A Competitive Test for Uniformity of Monotone Distributions

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1 A Compeve Tes for Unformy of Monoone Dsrbuons Jayadev Acharya Ashkan Jafarpour Alon Orlsky Ananda Theerha Suresh Unversy of Calforna San Dego Absrac We propose a es ha akes random samples drawn from a monoone dsrbuon and decdes wheher or no he dsrbuon s unform The es s nearly opmal n ha uses a mos On log n samples, where n s he number of samples ha a gene who knew all bu one b abou he underlyng dsrbuon would need for he same ask Conversely, we show ha any such es would requre Ωn log n samples for some dsrbuons INTRODUCTION Many a lvely debae rages over wheher some quany grows wh a possbly relaed parameer, or s ndependen of : cancer prevalence vs radaon exposure; dabees onse rae vs age; flu frequency vs negave emperaure; hear aacks vs sress; recovery speed vs drug dosage; produc falures vs me from manufacure; axes or educaon level or negave ncarceraon rae vs ncome; gender vs salary; pary afflaon vs ncome or age; grades vs preparaon me; qualy vs cos; and fnally, age before beauy We propose a es ha akes random samples generaed by a monoone dsrbuon and decdes wheher or no he dsrbuon s unform The es s near opmal n he followng sense A gene who knew he underlyng dsrbuon, could clearly decde f s unform or no whou any samples We show ha f a gene who knew all bu one specfc b abou he underlyng dsrbuon would requre n samples o decde on unformy, hen our es wll requre a mos On log n samples o make he same decson We also show ha Ωn log n samples are necessary In general, he parameer of neres can be dscree Appearng n Proceedngs of he 6 h Inernaonal Conference on Arfcal Inellgence and Sascs AISTATS 03, Scosdale, AZ, USA Volume 3 of JMLR: W&CP 3 Copyrgh 03 by he auhors ncome or connuous emperaure and can assume dfferen ranges However, every monoone dsrbuon over a fne connuous nerval, and every monoone dscree dsrbuon over a fne doman, can be convered o an equvalen monoone dsrbuon over [0, For example, he monoone dsrbuon p 06, p 03, p7 0, can be convered o 06 [0,/3 x+03 [/3,/3 x+0 [/3, x Hence we consder probably densy funcons, or dsrbuons for shor, over [0, Monoone dsrbuons over nfne ranges can be smlarly analyzed and wll be addressed n he full verson of hs paper Over [0,, he dsrbuon ux s unform A dsrbuon f s monoone f for all x < y < z, fy fx fz fy 0 Noe ha monoone dsrbuons can be eher ncreasng or decreasng Le M be he collecon of monoone dsrbuons, and M M {u}, he collecon of monoone non-unform dsrbuons We would lke o decde wheher a dsrbuon f M s unform or no based on ndependen samples generaes A es s a mappng : [0, ] {un, non} declarng wheher he observed samples are beleved o be generaed by a unform or a non-unform dsrbuon The error probably of for f M based on n samples generaes s P ef, n { P X n non, f f u, P X n un, f f M, where X n X,, X n are samples dsrbued ndependenly f As n several recen works, we are mosly neresed n he sample complexy, he number of samples requred o acheve a ceran error For a dsrbuon f M, es, and error ɛ > 0, le N ɛ f mn {n : P ef, n < ɛ and P eu, n < ɛ} be he smalles number of samples for whch has error < ɛ for boh f and he unform dsrbuon We requre s error o be small for boh f and u as a rval es acheves 0 error for jus one of hem

