ASYMPTOTIC EQUIVALENCE OF NONPARAMETRIC REGRESSION AND WHITE NOISE. BY LAWRENCE D. BROWN 1 AND MARK G. LOW 2 University of Pennsylvania

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1 The Aals of Sascs 996, Vol., No. 6, ASYMPTOTIC EQUIVALENCE OF NONPARAMETRIC REGRESSION AND WITE NOISE BY LAWRENCE D. BROWN AND MARK G. LOW Uversy of Pesylvaa The prcpal resul s ha, uder codos, o ay oparamerc regresso problem here correspods a asympocally equvale sequece of whe ose wh drf problems, ad coversely. Ths asympoc equvalece s a global ad uform sese. Ay ormalzed rsk fuco aaable oe problem s asympocally aaable he oher, wh he dfferece ormalzed rsks covergg o zero uformly over he ere parameer space. The resuls are cosrucve. A recpe s provded for producg hese asympocally equvale procedures. Some mplcaos ad geeralzaos of he prcpal resul are also dscussed. Iroduco. The prcpal resul of hs paper s ha o ay oparamerc regresso problem here correspods a whe ose wh drf problem whch s asympocally equvale. The mpac of hs asympoc equvalece s ha ay asympoc soluo o oe of hese problems wll auomacally yeld a correspodg soluo o he oher. I addo, here s a explc recpe for hs correspodece. For example, he opmal raes of covergece wll be equal as wll suably ormalzed local ad global asympoc rsks, ad kowledge of a mmax procedure or of a lear mmax procedure oe problem auomacally yelds he correspodg procedure he oher ad so forh. I parcular, may classcal fucoal esmao problems whch have prevously bee reaed separaely fall o hs framework. These problems clude:. Esmag he whole fuco, cosdered he whe-ose model by Psker 980. ad he regresso coex by Nussbaum 985. ad Speckma See also Dooho, Lu ad MacGbbo 990. ad Golubev ad Nussbaum Esmag a po fucoal such as f x. 0. For whe ose, see Ibragmov ad asmsk 98. ad for regresso, see Ibragmov ad asmsk 98.. See also Dooho ad Lu 99., pages 677ff ad 688 ad Dooho ad Low 99.. Receved Jauary 99; revsed Aprl 993. Suppored par by NSF Gras DMS-9-078, 93-08, Suppored par by NSF Gra DMS AMS 99 subjec classfcaos. Prmary 6G07; secodary 6G0, 6M05. Key words ad phrases. Rsk equvalece, local asympoc mmaxy, lear esmaors. 38

2 NONPARAMETRIC REGRESSION AND WITE NOISE Esmag a olear fucoal such as f x. dx, reaed for whe ose Fa 99b. ad earler refereces ced here ad for boh whe ose ad regresso Dooho ad Nussbaum Esmag he whole fuco based o drec observaos as Fa 99a.. The equvalece heory also covers may adapve suaos. If here s a parcular sequece of esmaors whch s asympocally mmax over a colleco of parameer spaces he whe-ose case, he here s a correspodg sequece based o he regresso model whch s also mmax over each of hese parameer spaces. Such a sequece of adapve esmaors for esmag he whole fuco has bee foud by Efromovch ad Psker 98. he whe-ose case ad by Golubev 987. for oparamerc regresso. See also Golubev 99.. For he problem of esmag a lear fucoal boh regresso ad whe ose, see Lepsk 99.. all ad Johsoe 99. dscuss examples of esmag opmal, possbly radom, badwdhs he regresso ad whe-ose coexs. These problems are also covered by he geeral equvalece heory developed hs paper. See Remark.5. The equvalece heory ca provde a addoal echcal advaage. Some proofs, for example hose volvg raes of covergece, may be much smpler he whe-ose model. Thus oe may use he whe-ose model o fgure ou he opmal rae va homogeey ad he he same rae holds oparamerc regresso. See, for example, Seco 7 of Dooho ad Low 99.. Aalogous equvalece resuls should be vald for some oher oparamerc problems. Ideed, Nussbaum 993. has very recely proved oe such resul for oparamerc desy esmao. The frs par of he paper coas ecessary backgroud. Ths cludes descrpos of he oparamerc regresso ad whe-ose problems, he defo of asympoc equvalece ad a dscusso of some of s geeral cosequeces. The secod par of he paper coas he ma equvalece heorems. Two cases are reaed separaely. I oe case, he depede varables are deermscally fxed; he oher, hey are a radom sample from a specfed dsrbuo. Par. Backgroud.. Noparamerc regresso. The oparamerc regresso model o be reaed hs paper s as follows: Le I be a possbly fe erval. Le f.: I ad.: I 0,. be wo measurable fucos ad le : 0, be a creasg c.d.f. The varables X, Y.,,...,, are observed. I he deermsc X vara he depede varables are gve by a deermsc scheme whch wll be gve by.. x.,,...,,

