Statistica Sinica 6(1996), SOME PROBLEMS ON THE ESTIMATION OF UNIMODAL DENSITIES Peter J. Bickel and Jianqing Fan University of California and U

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1 Statistica Siica 6(996), SOME PROBLEMS ON THE ESTIMATION OF UNIMODAL DENSITIES Peter J. Bickel ad Jiaqig Fa Uiversity f Califria ad Uiversity f Nrth Carlia Abstract: I this paper, we study, i sme ew ways, the estimati f uimdal desities. Several methds fr estimatig uimdal desities are prpsed: plug-i MLE, pregrupig techiques, liear splie MLE. Based the maximum likelihd methd, a autmatic prcedure fr estimatig a uimdal desity as well as its mde is prpsed. We als give asympttic thery fr the prpsed estimatrs. A imprtat csequece f this study is that havig t estimate the lcati f a mde des t aect the limitig behavir f the prpsed uimdal desity estimate. Simulati studies illustrate the prpsed methds. Key wrds ad phrases: Asympttic distributis, MLE, mdes, plug-i methds, pregrupig methds, uimdal desities.. Itrducti Nparametric desity estimati prvides a useful techique f examiig the verall structure f a set f data. A cmmly used techique is the kerel methd. The behavir f akerel desity estimate relies strgly the chice f smthig parameter (badwidth). Data-drive badwidth selecti methds have bee studied recetly. Oe tries t miimize the Itegrated Square Errr (ISE) r the Mea ISE (MISE) r ther related bjects, ad uses e f them as a measure f glbal eectiveess f a curve estimate. I practical desity estimati, hwever, features such as shape ad area uder mdes may bemre iterestig. ISE ad MISE are t gd criteria fr these purpses. Fr example, the ISE f tw curves ca be very small, while the shapes f the tw curves are quite dieret. Whe shape ifrmati is available, a alterative apprach is t estimate a curve uder shape restrictis. I this paper, we fcus a umber f appraches t the estimati f a uimdal desity with a ukw mde lcati. We describe ur results ad the pit t sme f the histrical backgrud f ur apprach. T estimate a uimdal desity, we rst begi byitrducig a plug-i maximum likelihd methd. Let ^f (x m) be the parametric maximum likelihd estimate uder the restricti that the ukw desity is uimdal with the

2 24 PETER J. BICKEL AND JIANQING FAN mde lcati parameterized by m. Let ^m be a csistet estimate f the true lcati m 0 f the mde. The, the plug-i versi f the estimate is ^f (x ^m). We shw, i Secti 2, that fr all csistet estimates ^m, ^f (x ^m) cverges at the same rate ;=3 with the same asympttic distributi. The implicati f this is that estimatig a ukw desity with ukw lcati f mde is t appreciably mre dicult tha estimatig a ukw desity with a kw lcati f mde. This pheme was als bserved by Birge (987c), wh shwed that amg ther prperties, =3 k ^f ( ^m) ; ^f ( m 0 )k cverges t zer fr a particular chice f estimatr ^m. Hwever, the curret result hlds fr ay csistet mde estimatr althugh the result is lcal rather tha glbal. This cclusi gives mre supprt t Birge's ti that the MLE is rbust t mde estimati. We the prpse a autmatic methd fr estimatig the mde ad the desity based the maximum likelihd methd. A rate f cvergece fr the mde estimate is derived. The maximum likelihd estimate f the desity is shw t have the same asympttic prperties as the case where the mde is kw. The graph f ^f (x ^m) is quite spiky ear the lcati ^m. Oe way f reducig the spikiess prblem is t use a pregrupig techique. The idea is t grup the data it a umber f grups rst, ad the t perfrm a frm f MLE. We the prve that if the grupig is t t crude, the pregrupig versi f the MLE des as well as the plug-i MLE i terms f pitwise weak cvergece. This pregrupig techique als saves cmputig csts. Ather way f reducig the peakig prblem is the maximum pealized likelihd methd. (See Wdrfe ad Su (993)). The disctiuity f the plug-i MLE is usatisfactry. T deal with this prblem, we itrduce a maximum likelihd liear splie estimate. We give explicitly the frm f the estimate. The asympttic distributi f the estimate is derived whe the mde is assumed kw. Sice t kwig the lcati f the mde is t a serius matter i estimatig a uimdal desity whe the MLE is used, we expect but have t yet shw that such a estimate shuld als wrk well whe we d t kw the lcati f mde. A ice feature f such a estimate is that the lcati f the mde is determied autmatically by the data. Agai, the pregrupig techique ca be used t guard agaist spikig prblems ad t reduce cmputati. Varius related issues are discussed i Secti 3. N thery is available as yet, but we give sme heuristics belw. A early wrk estimatig a desity uder shape restrictis is Greader (956), wh estimated a decreasig desity by usig a maximum likelihd