2 A Compeve Tes for Unformy of Monoone Dsrbuons The smalles number of samples requred by any es o acheve error probably < ɛ for f M and u s herefore, Nɛ f mn {Nɛ f} The es achevng he mnmum s denoed Example For > 0, le { + f x [0, fx, f x [, Snce boh f and u are consan over [0, and [,, all he nformaon X n conveys abou f and u s conaned n M { : X < }, he number of samples n [0, If he underlyng dsrbuon s u, hen M Bn, n, wh expeced value n and sandard devaon n If he underlyng dsrbuon s f, hen M Bn +, n, wh expeced value + n and sandard devaon n Consder he es ha decdes un f M < + n and non oherwse The es errs only f M srays from s mean by > n, namely by roughly n sandard devaons If n > C, hen he error probably wll be small, showng ha for any fxed ɛ, Nɛ f Nɛ f O For he lower bound, observe ha f n, hen he dfference beween he means of M when sampled under f and u wll be sgnfcanly smaller han a sandard devaon, and no es wll dsngush he wo alernaves Hence Nɛ f Ω, and herefore Nɛ f Θ Ths argumen s of course jus nuve A rgorous proof s based on Equaon and compues he l dsance beween he wo bnomal dsrbuons Clearly, he closer f ges o u, he larger N f becomes Prevous approaches o hs problem have herefore consdered he sub-collecon M M of all monoone dsrbuons whose l dsance from u s a leas Daskalaks e al, 0 showed ha a es smlar o n he las example sasfes Nɛ f O for all f M, and ha for some f M, Nɛ f Ω Therefore he hghes sample complexy of any dsrbuon n M s Θ However, he nex example shows ha some M dsrbuons have a far lower complexy Example Le fx x + where x s Drac Dela funcon Clearly l f, u Consder he es ha declares non f a sample a x 0 s observed, and un oherwse For he unform dsrbuon u, he es errs wh probably 0, and o ensure ha for f errs wh probably /3, we need 3 n Namely n log 3/ log log 3 samples suffce I can be easly shown ha roughly hs number s also necessary, hence N ɛ f Θ The wo examples show ha he l dsance of f from u s no an accurae measure for he number of samples needed o es s unformy, and n parcular ha some dsrbuons n M can be esed for unformy usng much less han Θ samples They also show ha achevng N ɛ f may requre advance knowledge of f as he es may depend on he underlyng dsrbuon In hs paper we derve a es ha performs almos as well as possble, no jus for he wors dsrbuon n a subclass of M, bu for every dsrbuon n M Is performance for every underlyng dsrbuon approaches ha of he bes es desgned wh knowledge of all bu one albe carefully chosen b abou he underlyng dsrbuon Specfcally, f a gene knew he underlyng monoone dsrbuon, could deermne wheher he dsrbuon s unform or no whou any samples a all We show ha f he underlyng dsrbuon s f, and a very knowledgeable gene, who knew ha he underlyng dsrbuon was eher f or u bu dd no know whch, n a sense mssng jus a sngle b abou he dsrbuon, hen f ha gene needed n samples o deermne wheher he dsrbuon was unform or no, our es, desgned of course whou knowledge of f, would requre a mos On log n samples Before formally sang he resuls, noe ha for many daa-based decson problems, ncludng hs one, once an error probably ɛ < / s acheved, one can repea he es several mes and ake he majory of all decsons, resulng n an error dmnshng exponenally fas n he number of repeons Specfcally, f we repea he es on T ndependen daa ses, can be easly shown ha he error probably of he combned es s a mos ɛ ɛt/ For smplcy herefore we consder he number of samples needed o ge error probably < /3 Any oher desred error can be acheved by repeang he expermen a consan number of mes For f M and es, le and N f N /3 f, N f N /3 f, be he number of samples ha, and he bes es, need o have error probably < /3 We desgn a sngle es c ha s nearly opmal regardless of he underlyng dsrbuon Namely, for

3 Jayadev Acharya, Ashkan Jafarpour, Alon Orlsky, Ananda Theerha Suresh every f M N c f O N f log N f We also show ha hs exra facor s necessary More precsely, for every es, here s a f M such ha N f Ω N f log N f Two observaons are n order Frs, all unformy ess make some nheren assumpon abou he underlyng dsrbuon For example, he es Daskalaks e al, 0 for dsrbuons n M assumes ha he esed dsrbuon s eher unform or n M If he dsrbuon s no n hese wo classes, namely 0 < l f, u <, hen he es s performance s no guaraneed Smlarly, for our problem, f n on f log N f, hen here s no guaranee on he performance of c In fac, we show ha n ha case, for suffcenly large n, c oupus unwh probably