3 386 L. D. BROWN AND M. G. LOW excep where oherwse oed. I he radom X vara he X are depe de radom varables.. X..d.,...,. I eher case he codoal dsrbuo of Y gve X s descrbed va.3. Y f x. x., N 0,. d.,,...,. The parameer space cosss of a possbly large se of choces of f. The c.d.f. ad he fuco are assumed fxed ad kow pror o expermeao. ere are some examples whch are subjec o laer resuls of hs paper. EXAMPLE.. s gve erms of a Lpshz codo as...., B ½ Ý. f : f x f x f x! B ad sup f x. B f, 5 xi. f : f x. f x. B f 0., B. holds for all x, x I. Laer we shall requre, as explaed Remark.7. EXAMPLE.. s gve by a Sobolev ype codo such as. 5 xi.. f : f d B ad sup f x B, B ½ for,..., where f. deoes he h dervave of f, assumed o exs he sese ha f. s absoluely couous. Noe ha whe I d, he,. B., B.. Whe ose. Assume wh o loss of geeraly ha 0 I. Le B : I deoe Browa moo o I codoal o B0 0, for example, B N 0,. ad B, 0, s depede of B, 0. Fx,.... Le Z deoe he Gaussa process whose whe-ose verso s repre- seed as dz. d. db '. The parameer space hs model s, as before, he se of possble mea fucos. Cosequely, he sascal whe-ose problem for gve s o observe he process Z defed above for some. I hs way a sequece of problems s defed for,,..., each of whch has he same parameer space.

4 NONPARAMETRIC REGRESSION AND WITE NOISE 387 Suppose Y : J ad Z : J are wo Gaussa processes, lke hose above, wh respecve mea fucos. ad. ad wh decal varace fucos. Le g Y ad gz deoe her respecve probably deses wh respec o ay domag measure. Le. L g. g. d.. Y Z Where covee we also use eher he oao L Y, Z. or L g, g.. Sadard calculaos yeld Y Z. L D. where Cosequely, L O D. D.... d. REMARK.. Le Z be as above. Assume s absoluely couous wh dd h ad assume h 0 a.e.. o I. Defe he Gaussa process V Z, 0,. The V has mea. ad varace fuco. gve by ,.. h. h. ece here s o sgfca loss of geeraly assumg I 0, ad s uform o I, alhough o do so wll affec he defo of, he maer suggesed by Sascal equvalece. Cosder wo sascal problems, P. ad.. P, wh sample spaces X,, ad suable -felds., respecvely, bu wh he same parameer space. Deoe he respecve famles of dsrbu-. os by G :. The followg paragraph descrbes Le Cam s merc for he dsace bewee wo such expermes see, e.g., Le Cam 986. or Le Cam ad Yag Le A be ay measurable. aco space ad le L: A 0,. deoe a. loss fuco. Le L sup L, a :, a A. wll be he geerc symbol for a radomzed. decso procedure he h problem. The rsk from usg procedure. whe L s he loss fuco ad s he rue value.. of he parameer s deoed by R, L,.. Le Cam s merc s 3.. P., P max f sup sup sup R, L, R, L,,.. L: L.... f sup sup sup R, L, R, L,... L: L