3 UNIMODAL DENSITY ESTIMATION 25 apprach. The asympttic distributi f the MLE at a pit was fud by Prakasa Ra (969), ad Greebm (985). Recet develpmets i estimatig a mte desity ca be fud i Birge (987a,b), wh gives the behavir f parametric miimax risks. Wegma (969, 970a,b) prpsed ad studied the estimati f a uimdal desity by dig the MLE fr a mdal iterval f legth ". I particular, he fud the pitwise asympttic distributi f the MLE except fr the mdal iterval, which the MLE is t eve csistet. We give a mre atural MLE methd, ad derive the asympttic distributi fr all pits except the mde itself. Mamme (99a,b) made a iterestig study f the shape restricted curve estimati i the ctext f the parametric regressi setup. Varius applicatis f the istic methd ca be fud i Barlw ad va Zwet (970), Barlw elal: (972), Rberts et al: (988), Wag (986), Ramsay (988), amg thers. 2. Prblems ad Mai Results Let f(x m) be a uimdal desity with mde lcati parameterized by m. Let X < < X be rder statistics. Suppse that X 0 ::: X0 are i.i.d. frm f(x m 0 ), where m 0 is the true lcati f the mde. If m 0 is kw, the parametric maximum likelihd estimatr ^f (x m 0 ) is such that whe x> m 0, ^f (x m 0 ) is the left derivative f the least ccave majrat f the empirical distributi fucti, ad whe x<m 0, ^f (x m 0 ) is the right derivative fthe greatest cvex mirat f the empirical distributi. (See Greader (956).) I applicatis, the true mde m 0 is typically ukw. Let ^m be a csistet estimatr f m 0. The, we use the estimatr ^f (x ^m) asaestimatrfthe ukw desity f(x m 0 ). We call such a estimatr the plug-i MLE. Therem. Let ^m be acsistet estimate f the mde m 0 f the true uderlyig desity, ad f 0 (x m 0 ) 6= 0 be the derivative f the desity f(x m 0 ) with respective t x. The, =3 2 f(x m 0)f 0 (x m 0 ) ; 3 ( ^f (x ^m) ; f(x m 0 )) L ;! 2Z where the radm variable Z is distributed as the lcati f the maximum f the prcess (W (u) ; u 2 u2<), ad W () is a stadard tw-sided Brwia mti the real lie < rigiatig frm zer (i.e. W (0) = 0). Remark. A strikig feature f Therem is that fr ay csistet estimate ^m, the plug-i MLE ^f (x ^m) has the same asympttic distributi as ^f (x m 0 ). Birge (987c) shwed that fr a particular chice f ^m, thel rm f ^f (: ^m) ; ^f ( m 0 ) is als f rder P ( ;=2 ). Less satisfactry is the lack f real ifrmati abut ^f(m 0 m 0 ).

4 26 PETER J. BICKEL AND JIANQING FAN We w prpse a mde estimate based the maximum likelihd methd. Estimatig the mde by the kerel methd (Parze (962), Eddy (980)) ad greatest \clusterig" methd (Cher (964), Veter (967)) requires a chice f smthig parameters. Ulike these traditial appraches, the maximum likelihd methd is fully autmatic. Let ^f j ( X j ) be the maximum likelihd estimate fr the data fx i i6= jg with mde lcati X j. Let ^j = arg maxj Pi6=j lg( ^f j (X i X j )). The, the prpsed estimate f the mde ad f the desity are ^m MLE = X^j ^fmle () = ^f^j ( ^m MLE ): Herewith is a csistecy result, which shws that ^m MLE ca be used i Therem. Hece, the estimated desity ^f MLE has the same asympttic prperty as i Therem. Therem 2. Suppse that the tail f the uderlyig distributi satises F (x); F (;x) =; (x ;= ) as x! + fr sme >0 ad that the desity f(x m 0 ) is buded ad uimdal with mde m 0. If the mde is uiquely deed, the ^m MLE is a csistet estimate f the mde m 0. Remark 2. We shw, i fact, that i additi t the cditis give i Therem 2, if there exists a psitive cstat k adc>0 such that i a eighbrhd f m 0, jf(y m 0 ) ; f(z m 0 )jcjy ; zj k fr y z < m 0 ad y z > m 0 (2:) ad the desity is Lipschitz ctiuus at m 0, the ^m MLE ; m 0 = P ;=2 lg 2 () =(2k+) : (2:2) We cjecture that the estimates leadig t (2.2) are t crude ad that the rate is ;=(2k+). The heuristic basis f the cjecture is give i Secti 5. The truth f this cjecture wuld imply that this estimate has cvergece rate O( ;=5 ), the same rate as kerel based desity estimate (Eddy (980)), if f 00 (m 0 m 0 ) < 0 ad has rate O( ;=3 ) if the desity has a wedge (e.g. the triagular desity). Wag (994) shwed that Birge's (987c) result ca be exteded t ay ^m which cverges t m 0 slwer tha O P ( ;=(2k+) ). Thus, if the cjecture is crrect ^m MLE give the apprpriate rate fr the L -rm: k ^f MLE () ; f( m 0 )k = O P ( ;=3 ). Numerical supprt fr the cjecture is give i Figures 5. ad 5.2 ad sme heuristics fr the cjecture are give fllwig the prf f (2.2) i Secti 5. It has bee bserved empirically that the MLE fr estimatig a uimdal desity appears t be spiky ear the estimated mde. We suggest a pregrupig