a leas 3 Second, wh he saed number of samples, c has error /3 If a smaller error ɛ s desred, hen as above, he number of samples can be mulpled by 7 ln ɛ Noe however ha n ha case, we would be comparng /3 error for he gene wh ɛ error for c If we requred he gene o acheve ɛ error oo, he dependence on ɛ n he consan wll decrease and perhaps vansh RELATED WORKS The sascs leraure offers varous unformy ess, see Woodroofe and Sun, 999 and references heren The compuer scence communy has consdered he problem more recenly, ypcally addressng dscree dsrbuons Whou loss of generaly, assume he underlyng dsrbuon s over [k] {,, k} Pannsk, 008 showed ha esng f he dsrbuon s unform or ɛ far away from unform n l dsance, requres Θ k ɛ samples Bau e al, 00 showed ha esng f he dsrbuon s monoone or far from all monoone dsrbuons n l dsance requres Õ k samples, where he mpled consan s an nverse polynomal n he l dsance They also showed ha esng f wo monoone dsrbuons are close n l dsance requres Opolylogk samples Daskalaks e al, 0 exended hese resuls o m-modal dsrbuons In parcular, hey showed ha esng f a monoone dsrbuon equals a prespecfed dsrbuon requres O log k log log kɛ 5 samples and he dependence on k s opmal up o a Olog log k facor Alhough no drecly relaed o hs paper, muldmensonal dsrbuons were consdered as well Rubnfeld and Servedo, 009 showed ha esng wheher a monoone dsrbuon over he n dmensonal boolean hypercube {, } n s unform or far from unform n l dsance, requres Θn samples, where Θ represens Θ wh possble poly-logarhmc facors Smlarly, Adamaszek e al, 00 showed ha esng unformy of monoone dsrbuons over he connuous [0, ] n hypercube requres Θn/ɛ samples All he above resuls consder he sample complexy of he wors dsrbuon n a class A dfferen, compeve, approach ha compares he performance for each dsrbuon o he bes possble, was consdered as well Acharya e al, 0 and Acharya e al, 0 consdered closeness esng, where we would lke o deermne wheher wo sequences are drawn from he same or from dfferen dsrbuons, and classfcaon where we are gven wo ranng sequences generaed by unknown dsrbuons and would lke o deermne whch of he wo dsrbuons generaed a new es sequence For boh problems, hey proposed ess ha requre Õn3/ samples, where n s he number of samples requred by he opmal es, desgned wh knowledge of he underlyng dsrbuon mulse 3 PRELIMINARIES 3 Posson samplng Consder any paron of he range [0, no dsjon nervals When a dsrbuon on [0, s sampled exacly n mes, he number of mes elemens appearng n he nervals are dependen for example, hey sum o exacly n, complcang he analyss of many properes A sandard approach o overcome he dependence, eg, Mzenmacher and Upfal, 005 s o sample he dsrbuon a random pon mes, he Posson dsrbuon wh parameer n, resulng n sequences of random lengh close o n Posson al bounds show ha for any α > 0, wh hgh probably a random varable pon + α s larger han n Thus, any es wh error probably 3 ɛ for n samples, can be modfed o work wh error probably < 3 for pon + α samples Snce α can be any posve consan, he es works wh a fraconally larger Posson parameer For a dsrbuon f, he probably ha pon samples generaed accordng o f wll resul n samples x

4 A Compeve Tes for Unformy of Monoone Dsrbuons s fx e n n x x! In parcular, for he unform dsrbuon u, ux e n n x x! We wll also use an equvalen formulaon of Posson samplng Le h be a non-negave funcon over [0,, and le H hx denoe s negral Then a dsrbuon over [0, s ponh accordng o h 0 ff The number of samples n any wo dsjon subses are ndependen For any A [0,, he number of samples n A s pon x A hxdx 3 Smulaon Anoher useful propery of Posson samplng s ha n some cases one can use samples generaed by one dsrbuon o smulae samples generaed by anoher, whou necessarly knowng he underlyng dsrbuon Le h and f be any non-negave funcons over [0, such ha h s furher from han f s, namely for all x, hx /fx