varα j, fro Eqs. A. and A.2: n j = 0, + n Varα j, = 2 20 n 8 + n + 69 n n µ n, 256 A n 8 µ n, 2 j =,...,, where µ n, E. 4 N j, I {N j, >0} Siilarly, we can calculate the covariance Covα j,, α j2, between any two coponents. An additional fact used in the calculation is the following lea: Lea A.3. For distinct j, j 2, given N j, and N j2,, the corresponding coponents α j, and α j2, are conditionally independent. In addition, α j, is only dependent on N j,. Using Lea A.3, we iediately obtain Covα j,, α j2, = Eα j,α j2, Eα j,eα j2, = E [ Eα j, N j,, N j2,eα j2, N j,, N j2, ] E [ Eα j, N j, ] E [ Eα j2, N j2, ] = E [ Eα j, N j,eα j2, N j2, ] E [ Eα j, N j,]e[eα j2, N j2, ]. The rest of the calculation of Covα j,, α j2, is siilar to that of Varα j,, obtaining the forulas n j + n 2 = 0, j 2 = + n Covα j,, α j2, = 3 n 9 + n 2 + n 6 9 δ n, j = 0, j 2 69 n 8 + n 2 + n 46 8 δ n, ν n, µ 2 n, j, j 2 =,...,, where δ n, E 4 I N {N j 2, j,>0,n j2,>0} n + n E 4 I N {N j 2, j2,>0}, ν n, E 4 I N {N j,+n j2, j,>0,n j2,>0}. A.4 A.2. Expressions in ters of series In the previous sub-section, we converted the variance and covariances of the coponents into expressions deterined by µ,n, δ n, and ν n,. Our next step is to copute series expressions for µ,n, δ n, and ν n,. For µ n,, we have [ ] µ n, = E 4 N I{Nj, 0} I {Nj, =0} j, = E PN 4 N j, = 0 = E j, 4 N j, + n. Fro Lea A., we have E 4 L N j, = l j, = j, n 4 q n l q j, q l j, n q. Use the distribution of L j, fro Lea A.2 to write µ n, as an integral: n µ n, = l q 4 q j, l j, n q l j, dl j, q + n. 0 0 q n

9 P. Xiang, J.C. Wieran / Statistics and Probability Letters Interchange integration and suation, and integrate, to obtain µ n, = n n! + n q! + n 4 q n q! + n!. A.5 Siilarly, δ n, = n n! + n q 2! n 2 q n 4 q n q! + n! + n + n 2 + n, and ν n, µ n, 2 = n + n n n! + n q 2! + n 2 + n n 4 q n q! + n 2! 2 + nq q n + n n n! + n q! 4 q n q! + n! 2 + { n n! + n q 2! 4 q n q! + n 2! q nq 2 + n + n }. A.6 A.3. Asyptotic results In the calculation of the exact liiting values µ r, δ r, ν r, we rely heavily on the following version of the doinated convergence theore DCT. Theore A.2. If D n q n Dq, and D n q D q where D q <, then D nq n Dq. For µ n,, let D n q = 4 n! + n q! { q n q! + n! can be easily checked that D n q n q n, then µ n, can be written as 0 q > n 4r+ as /n r, the liiting value of µ n, is r q = r r + 4r + r + 4r+ thus, µ n, = r r + 4r o. +n D nq. If /n r, then it q, and D n q 4 q, where 4 q = 4 <. Therefore, by Theore A.2, 3 By the sae technique, but a ore coplicated calculation, we obtain δ n, = r 4r r + 3 4r n + o, n ν n, µ n, 2 = r4r r + 3 4r n + oo Recalling that Γ n, = α j,, we have n, A.7 A.8. A.9 VarΓ n, = 2Varα 0, + Varα, + Covα 0,, α, + 2 Covα 0,, α, + Covα,, α 2,. By plugging Forula A.7 A.9 into Eqs. A.3 and A.4, and substituting the generated expressions for Varα j, and Covα j,, α j2, into the above equation, we can finally get the desired result stated in Theore.3.

10 232 P. Xiang, J.C. Wieran / Statistics and Probability Letters Appendix B. Proof of negative association of {N j, : 0 j, } To prove the rando vector N 0,,..., N, is NA for any, we need the following lea: Lea B.. If F X = F Y = U[0, ], then L 0,,..., L, is NA. Proof. Recall that L j, = Y j+ Y j and suppose that Z 0,..., Z are i.i.d. rando variables with an exponential distribution, where {Z 0,..., Z } are independent of {L 0,,..., L, }. Since the exponential distribution is log-concave, fro Proposition.3 we know that given Z i, the rando vector Z 0,..., Z is NA. Hence by Definition., we know that for any pair of disjoint subsets I, J of {0,..., } and any increasing functions f I and f J such that the following covariance exists, { } Cov f I Z i, i I, f J Z j, j J Z k = a 0, where a > 0. Since f I Z i, i I and f a J Z j, j J are still increasing functions of Z a i, i I and Z j, j J, respectively, we have { } i.e., Cov Cov f I f I Zi a, i I, f J Zj a, j J Z i, i I, f J Z k Z k = a Z j, j J Z k 0, Z k = a 0. Note that given Z Z k = a, the distribution of 0, i=0 Z i yields Z Cov f I i, i I, f Z j J, j J 0. Z k Z k Therefore, the rando vector Z 0 Z,,..., Z i Z i i=0 i=0 is NA. However, NA. Z 0 i=0 Z i, Z Z i i=0 Z,..., i=0 Z i Z,..., i=0 Z i Z is independent of a, so the above inequality i=0 Z i Z and L i=0 Z 0,,..., L, have the sae distribution, so L 0,,..., L, is also i Lea B.2. If F X = F Y = U[0, ], then the rando vector N 0,,..., N, is NA. Proof. It is easy to show that given L j, = l j,, j =,...,, the rando vector N 0,,..., N, is ultinoially distributed, hence it is NA Proposition.2. Fro the definition of negative association, we know that for any disjoint subset I, J of {0,..., } and increasing functions f I, f J, the following inequality holds: [ E f I N i,, i I f J N j,, j J L k,, k = 0,..., ] E [ f I N i,, i I L k,, k = 0,..., ] E [ f J N j,, j J L k,, k = 0,..., ]. Note that given L k, = l k,, k = 0,...,, the joint distribution of {N i,, i I} only depends on L i,, i I, thus E [ f I N i,, i I L k,, k = 0,..., ] = E [ f I N i,, i I L i,, i I ] ; siilarly, E [ f J N j,, j J L k,, k = 0,..., ] = E [ f J N j,, j J L j,, j J ]. Therefore, E [ f I N i,, i I f J N j,, j J L k,, k = 0,..., ] = E [ f I N i,, i I L i,, i I ] E [ f J N j,, j J L j,, j J ].