5 388 L. D. BROWN AND M. G. LOW. Thus, f, hs meas ha for every procedure problem here s a procedure j. problem j, j, wh rsk dfferg by a mos, uformly over all L such ha L ad.. Two sequeces of.. problems P :,... ad P :,... are asympocally equvale.. f P, P. 0 as. I hs case, for ay sequece of procedures. problems P.,,..., here s a sequece. problems P. for whch.... lm sup sup R, L, R L,.. 0. L: L Such sequeces of procedures are sad o be asympocally equvale. I our proofs of sascal equvalece he key sep s o arrage maers so ha X. X.. The defe L P, P. sup g x. g x. dx., where domaes G. ad g. dg. d,,. The followg wellkow fac ca he be used o esablsh asympoc equvalece. TEOREM R, L,. R, L,. L P, P L. Cosequely, 3.. P., P.. L P., P... Also, PROOF. P., P.. 0 f L P., P s jus a resaeme of he sadard equaly... h x g x g x dx ž /... sup h x. g x. g x. dx.. Two oher echques are also used repeaedly: oe s reduco by suffcecy; he oher volves relaos bewee he L orm ad he ellger merc. The use of suffcecy s based o he followg. LEMMA 3.. Le X ad s -feld be a Polsh space wh s assocaed Borel feld. Le P. deoe a experme wh sample space X. Le S: X Y. be a suffce sasc ad le P deoe he experme whch Y S X... s observed. The P, P. 0. A Polsh space s oe whch s locally compac, merzable ad secod couable. All he measurable spaces ecouered hs paper possess hese properes.. PROOF. The lemma follows from he fac ha uder he hypohess here exss a measurable map dx y. such ha BS x.. G dx. G B.. See Le Cam 986..

6 NONPARAMETRIC REGRESSION AND WITE NOISE The ellger merc G, G s defed by I provdes a boud for L.... G, G g x g x dx. sce 3.5. L G., G.. G., G... Aoher useful fac s ha f he G. are produc measures, G. Ł m jgj., he.. j j m G, G G, G. Ł j Le Cam 986. Fally, drec calculao yelds N,, N,. exp. Par. Equvalece resuls.. Deermsc X. Le I,,. Le 0 be a gve absoluely couous fuco o I such ha.. l. C, I. for some C. Suppose.. sup f. : I, f B ad also.5, below. Assume s absoluely couous o I ad.3.. h. 0 a.e. o I. Wh x as.. defe he sep fuco.. f. ½ f x.,,,...,, f x.,, where.. The depedece of o s suppressed for cove- ece.. Assume..5. lm sup f. f. h. d 0. f REMARK.. Equao.5. s a uform smoohess codo o f. I s sasfed a wde varey of examples. I parcular s sasfed Examples. ad. whe. See Remark.7 for dscusso of he case..

7 390 L. D. BROWN AND M. G. LOW For example, o verfy.5. Example. for he case, I 0,, uform ad we oe ha f. f. B., where.. ece. Ý 0. B ž / 0 f f d B d sce. ece.5 s sasfed. B d 0 TEOREM.. Uder assumpos...5. he deermsc X oparamerc regresso model s asympocally equvale o he whe-ose model. havg. f.,.. h.. PROOF. Le Z be he whe-ose model descrbed by. dz f. d db. ' Noe ha sup.: If.: I because of.. ad.3.. ece.5.,. ad Theorem 3. show ha Defe. Z, Z 0. x. dz.6. K ad S K.. d.. The varables S :,..., are suffce for Z : I. ece. Z, S 0 by Lemma 3.. The varables S,,...,, are depede wh K d V S x.,. f E S. K d..7. h. ' x. f x. d. x. f x.,.

8 NONPARAMETRIC REGRESSION AND WITE NOISE 39 where. The exsece of follows from he mea value heorem sce h. d. Assumpo.. he yelds E S f x. O. uformly f, ad. Noe also ha he O erm s zero f, a cosa. The above, ogeher wh 3.6. ad 3.7., yelds.8.. S, x, Y ž Ý.. / O f x. uformly over f. B b a. O sup. :,..., o by. ad.3. ece lm S, x, Y. 0 by 3.5. ad Theorem. I follows from he precedg facs ha lm Z, x, Y. 0. TECNICAL NOTE. The codos...3. are clearly sroger ha ecessary. They are used order o yeld he cocluso o. u-. formly over f whch appears.8. For example, f, a cosa, I 0, ad h. o 0,, he.3. holds ad we may also drop codos.. ad... The above heorem yelds a prescrpo for producg, from a sequece of procedures oe problem, a asympocally equvale sequece he oher. The followg corollary gves a precse recpe. The corollary apples o eher oradomzed or radomzed procedures. COROLLARY.. Le be a sequece of procedures he regresso model of Theorem.. Defe he correspodg whe-ose problem by dz..9. Z. S. where S K as.6.. The s asympocally equvale o. Coversely, suppose s a gve sequece of procedures he whe-ose problem. The s a asympocally equvale sequece he asympo- cally equvale regresso problem, where s he radomzed procedure descrbed for ay measurable A A as follows: dz.. ž /.0 A s E A Z K d s,,...,. PROOF. The corollary s mplc he proof of Theorem 5.. Noe ha.0 s well defed sce s depede of he drf parameer f. of he Z process because for each, S s suffce for Z.