5 UNIMODAL DENSITY ESTIMATION 27 techique t reduce the spikiess ad cmputati. The idea is t grup the data rst, ad the apply the plug-i techique. Let fi j = (;t j t j+ ] j = 0 2 :::g be a partiti f the real lie, where ft j g is a sequece f icreasig cstats. Dee a mdied versi f the empirical distributi fucti by F (x) = (# f X0 is t j+ ) whe x 2 (t j t j+ ]: Let ^f (x m) be the left derivative f the least ccave majrat f F (x) whe x>m, ad the right derivative fthegreatest cvex mirat ff (x) whe x<m. Let ^m be a csistet estimate f m 0. We call ^f (x ^m) a \pregrupig" versi f the plug-i MLE ^f (x ^m). Nte that the estimatr ^f (x ^m) is the plug-i MLE f the gruped data: takig all data i the iterval (t j t j+ ]tbe t j+. Ituitively, the carser the partiti f the iterval, the less spiky the MLE. A atural questi is hw crude a partiti ca be s that the pregrupig MLE preserves the asympttic prperties f the usual MLE. Therem 3. Let ^m be a csistet estimate f the mde m 0. Suppse that the fucti f( m 0 ) is buded, ad f 0 (x m 0 ) is zer at the pit x. If max j jt j+ ;t j j = ( ;=2 ), the the cclusi f Therem hlds with ^f (x ^m) replaced by ^f (x ^m). Nte that ther smthig methds shuld als yield the same behavir. Fr example, the kerel smthig estimate fr estimatig a decreasig desity wuld be the desity f the least ccave majrat f the smthed empirical distributi (Mamme (99a)). The MLE, beig a radm bi width histgram, is t smth. We ca btai a smther estimate by dig the MLE satisfyig the mticity restrictis amg liear splies. The prblem, f curse, already appears i estimatig a decreasig desity. Let X ::: X 0 0 be a radm sample frm a decreasig desity f ad let FL D be the class f ctiuus liear splie decreasig desities [X X ] with kts at the data pits. We wishtd: arg max f2f D L Y j= f(x 0 j): (2:3) The sluti t prblem (2.3) ca be cmputed explicitly by istic regressi techiques. Let ^f aj = 8 >< >: mi a+t>j max sj mi asj max t>j max max sa t;s (z t;z s) t;s (z t;z s) a;s+ (z a+;z s) max t>a whe j<a, whe j>a, t;a (z t;z a) whe j = a, (2:4)