Le Y be pon samples ha are generaed accordng o eher h or u We show ha whou he knowledge of he underlyng dsrbuon, one can conver Y o X such ha f he underlyng dsrbuon s h hen X f and f Y u hen X u Lemma 3 There exss an algorhm wh npu Y, n and oupu X such ha If Y h, hen X f, If Y u, hen X u Proof Generae Z u of lengh pon Consruc he se of X as follows Add any sample Y o he nally empy se X wp fx /hx and add any sample Z o he se wp hx fx /hx One can show ha for any bn A, and he lemma wll follow from f Y h, hen E[ X A ] n x A fx, f Y u, hen E[ X A ] n A 33 Error bounds One can consder wo ype of error probables for any es The average error, whch s he error of he es when he samples are eher from f or u wh equal probably, P ef, n P ef, n + P eu, n The wors-case error of whch s he larger of he errors for f and u, ˆP ef, n maxp ef, n, P eu, n As wh all hypohess esng problems, Clearly, and P ef, n mn Pef, n + Peu, n mn f u,n ˆP ef, n P ef, n f u,n, ˆP e f, n mn P e f, n x mnfx, ux f u,n Where he l dsance beween f and u s f u,n x UPPER BOUND Tes fx ux We propose a es ha dsngushes every f from u wh Posson n ON f log N f samples wh error probably 3, whou he knowledge of f For brevy, le n N f Le k log n + 5 Paron he nerval [0, no k bns I,, I k as follows, I [0, k, [ k, k,, k, [ k, k k+,, k, [ k, k

5 Jayadev Acharya, Ashkan Jafarpour, Alon Orlsky, Ananda Theerha Suresh 0 Fgure : Paron of [0, no I,, I k Noe ha he bn szes ncrease exponenally from I o I k and decrease exponenally from I k+ o I k Ths paron s shown n Fgure The es c, descrbed below, performs a varaon of he χ -es on he number of samples appearng n each bn, f eher one of hem, or her sum, s large, he es declares f o be non Noe ha snce he varaon of he χ -es we use can be negave, we need o check boh he ndvdual values and her sum Tes c Inpu: Sequence x of lengh pon 000n k Oupu: un or non for k, le ν {x : x I } and n I f s reurn non else reurn un ν ν 5 k or ν ν 5 k In he remander of hs secon, we prove he compeveness of c Bounds on sample complexy Suppose a monoone dsrbuon f s dsngushable from u usng n samples In hs subsecon we show ha wh a consan facor more samples we can dsngush f from u by only usng he number of samples whn each bn Also, we show ha f f can be dsngushed from u, hen a varaon of χ dsance on he number of samples n he bns s large Conversely, we show ha f f canno be dsngushed from u wh n samples, hen χ dsance s small Defne a sarcase dsrbuon g, shown n Fgure, ha assgns he same probably as f o each bn Namely, for x I gx I y I fy The followng heorem uses Lemma 3 o relae he error probably of f o ha of g Theorem N g N f Proof Whou loss of generaly, suppose ha f s non-ncreasng We show ha from samples of g or u, we can generae samples ha are dsrbued accordng 0 0 Fgure : g and f o f or u Then we run he same opmal es for f on he nduced samples of g Le Y be pon samples from g Paron Y no wo sequences Y and Y, where each sample n Y appears n eher Y or wh equal probably The se of elemens n Y and Y are wo ndependen copes of Posson n samples from g and her unon s Y Snce g s no furher away from han f, Lemma 3 canno be appled drecly Insead, observe ha he average of f and hence g n I s hgher han all he values of f n I + and s less han all he values of f n I Also observe ha I, I + I and we are akng wce he number of samples from he dsrbuon g By scalng and ransformng and usng he followng seps we generae samples X whn each bn For any 3, f x I, fx hen samples of X belong o I are generaed from samples of Y whn I Smlarly f x I, fx, hen we use samples of Y from bn I o generae samples n I For any k, f x I, fx hen samples of X belong o I are generaed from samples of Y whn I + Smlarly f x I k, fx, hen we use samples of Y from bn I k o generae samples n I k 3 If fx changes sgn nsde I where {,, k } for he poron of he I where fx use em or and for he poron where fx use em 3 or o generae he samples of X whn I The above procedure s llusraed n Fgure 3 for k An arrow from I o I j ndcaes ha he samples of X n bn I j are generaed from samples of Y whn bn I The label of he arrow ndcaes he sep used Smlar o he proof echnque n Lemma 3 can be shown ha he samples X are dsrbued accordng o f or u n bns I,, I k and has no samples n I I k