11 P. Xiang, J.C. Wieran / Statistics and Probability Letters Fig. B.. A coupling arguent with respect to Inequality B.. Taking expectation on both sides of the above inequality yields E [ E [ f I N i,, i I f J N j,, j J L k,, k = 0,..., ]] E [ E [ f I N i,, i I L i,, i I ] E [ f J N j,, j J L j,, j J ]]. Since Lea B. showed that L 0,,..., L, is NA, and E [ f I N i,, i I L i,, i I ] and E [ f J N j,, j J L j,, j J ] are actually increasing functions of {L i,, i I} and {L j,, j J}, respectively see Reark B, applying the definition of NA rando vectors yields E [ E [ f I N i,, i I L i,, i I ] E [ f J N j,, j J L j,, j J ]] E [ E [ f I N i,, i I L i,, i I ]] E [ E [ f J N j,, j J L j,, j J ]]. Connecting the two inequalities above produces E [ E [ f I N i,, i I f J N j,, j J L k,, k = 0,..., ]] thus E [ E [ f I N i,, i I L i,, i I ]] E [ E [ f J N j,, j J L j,, j J ]], E [ f I N i,, i I f J N j,, j J ] E [ f I N i,, i I ] E [ f J N j,, j J ]. Reark B. It suffices to show that for any subset I = {i,..., i I } of {0,..., }, if l it, < l i t,, t I, then E[f I N i,, i I L i, = l i,, i I] E[f I N i,, i I L i, = l i, for i I {i t }, L it, = l i t, ]. B. As illustrated in Fig. B., suppose n X-points are independently uniforly distributed in [0, ], and denote N i,, i I as the nuber of X-points falling in the interval with length L i, = l i,. If the length { l it, of the ost right interval increases to l i t,, then N i t, will not decrease possibly increase. This eans that when L i, = l i, for i I {i t }, L it, = l i t,}, the rando variable N j, is stochastically larger than the original one when L i, = l i,, i I. By considering the fact that f I is an increasing function of N i,, i I, it follows that Inequality B. indeed holds. References Cannon, A., Cowen, L., Approxiation algoriths for the class cover proble. In: 6th International Syposiu on Artificial Intelligence and Matheatics. Ceyhan, E., Priebe, C.E., The use of doination nuber of a rando proxiity catch digraph for testing spatial patterns of segregation and association. Statist. Probab. Lett. 73, Ceyhan, E., Priebe, C.E., On the distribution of the doination nuber of a new faily of paraetrized rando digraphs. Model Assisted Statist. Appl., Ceyhan, E., Priebe, C.E., Marchette, D.J., A new faily of graphs for testing spatial segregation. Canadian J. Stat. 35, Ceyhan, E., Priebe, C.E., Wieran, J.C., Relative density of the rando r-factor proxiity catch digraph for testing spatial patterns of segregation and association. Cop. Stat. Data Anal. 50, DeVinney, J.G., Priebe, C.E., A new faily of proxiity graphs: Class cover catch digraphs. Discrete Appl. Math. 54, DeVinney, J., Wieran, J.C., A SLLN for a one-diensional class cover proble. Statist. Probab. Lett. 59, Eveland, C.K., Socolinsky, D.A., Priebe, C.E., Marchette, D.J., A hierarchical ethodology for one-class probles with skewed priors. J. Classification 22, Joag-Dev, K., Proschan, F., 983. Negative association of rando variables, with applications. Ann. Stat., Marchette, D.J., Priebe, C.E., Characterizing the scale diension of a high-diensional classification proble. Patt. Recog. 36, Newan, C.M., 984. Asyptotic independence and liit theores for positively and negatively dependent rando variables. In: Inequalities in Statistics and Probability. In: IMS Lecture Notes, vol. 5. pp Priebe, C.E., DeVinney, J., Marchette, D.J., 200. On the distribution of the doination nuber for rando class cover catch digraphs. Statist. Probab. Lett. 55, Priebe, C.E., Marchette, D.J., DeVinney, J., Socolinsky, D., Classification using class cover catch digraphs. J. Classification 20, Taylor, R.L., Patterson, R.F., Bozorgnia, A., A strong law of large nubers for arrays of row-wise negatively dependent rando variables. Stochastic Anal. Appl. 20, Wieran, J.C., Xiang, P., A general SLLN for the one-diensional class cover proble. Statist. Probab. Lett. 78, 0 8. Xiang, P., Liit theory for the doination nuber of rando class cover catch digraphs. Ph.D. Dissertation. Johns Hopkins University, Departent of Applied Matheatics and Statistics.