9 39 L. D. BROWN AND M. G. LOW The cosruco.0. s o erely felcous for wo reasos:. wll, geeral, be radomzed eve whe s o; he codoal expeca- o may be hard o evaluae. The followg wo remarks show ha hese dffcules ca somemes be parally allevaed. REMARK.. Jese s equaly ca somemes be used o mprove.0.. ere s a precse saeme. Suppose each A s a closed covex subse of a separable Baach space, ad suppose each L,. s a covex lower sem- couous fuco sasfyg lm a L, a.. Defe as he oradomzed procedure akg he values. s a da s. The s asympocally a leas as good as. Eve whe L s o covex may be ha.. sup R, L B,. R, L B,. 0. whch case ad are asympocally equvale sequeces for L. See Remark.3 for oe such suao. A real-valued. lear esmaor he whe-ose problem s a oradomzed esmaor whch ca be expressed he form.3. Z. dz, wh. d. I follows ha Z. has a ormal dsrbuo wh mea.. d ad varace.. d. A lear esmaor he regresso problem s, smlarly, oe whch ca be expressed he form. s s. Ý As above, such esmaors are ormally dsrbued. REMARK.3. The equvalece rasformao s especally covee for lear esmaors. I order o demosrae hs, s easer o wre.. a alerae form. As before, for coveece, specalze o he case I 0,, s uform, ad also assume. Le e. deoe he ormalzed learzed cumulave sums of s ; ha s, ½ Ý m m. 5 k e s s s,,,...,.

10 NONPARAMETRIC REGRESSION AND WITE NOISE 393 The he geeral lear esmaor. ca be rewre he form.5 s de. 0 Noe ha dz. e E Z K d s,,...,. ere, ad K. ece, suppose a lear esmaor.3. s gve he whe-ose problem. The he esmaor of.. s gve by.5. wh... The precedg resuls apply o regresso problems o a bouded erval. They ca be exeded whou much addoal complcao o also apply o ubouded ervals f a somewha dffere scheme for deermg he X values s roduced, as follows: Le I a, b. ad I,, a b ad le be a creasg c.d.f. o,. Choose he depede varables accord- g o.6. x.. The cosas whch also deped o. are defed as before, bu wh place of, ad f s defed by.. wh he addoal coveo f. 0,,. The assumpo.5 s he replaced by.7 lm sup h f f. d 0. f ere s a formal saeme of he equvalece resul. Is proof volves oly mor modfcaos of he proof of Theorem., ad wll be omed. COROLLARY.. Defe x as above. Assume.... are sasfed o, ad sasfes.3. Assume.7 s sasfed ad eher, a cosa, or max o. ere., 0 ad. As usual, he depedece of o s suppressed he oao. The he deermsc X oparamerc regresso model s asympocally equvale o he whe-ose model. havg. f. ad cremeal varace fuco depedg o ad gve by.. h.. REMARK.. There s a addoal queso whch arses he preced- g suao: Suppose a b ad wh h. Le Z be he whe ose wh drf f.. ad wh varace fuco. h. Is rue ha x, Y, Z. 0?