6 28 PETER J. BICKEL AND JIANQING FAN where z j =(X j + X j; )=2 with the cveti that X 0 = X, ad X + = X. Let ^f L (x a) be the fucti cectig the pits (X j ^f aj )by usig lies, ad 0 whe x is ut f the data rage [X X ]. The fllwig tw therems describe the sluti ad the asympttic behavir f the liear splie MLE. Therem 4. The sluti t prblem (2:3) is give by ^f L (x ). Therem 5. Suppse that X ::: X 0 0 are idepedet bservatis frm a decreasig desity f [0 ), which has a zer derivative f 0 (x) at a pit x 2 (0 ). The =3 2 f(x)f 0 (x) ;=3 ( ^f L (x ) ; f(x)) where the radm variable Z was deed i Therem. L ;! 2Z Liear splies ca als be applied t the uimdal case. Let FL U be the class f liear splie uimdal desities [X X ] with kts at the data pits. We wish t d arg max f2f U L Y j= f(x 0 j): (2:5) It will be shw i the prf f Therem 6 that ^f L (x a), deed abve, is a desity i F U L with mde lcati X a. Let ^f L (x ^a) be the maximizer f the likelihd fucti amg the pssible chices f desities ^f L (x a), a = :::. The, we have the fllwig result. Therem 6. The sluti t prblem (2:5) is give by ^f L (x ^a). Let us give a gemetric iterpretati f this result. Dee a mdied empirical distributi (strictly speakig, it is t a cdf) ^F (x) = + X j= I fzjxg (2:6) where I A is the idicatr f the set A. Let ^f a(x) be the left derivative f the least ccave majrat f ^F (x) whe x>z a. The we have fr j>a, ^f aj = ^f a(z j+ ): (2:7) I ther wrds, ^f L (x a) is a ctiuus versi f ^f a(x): ^fl (x a) is btaied by cectig pits (X i ^f a (X i )) by lies. This idetity gives a simple way f cmputig ^f aj by usig the \pl-adjacet-vilatrs" algrithm, ad a idicati that MLE liear splie shuld t be very dieret frm the MLE itself.

7 3. Discussi UNIMODAL DENSITY ESTIMATION 29 We have prpsed the maximum likelihd methd t estimate uimdal desities, a pregrupig techique t reduce peakig prblems ad t save cmputatial cst, ad a liear splie apprach t prduce ctiuus pictures. Here are sme cmputatial details. Amut f Pregrupig. I practice, we typically take the partiti ft j g t be equally spaced grid pits with spa l. Therem 3 suggests that the chice f l be t t large. Practically, we recmmed chsig l such that the data are gruped it grups (Recall 5 5 bis are suggested fr histgrams i may textbks we eed mre detail tha that), depedig the umber f data pits. Our experiece i simulatis shws that such a resluti is detailed eugh fr practical purpses. Bayesia Estimati f Mde. Let ^f j ( X j ) be the maximum likelihd estimate fr the data fx i i6= jg with mde lcati X j ad L(j) bethelikelihd f this estimate: Y L(j) = ^f j (X i X j ): (3:) i6=j Dee the Bayesia estimate f the mde by ^m B = X i L(i) Pj L(j)X i: (3:2) Our empirical experiece via simulati shws that this estimatr has a mre stable variace tha ^m MLE. Smthed MLE. As idicated at the ed f Secti 2, higher rder splie MLE such as liear splie MLE des t prduce a qualitatively dieret curve frm the MLE itself. Oe pssible way t prduce a smthed uimdal desity is t impse a smthess pealty the likelihd fucti ad the t maximize the pealized likelihd subject t the uimdality cstraits. We d t explre i this directi because we d t kw a simple ptimizati algrithm. A alterative way is t d a smthed curve that basically (i a least squares sese) passes thrugh the midpits f the MLE histgram estimate. Ufrtuately, the resultat curve is t ecessarily uimdal. Herewith is ur smthig prcedure. Let (x z ) ::: (x N z N ) dete the midpits f the MLE histgram estimate ^f MLE (i.e., x i is the midpit ftheith histgram bi ad z i is the height). Let us take x 2 x 6 ::: x 4m+2 (m =[(N ; 2)=4]), as iitial kts that may be

8 30 PETER J. BICKEL AND JIANQING FAN deleted. Let crrespdig pwer bases be ( Bj (x) = (x ; x 4j+2 ) 3 + j =0 ::: m B m+ (x) = B m+2 (x) =x B m+3 (x) =x 2 B m+4 (x) =x 3 : Let lg(f s (x)) = P m+4 k B k (x). Use the usual least squares t d k that miimizes NX [lg(z i ) ; m+4 X k B k (x i )] 2 w i (3:3) where w i is the area f the histgram estimate the ith bi. Dete the least square estimate f (3.3) by ^ j with stadard errr SE(^ j ). The, delete the j 0 th kt ( j 0 m) havig the smallest abslute t-value: j^ j j=se(^ j ), ( j m). Repeat the abve deletig prcess (at each step delete e kt) util the abslute t-value is smaller tha 3. Let ^x ::: ^x^j be the remaiig kts with bases Bj (x) = (x ; ^x j ) 3 +, j = ::: ^j, ad B (x) =, (x) = x, ^j+ B^j+2 B ^j+3 (x) =x2, ad B ^j+4 (x) =x3, ad estimates ^ j j = ::: ^j +4. Nw, frm the fucti ^f (x) = exp 0 ^ j B j (x) A : Nrmalize ^f (x) t be a desity ad dete the resultig fucti by ^f (x). The, ^f (x) is a smthed versi f MLE, which will be preseted i the ext secti. This kid f kt deleti idea was used i CART by Breima et al. (983). 4. Simulatis I this secti, we use 4 simulated examples t illustrate the prpsed prcedures ad t cmpare them with the kerel desity estimate. Fr each example, we use sample size = 200 ad umber f simulatis 500. Fr 500 simulatis, it is t pssible t plt here all f these estimated curves. Istead, we select a represetative simulati the simulati whse average L -lss f the MLE at data pits is media amg 500 replicatis. The fur simulated examples are Example. expetial distributi: f(x) = exp(;x)i fx>0g (4.) Example 2. Gaussia distributi: f(x) = p 2 exp(;x 2 =2) (4.2) Example 3. Asymmetric distributi: f(x) = 2(exp(2x)I 3 fx0g +exp(;x)i fx>0g ) (4.3) Example 4. Triagular distributi: f(x) =(;jxj) + (4.4)