6 A Compeve Tes for Unformy of Monoone Dsrbuons 3 3 I I I 3 I I 5 I 6 I 7 I 8 Fgure 3: Generaon scheme Le be a es such ha Pe f, n< 3 and Pe u, n< 3 The followng es, has error probably less han 0 If Y has samples n I or I k hen Y non Run he es on samples X When he underlyng dsrbuon s u, afer akng Posson n samples, wp e 6 no sample appears n I I k Snce he same es s used, he oal error n hs case s < 3 Addng he probably of havng sample n he I I k we have Peu, n < 3 + e 6 < 0 When he underlyng dsrbuon s nonunform, f a sample appears n I I k, hen he error s zero whle n he oher case runs he es and has error probably less han 3 So far we have shown ha N 0g n Hence, from Equaon, 0 g u,n 3 The followng observaon helps us evaluae he l dsance beween g and u Observaon 5 Snce g and u are consan whn each bn, her number of samples whn each bn s a suffcen sasc o dsngush hem Le ν denoe he number of samples n I Le E[ν ] when he samples are dsrbued accordng o u, and λ E[ν ] when he samples are dsrbued accordng o f Noe ha and λ are funcons of number of samples and for he ease of noaon, parameer n s omed When he underlyng dsrbuon s u, ν s Posson dsrbued wh mean a and when he underlyng dsrbuon s f or g, ν s Posson dsrbued wh mean a λ Le f be he average of f n I, and ν ν,, ν k Le uν be he dsrbuon of ν when X u and fν be he dsrbuon of ν when X f Then, uν e λ ν ν! and fν e λ λ ν ν! Le E h denoe he expecaon wh respec o h The followng lemma, s proof omed, bounds he l dsance beween wo dsrbuons Lemma 6 For any wo dsrbuons h and h, [ ] l h, h h E h h The nex lemma lower bounds a varan of he χ dsance beween f and u Lemma 7 For n N f, wh pon samples, λ > 0 Proof Usng Observaon 5 and Equaon 3, fν uν 0 Subsung fν and uν and usng Lemma 6, 6 k e λ λ ν e λ λ λ ν ν ν! λ exp, where he las nequaly follows from generang funcon of Posson random varables Hence, λ ln 6 > 0 Nex we upper bound he χ dsance beween wo dsrbuons Lemma 8 For n N f, wh pon samples, λ + λ ln 3 Proof Snce samples from f can be used o generae samples from g, he error of f s a mos ha of g, hence N g N f From Equaon, Hence, 3 ν ν exp exp mnuν, fν uνfν λ + λ λ λ + λ ln 3

7 Jayadev Acharya, Ashkan Jafarpour, Alon Orlsky, Ananda Theerha Suresh 3 Proof of compeveness Theorem 9 If n > 000n log n, hen Pe c u, n c and Pe f, n 3 3 Proof Le ν be dsrbued accordng o f, and One can show ha E[α] var[α] α ν ν, α α λ n n, λ + λ λ + 3λ + Unform case: If ν s dsrbued accordng o u, hen E[α] 0 and var[α] k Therefore, by Chebyshev s Inequaly, Pe c u, n 3 Non-unform case: Observe ha all of he s are larger han 000 k/6 Hence by Chebyshev s Inequaly, f s λ / 50 k, hen ν ν 5 k wp a leas /3 Oherwse can be shown ha, var[α] 3 0 E[α] +8k and by Chebyshev s Inequaly he error probably s < 3 Nex we show ha f he number of samples s small, hen c errs on samples of f wh probably > 3 Theorem 0 For suffcenly large n, f n on log n hen P c e f, n 3 Proof Skech By Lemma 8, he χ dsance beween f and u s o log n Suppose he underlyng dsrbuon s f The es c compares a varan of χ quany, α, o he hreshold Θ log n Smlar o he proof of Theorem 9, can be shown ha α concenraes around s mean, whch s smaller han he hreshold Hence, he es c declares un wh probably a leas 3 5 LOWER BOUND Smlar o he prevous secon, we ne bns whose szes ncrease exponenally and consruc a famly C n of monoone decreasng dsrbuons ha are fla whn each bn We show ha for any f C n, N f n, whle no sngle es can dsngush u from all f C n wh on log n samples Le m n log, for smplcy assume m s even, and k m