11 39 L. D. BROWN AND M. G. LOW A affrmave aswer o he precedg queso appears o requre some codo o he covergece of o. I some examples s o hard o supply such a codo. Cosder, for example, he suao Example. wh. Assume for smplcy of exposo ha. Make he oher assumpos Theorem. ad assume also ha.8. sup.. o '. as. The he aswer s affrmave. To esablsh he precedg clam, le x, Y deoe he deermsc X oparamerc regresso based o. The x, Y, x, Y. Ý f ž f B 0. ž // ž ž //. The apply Theorem. o see ha x, Y, Z 0. REMARK.5. The precedg mehodology ca be used whe he loss fucos also deped o he observaos. Thus, suppose he loss he. regresso problem s L : A x, y. 0, B ad he whe-ose. problem s L : A Z 0, B. For smplcy assume, a cosa. Defe T from Z he same maer as S was defed from Z see.6.. Assume he wo loss fucos asympocally agree he sese ha for each f, E L f, a, Z L f, a, x, T 0 f uformly a A. The proof of Theorem. ca he easly be adaped o prove ha correspodg procedures he wo problems have asympocally equal rsk fucos uder he respecve losses L. ad L.. The precedg observao ca be used o prove equvalece he problem of opmal badwdh seleco as formulaed all ad Johsoe 99.. The codo.9. s relavely sraghforward o check her coex. The resulg cocluso s ha her asympoc resul, oce proved he whe-ose seg, s he also vald he oparamerc regresso seg. REMARK.6. Lookg a Example. he case where shows wha ca happe whe he key regulary codo.5. fals. Le I 0,, h ad. I he whe-ose problem ' Z d N 0, 0

12 NONPARAMETRIC REGRESSION AND WITE NOISE 395 dsrbuo uformly over. I fac he lef sde s exacly N 0,.. owever, he regresso problem of Seco here cao exs a esmaor d. such ha.,. '..0 d x, Y f d N 0, 0 uformly over f,... By vrue of Remark 3., hs shows ha he wo problems are o asympocally equvale. To verfy he mpossbly of.0., le ', where. s defed by.. m j.: j,...,. The f. 0 ad f.. are. boh, ad x, Y.,. has he decal dsrbuo boh cases. ' owever,. d ' 3. ece valdy of.0. for f. 0 0 mples s falure for f... Alhough he wo sequeces of problems are o asympocally equvale he srog sese of Theorem., everheless may specal cases hey are asympocally equally useful. For example, hey wll have he same local asympoc mmax rsks for esmag f x. 0 uder squared error loss or for esmag f. uder egraed squared error loss. 5. Radom X. Resuls aalogous o hose of he precedg seco ca be derved for he radom X oparamerc regresso problem, as defed... I hs case he oparamerc regresso problem s asympocally equvale o a whe-ose model whch he drf depeds o he observed values of X, as well as he ukow parameers. To descrbe hs whe-ose model o I,,, le x.,..., x deoe he ordered values of X ad le x0., x. ad le 5.. x... x x... f x. x.,,...,. Now defe ad defe f by. wh place of. Le h. deoe he lef-had dervave of a. I place of.5. assume lm prob sup f. f. h.. d 0. f REMARK 5.. Assumpo 5.. s usually o much harder o verfy ha s.5.. For example, f I 0,, s he uform dsrbuo,. ad Example., he 5.. follows sce. ž../ f f h d f v f v dv

13 396 L. D. BROWN AND M. G. LOW ad ½ ž / 5 j. j.. E f v f v B E x x 0. uformly v,, where j m : x v. I he precedg suao defe.. h. ad. X, dz f. d db. ' X,. Noe ha he dsrbuo of Z depeds o he acllary observaos X hrough he local varace fuco. TEOREM 5.. Assume, f, sasfy...3. ad 5... The he radom X oparamerc regresso model s asympocally equvale o he whe-ose model PROOF. Aalogously o Theorem., le. X,. dz f. d db. ' The remader of he proof exacly follows he paer of proof of Theorem. wh Z place of Z ad f place of f. Also.8. eeds o be modfed slghly o beg as ž /. ž Ý / 5.. X, S, X, Y O E f x.. ad so forh, sce x ad are radom. A alerave proof could be based drecly o Corollary... REMARK 5.. A cospcuous feaure of he precedg resul s ha kowledge of s o requred o cosruc Z X,. or o carry hrough he proof of he heorem. Thus, suppose s oly assumed ha, where.. lm prob sup sup f f h d 0 f ad also ha eher or.3. s modfed o requre he exsece of a h sasfyg 0 f h. : h h. 0 a.e. o I. The he equvalece assero of Theorem 5. remas vald. Noe also ha he cosruco Corollary. of equvale procedures ogeher wh he coes of Remarks.. ca easly be carred over o he curre suao. 0