9 UNIMODAL DENSITY ESTIMATION 3 I the MLE ttig, we ly assume that the desity is uimdal with ukw mde, althugh desity (4.) is ideed decreasig. These desities represet dieret degrees f skewess ad dieret weights f tails. The kerel desity estimate is deed as Xi ; x ^f(x) = h i= with the badwidth determied by the rmal referece rule (see Silverma (986)): h =:06s ;=5 (4:5) where K() is the stadard Gaussia desity ad s is the sample stadard deviati. Nte that this chice f badwidth is asympttically ptimal if the true desity is rmal. Thus, the kerel desity perfrms well uder mdel (4.2). I geeral, the abve chice f badwidth teds t versmth. Hece, it fte prduces a uimdal desity ad gives a gd estimati f the mde lcati fr symmetric desities. Fr these reass, we wuld expect that the kerel desity estimate with badwidth (4.5) perfrms well fr symmetric distributis. Figures -4 depict the simulati results: The pregruped MLE estimate with mde estimated by ^m MLE, smthed MLE prpsed i Secti 3 ad the kerel desity estimate. The kerel desity estimate des t estimate the tail f desities well ad mis-estimates the peak whe the distributi is asymmetric (e.g desities (4.) ad (4.3)). Fially, we cmpare mde estimati by the MLE ad by kerel desity estimati. As we aticipated, the kerel desity estimate perfrms better fr symmetric desities ad wrse fr asymmetric desities. I a attempt t uderstad the cvergece rates, we simulated 500 times frm (4.3) ad (4.4) fr = 50 2 j (j = 0 ::: 5), ad cmputed the MSE f the mde estimati fr three estimatrs: ^m MLE, ^m B, ad the kerel desity estimate. Figure 5 plts the lgarithm f MSE agaist lg 2 () (hece the slpe idicates the rate f cvergece). Fr the symmetric desity (4.4), the MLE methd seems t have a rate cmparable t the kerel desity estimate except that the cstat factrs are larger. Fr the asymmetric distributi (4.3), the mde estimati by kerel has a much slwer rate f cvergece. Overall, the Bayesia estimati f mde (3.2) seems t have a smaller cstat factr tha the ^m MLE (the rates are the same because the curves are parallel). Fr the symmetric desity (4.4) the bias i the mde estimati is egligible (abut 0 t 00 times smaller tha the variace), whereas fr the asymmetric desity (4.4), the bias is t egligible. Figure 5.3 shws the bias ad variace ctributi i the lgarithmic scale. K h

10 32 PETER J. BICKEL AND JIANQING FAN y x Figure.. Example : Desity Estimati lg-likelihd x Figure.2. Example : Prle lg-likelihd y x Figure 2.. Example 2: Desity Estimati

11 -.3 UNIMODAL DENSITY ESTIMATION 33 lg-likelihd x Figure 2.2. Example 2: Prle lg-likelihd 0.6 y x Figure 3.. Example 3: Desity Estimati -.8 lg-likelihd x Figure 3.2. Example 3: Prle lg-likelihd

12 34 PETER J. BICKEL AND JIANQING FAN y x Figure 4.. Example 4: Desity Estimati lg-likelihd Figure Captis x Figure 4.2. Example 4: Prle lg-likelihd Figures -4. A represetative estimated curve based MLE ad kerel desity estimate with sample size = 200. Figures.-4. are the estimated curves. Slid curve true desity slid step fucti pregruped MLE estimate dashed lie kerel desity estimate with badwidth (4.5) dtted lie smthed MLE. Figures are plt f the lgarithm f the prle likelihd: fx j g agaist ; lg L(j) with L(j) deed by (3.). Figure 5. Plt f the lgarithm f mea square errrs agaist the lgarithm f the sample sizes fr mde estimati. The slpe i this lg-lg plt idicates the rate f cvergece. Figure 5. is fr asymmetric desity (4.3) ad Figure 5.2 is fr symmetric desity (4.4). Figure 5.3 gives the bias ad variace decmpsiti fr the asymmetric desity (4.3) i the lgarithmic scale. Bias N variace. Fr the symmetric desity (4.4), the bias is egligible.