Defne, I { [ 3n, [ k 3n 3n,, k, k + Then, I n for,, k Le Ix be he ndex of he bn conanng x Le S {,, m } {m +,, 3m } {k m +,, k m } Then, S m m For any j j,, j m S, { + m jr x / I k+ f j x m r r:j r Ix m jr jr 3n k x I k+ The dsrbuon f j s nduced from u, by removng some probably mass from I k+ and spreadng across I, I jr for each j r The amoun of probably mass shfed s proporonal o he sandard devaon of he number of samples appearng n I jr Ths ensures ha f j s monoone and has exacly m+ jumps Le {f j : j S} Thus, C n S m m Frs, we show ha N f j n C n Lemma f j C n, N f j n Proof For ν s and λ s ned n he prevous secon, le m β ν j λ j λj We show ha β concenraes around dfferen values for u and f j, and use ha o prove ha hey can be dsngushed wh n samples When u s sampled pon mes, E[β] 0 var[β] whereas f f j s sampled, E[β] m m λ j k λj k m n I j n I j m, m j k m m m m m var[β] + < 6m, m j k

8 A Compeve Tes for Unformy of Monoone Dsrbuons where he las nequaly holds for m Chebyshev s Inequaly, u : f j : hence, j S, N f j n P β 3m m 3m 3, P β 6m 3m 3 m 3, Usng The followng lemma helps o lower bound he number of samples necessary o dsngush u from all dsrbuons n C n Lemma For posve x,, x n, n e x n + exp n x The nex heorem lower bounds he number of samples needed o dsngush all dsrbuons n C n from u Theorem 3 For any es, f j C n such ha, N f j Ω N f j log N f j Proof We fnd a lower bound on n such ha pon samples are necessary o dsngush all f j C n from u wh error 3 Consder he average funcon and le f ave x m m f j x, λ rj λ r + λ r l:j l r j S m j l Nex we relae he maxmum error of all dsrbuons conaned n C n, o he error of her mxure 3 mn max maxp ef, n, Peu, n f C n mn maxp ef ave, n, P eu, n f ave u,n, where he las nequaly follows from Equaon By Lemma 6, 9 f ave u,n E fave [ ] fave x ux Afer movng he consans o lef hand sde we have, [ ] 3 9 E fave x f ave ux a k+ e λ ν k+ νr λrj S e λr λ rj ν ν! λ j S r r b k+ e λ ν k+ λrj λ νr rk S ν j S k S ν! λ r r c 6n m m S exp j k mn j S k S d 7n m S exp j k mn j S k S e m m/ m m/ m 7n S exp mn j k jm m+ km m+ f m m/ 7n S exp mn j m m j m/ 7n exp S mn j m m j g 8n m m/ + exp S mn exp exp 8n mn, where a follows snce all f j s and u are consan whn each I and he number of samples whn each bn s a suffcen sasc for f ave x and ux, b follows from k+ r eλr λ rj, c by subsung λrj and λ rk, d snce s he domnan erm and ohers are exponenally decreasng wh rao a mos /, e from rewrng he sum of producs as produc of sums, f from replacng each erm by he maxmum value, whch occurs for k m + m, and g from Lemma Smplfyng, we have for any es, he number of samples necessary s Ωnm where m [ log n ] References J Acharya, H Das, A Jafarpour, A Orlsky, and S Pan Compeve closeness esng JMLR - Proceedngs Track, 9:7 68, 0 J Acharya, H Das, A Jafarpour, A Orlsky, S Pan, and A T Suresh Compeve classfcaon and closeness esng JMLR - Proceedngs Track, 3: 8, 0 M Adamaszek, A Czumaj, and C Sohler Tesng monoone connuous dsrbuons on hgh-dmensonal real cubes In SODA, pages 56 65, 00

9 Jayadev Acharya, Ashkan Jafarpour, Alon Orlsky, Ananda Theerha Suresh T Bau, R Kumar, and R Rubnfeld Sublnear algorhms for esng monoone and unmodal dsrbuons In Proceedngs of he hry-sxh annual ACM symposum on Theory of compung, STOC 0, pages , New York, NY, USA, 00 ACM ISBN C Daskalaks, I Dakonkolas, R A Servedo, G Valan, and P Valan Tesng $k$-modal dsrbuons: Opmal algorhms va reducons CoRR, abs/5659, 0 M Mzenmacher and E Upfal Probably and compung - randomzed algorhms and probablsc analyss Cambrdge Unversy Press, 005 ISBN L Pannsk A concdence-based es for unformy gven very sparsely sampled dscree daa IEEE Transacons on Informaon Theory, 50, 008 R Rubnfeld and R A Servedo Tesng monoone hghdmensonal dsrbuons Random Sruc Algorhms, 3:, January 009 M Woodroofe and J Sun Tesng unformy versus a monoone densy The Annals of Sascs, 7:pp , 999