14 NONPARAMETRIC REGRESSION AND WITE NOISE 397 REMARK 5.3. Suppose s kow. A ssue whch he arses s wheher hs kowledge ca serve place of observao of he acllary sascs X,..., X. More precsely, le Xj Xj deoe he ordered values of X :,..., ad le Y,..., Y deoe he correspodg values of Y j j. The gve kowledge of, s he experme Y :,..., j asympo- cally equvale o X, Y.:,...,? Because of Theorem 5., hs s X,. equvale o askg wheher he experme Z : I s asympocally X,. equvale o, Z.: I. The geeral aswer o hs queso s No. Suppose I 0,,. for I ad f: f. c, c. The he experme Z X,. he X,. UMVU ad mmax esmaor s dz, whch has varace ž / 5.5 d Var d 3. Meawhle, for he experme, Z X,. he UMVU ad mmax esma- X,. or s dz, whch has varace. Ths shows he wo exper- mes are o asympocally equvale. O he oher had, for may purposes he wo expermes are equally useful asympocally. Roughly speakg hs happes whe he rae of covergece of he esmaor used s slower ha he classcal ' rae. eurscally, wha happes such problems s ha he mprecso ro- s represeed by a erm aalogous o he secod duced by o observg oe o he lef of 5.5., whch s of order. Whe he esmaor coverges ' a slower ha rae, so ha s varace coverges more slowly ha, hs secod erm s asympocally eglgble. Such behavor ca be deduced examples lke hose ced Dooho ad Lu 99. ad Low 99.. The auhors hak Luce Le Cam for helpful co- Ackowledgme. versaos. REFERENCE DONOO, D. L. ad LIU, R. C Geomerzg raes of covergece III. A. Sas DONOO, D. L., LIU, R. C. ad MACGIBBON, B Mmax rsk for hyperrecagles. A. Sas DONOO, D. L. ad LOW, M. C Reormalzao expoes ad opmal powse raes of covergece. A. Sas DONOO, D. L. ad NUSSBAUM, M Mmax quadrac esmao of a quadrac fucoal. J. Complexy EFROIMOVIC, S. Y. ad PINSKER, M. S A learg algorhm for oparamerc flerg. Auoma. Telemekh I Russa.. FAN, J. 99a.. O he opmal raes of covergece for oparamerc decovoluo problems. A. Sas FAN, J. 99b.. O he esmao of quadrac fucos. A. Sas GOLUBEV, G. K Adapve asympocally mmax esmaes of smooh sgals. Problemy Peredach Iformas I Russa.. GOLUBEV, G. K LAN oparamerc fucoal esmao problems ad lower bouds for quadrac rsk. Teorya Veroyaos. Prmee I Russa..

15 398 L. D. BROWN AND M. G. LOW GOLBEV, K. ad NUSSBAUM, M A rsk boud Sobolev class regresso. A. Sas ALL, P. ad JONSTONE, I. M Emprcal fucoals ad effce smoohg parameer seleco. J. Roy. Sas. Soc. Ser. B IBRAGIMOV, I. A. ad ASMINSKII, R. Z Bouds for he rsk of oparamerc regresso esmaes. Theory Probab. Appl Eglsh raslao.. IBRAGIMOV, I. A. ad ASMINSKII, R. Z O oparamerc esmao of he value of a lear fucoal Gaussa whe ose. Theory Probab. Appl Eglsh raslao.. LE CAM, L Asympoc Mehods Sascal Decso Theory. Sprger, New York. LE CAM, L. ad YANG, G. L Asympocs Sascs: Some Basc Coceps. Sprger, New York. LEPSKII, O. V Asympocally mmax adapve esmao I. Upper bouds. Opmally adapve esmaes. Theory Probab. Appl LOW, M. G Reormalzao ad whe ose approxmao for oparamerc fucoal esmao problems. A. Sas NUSSBAUM, M Sple smoohg regresso models ad asympoc effcecy L. A. Sas NUSSBAUM, M Asympoc equvalece of desy esmao ad whe ose. A. Sas PINSKER, M. S Opmal flerg of square egrable sgals Gaussa whe ose. Problems Iform. Trasmsso Eglsh raslao.. SPECKMAN, P Sple smoohg ad opmal raes of covergece oparamerc regresso models. A. Sas DEPARTMENT OF STATISTICS UNIVERSITY OF PENNSYLVANIA 360 LOCUST WALK PILADELPIA, PENNSYLVANIA 90 lbrow@compsa.wharo.upe.edu lowm@compsa.wharo.upe.edu