13 UNIMODAL DENSITY ESTIMATION lg(mse) MLE Estimatr (3.2) Kerel lg 2(=50) Figure 5.. Example 3: MSE fr mde estimati -3-4 lg(mse) -5-6 MLE Estimatr (3.2) Kerel lg 2(=50) Figure 5.2. Example 5: MSE fr mde estimati -2-3 lg(mse) MLE Estimatr (3.2) Kerel lg 2(=50) Figure 5.3. Example 3: Bias ad Variace mde estimati

14 36 PETER J. BICKEL AND JIANQING FAN 5. Prfs Prf f Therem. We give the prf fr x > m 0 the ther case ca be treated similarly. First te that if x>m m 2, the ^f (x m ) ^f (x m 2 ) (5:) by the deiti f the estimatrs. Let l(x) =j f(x m 2 0)f 0 (x m 0 )j ;=3. Fr ay ">0, ad m 0 + "<x,by the csistecy f ^m, P 3 l(x)( ^f (x ^m) ; f(x m 0 )) t = P 3 l(x)( ^f (x ^m) ; f(x m 0 )) t j ^m ; m 0 j" + (): (5:2) Nte that f(x m 0 ) is decreasig whe x m 0 + ". By a result f Prakasa Ra (969) ad Greebm (985), we have P 3 l(x)( ^f (x m 0 + ") ; f(x m 0 )) t ;! P f2z tg 8t 2 (; +): (5:3) Thus, the cmbiati f (5.), (5.2) ad (5.3) leads t lim if P 3 l(x)( ^f (x ^m) ; f(x m 0 )) t lim if P 3 l(x)( ^f (x m 0 + ") ; f(x m 0 )) t Similarly, by (5.) ad (5.2), we have = P f2z tg : (5:4) lim sup P 3 l(x)( ^f (x ^m) ; f(x m 0 )) t lim sup P 3 l(x)( ^f (x m 0 ; ") ; f(x m 0 )) t : (5:5) The prf is cmpleted if we shw that (5.5) has a limit (5.4). Let f " = f(y m 0 )=( ; F (m 0 ; ")) ad f" () be the sluti t the prblem: max g() is a decreasig desity [m 0;" ) Z m 0;" [lg g(y)]f " (y)dy: The, f" (y) = f " (a) fm0;"yag + f " (y) fy>ag, where a is chse s that f" is a desity fucti. See Bickel ad Fa (990) fr a prf. Thus, fr each xed x>m 0, there exists " 0 such thatf" (x) =f " (x) fr "<" 0. By the argumet f Greebm (985), e ca shw that P N =3 j 2 f " (x)f 0 " (x)j ;=3 ( ^f N (x) ; f " (x)) t ;! P f2z tg

15 UNIMODAL DENSITY ESTIMATION 37 where ^f N () is the MLE ver the class f decreasig desities [m 0 ; " ) baseddatax j m 0 ; ", adn = [ ; ^F (m 0 ; ")]. Usig the fact that ^f (x m 0 ; ") = N ^f N (x) 8x >m 0 ; " we have fr "<" 0, P =3 l(x)[ ^f (x m 0 ; ") ; f(x m 0 )] t ;! P f2z tg : This, tgether with (5.4) ad (5.5), leads t the desired cclusi. We eed the fllwig tw lemmas t prve Therem 2. Lemma. Let f(x m 0 ) be a uimdal desity with mde m 0 ad Z G(m) = sup lg g(x)f(x m 0 )dx (5:6) g2f m where F m is the class f uimdal desities with mde m. The G(m) is icreasig whe m < m 0 ad is decreasig whe m > m 0. If cditi (2:) is satised, the fr m i a eighbrhd f m 0, fr sme c > 0. G(m 0 ) ; G(m) >c jm 0 ; mj 2k+ (5:7) Prf. Withut lss f geerality, we prve this lemma fr the case m < m 0. First, the sluti t the ptimizati prblem (5.6) is give by f m (x) =h m fmxmmg + f(x m 0 ) fx<m r x>m mg (5:8) where h m = f(m m m 0 )adm m is a cstat such that f m (x) is a desity: Z Mm m f(x m 0 )dx = h m (M m ; m): (5:9) (See Bickel ad Fa (990) fr a prf.) Give m 2 <m <m 0,sicef m2 2F m, we cclude that G(m ) G(m 2 ). Therefre, G(m) is icreasig whe m m 0. Next, we prve (5.7). First f all, by (5.8), we have G(m 0 ) ; G(m) = Z Mm Evidetly, asm! m 0, M m! m 0 ad m lg(f(x m 0 )=h m )f(x m 0 )dx: sup mxm m jf(x m 0 )=h m ; j!0:

16 38 PETER J. BICKEL AND JIANQING FAN By Taylr's expasi, we btai G(m 0 ) ; G(m) = = Z Mm m Z Mm m (f(x m 0 )=h m ; )f(x m 0 )dx ( + (m 0 ; m)) (f(x m 0 ) ; h m ) 2 dx=h m ( + (m 0 ; m)) (5:0) where the last equality fllws frm (5.9). Let m 2 (m m 0 ) be the pit such that f(m m 0 )=f(m m m 0 ). Whe m is clse t m 0,by (2.) we have G(m 0 ) ; G(m) = Z Mm (f(x m 0 ) ; f(m m 0 )) 2 dx 2f(m 0 m 0 ) m Z m0 ; cjx ; m j k 2 dx 2f(m 0 m 0 ) m c 2 ; jm0 ; m j 2k+ + jm ; mj 2k+ 2(2k +)f(m 0 m 0 ) c 2 m 2 2k+2 0 ; m 2k+ : (2k +)f(m 0 m 0 ) The cclusi fllws frm the last iequality. Recall that X < <X dete the rder statistics. Lemma 2. Suppse that the tail f the uderlyig distributi satises F (x) ; F (;x) =; (x ;= ) as x! + fr sme >0 ad that the desity f(x m 0 ) is buded. The, the miimum ad maximum spacig satisfy fr all >0. P fmi i (X i ; X i; ) > ;2; g! P fx ; X g! Prf. Accrdig t Pyke (965), the uifrm spacig has the fllwig represetati: (F (X 2 ) ; F (X ) ::: F(X ) ; F (X ; )) =( d + ::: ; )= i where ::: + are i.i.d. stadard expetial radm variables. Thus, P f 2+=2 mi i (F (X i+ ) ; F (X i )) > g = P fmi i X X i= + i > ;;=2 i =g i= P fmi i i > 2 ;;=2 g + ()! :

17 UNIMODAL DENSITY ESTIMATION 39 Sice sup x f(x) mi i (X i+ ; X i ) mi i (F (X i+ ) ; F (X i )), we have P fmi(x i ; X i; ) > ;2; g!: i It is easy t check, uder ur assumpti F,that P fx > =2g! ad P fx < ; =2g!: Thus, with prbability tedig t e, X ; X : This cmpletes the prf. Prf f Therem 2. Dete the lg-likelihd by G (X j )= sup g2f Xj X i6=j lg g(x i ): Sice the maximum likelihd estimate ^f(x X j ) is the right derivative f the greatest cvex mirat f the empirical distributi whe x < X j, ad the left derivative f the least ccave majrat f the empirical distributi whe x>x j, the by Lemma 2, with prbability tedig t e, we have max i6=j ^f(x i X j ) < 3 ad mi i6=j ^f(x i X j ) > ;; : Dete this set by. The previus statemet is equivalet t P ( )!. Thus, fr! 2, G (X j )= sup fk lg gkd lg g2f Xj g = sup fk lg gkd lg g2f Xj g Z X i6=j lg g(x i ) lg gdp + O(lg =) where P is the empirical prcesses ad d = maxf3 +g. Let C be the class f uimdal fuctis whse suprm is buded by.the, fr! 2 max j jg (X j ) ; G(X j )jd lg sup g2c Z g(x)(dp ; dp ) + O(lg =): By empirical prcess thery (Therem 37, Pllard (984)) Z g(x)(dp ; dp ) a (lg =) =2 almst surely sup g2c = fr ay sequece a!. Takig a =lg 0:25 (), say, wehave max jg (X j ) ; G(X j )j = P (lg :75 ()= p ): (5:) j

18 40 PETER J. BICKEL AND JIANQING FAN If the mde m 0 is uiquely deed, the fr small " > 0, f m0;"() ad f m0;2"() deed by (5.8) ca t be idetical. Thus, by the uimdality fg() i Lemma, ad hece G(m 0 ; ") >G(m 0 ; 2") ad G(m 0 + ") >G(m 0 +2") if G(m) > sup G(m): jm;m 0j" jm;m 0j2" Usig this ad (5.), the X^j 2 (m 0 ; 2" m 0 +2") with prbability tedig t e. That is, ^m MLE is a csistet estimate f m 0. Prf f (2.2). Let " =(lg :75 ()= =2 ) =(2k+). I the sequel, we shw that with prbability tedig t e, it is t pssible t have ^m MLE lies utside the iterval (m 0 ; " m 0 + " ). By Lemma, G(m 0 ) maxfg(m 0 ; " ) G(m 0 + " )g + c " 2k+ = sup G(m)+c " 2k+ : jm;m 0j" Sice f() is Lipschitz ctiuus at m 0, it ca easily deduced frm (5.0) that 0 G(m 0 ) ; mifg(m 0 ; lg =) G(m 0 +lg=)g O(lg =): Csequetly, whe is large, It is easy t shw that if G(m) > sup G(m)+c " 2k+ =2: (5:2) jm;m 0jlg = jm;m 0j>" P fat least e data pit fallsi (m 0 ; lg = m 0 + lg =)g!: Let X be a data pit i(m 0 ; lg = m 0 +lg=). By (5.), G (X ) G(X )+ P lg :75 () ;=2 : Fr X j such that jx j ; m 0 j", the by (5.) ad (5.2), whe is large, we have G (X ) G(X j )+c " 2k+ =2+ P lg :75 () ;=2 G (X j )+c " 2k+ >G (X j ): =2+ P lg :75 () ;=2 Thus, with prbability tedig t e, the maximum f G () ca t be achieved at the pit X j such thatjx j ; m 0 j >". Hece, P fj ^m MLE ; m 0 j" g!

19 ad The cclusi fllws. UNIMODAL DENSITY ESTIMATION 4 ^m MLE ; m 0 = O p (" )= P (lg 2 =) =(2k+) : Heuristic basis f cjecture. We base ur cjecture the cjectured apprximati, i= lg ^f (X i m) ^f (X i m 0 ) = i= lg f m(x i ) f m0 (X i ) + O P ( ; + jm ; m 0 j 3 ) (5:3) uifrmly fr m m 0 62 fx i : i g where f m () is deed by (5.8)ad the cjecture that ^m behaves like the maximum f the left had side f (5.3) fr m as specied. If (5.3) hlds it is easy t see frm lemma that, if k, supf i= lg f m(x i ) f m0 (X i ) : jm ; m 0jg = O P ( (2k+)=2 ;=2 )+C 2k+ where C > 0: (5:4) Therefre, if k the sup i (5.4) must be achieved fr jm ; m 0 j = O( ; 2k+ ). Sice the remaider i (5.3) is als O P ( ; ) fr k the cjecture fllws. Our belief i (5.3) is based the behavir i the crrespdig parametric situati where f = f( ), the truth is f( 0 0 ), ^() is deed by ^() = max ; i= lg f(x i ) ad () bymax ; R lg f(x )f(x 0 0 )dx. If we Taylr expad i= i= lg f(x i ()) f(x i ^()) abut ^() ad^() is assumed i the iterir the, if j ; 0 j = (), we expect lg f(x i ^()) f(x i 0 ^( 0 )) ; lg f(x i ()) f(x i 0 ( 0 )) = O P (fj^() ; ()j 2 ;j^( 0 ) ; 0 j 2 g): (5:5) Fially, it seems plausible that (^() ; ()) ; (^( 0 ) ; ( 0 )) = O P (j^( 0 ) ; ( 0 )jj ; 0 j): (5:6) If we cmbie (5.5) ad (5.6), idetify with the shape f f, ad te that i ur case we expect ^() ; () =O P ( ;=3 ) the (5.3) fllws. Of curse, there

20 42 PETER J. BICKEL AND JIANQING FAN is much wrg with this argumet. We d t have ay assurace that buds like (5.5) ad (5.6) are valid sice we kw that ^() is achieved the budary s that we cat use Taylr expasis i fucti space. Nevertheless the cjecture lks prmisig t us. Lemma 3. Let X 0 ::: X 0 be i.i.d with a desity f(x). If f is buded ad the maximum spa f the partiti satises the cditi f Therem 3, the sup j ^F (x) ; F(x)j = p ( ;=2 ) x where ^F is the empirical cdf f X 0 ::: X0. We mit the prf f Lemma 3 (but see Bickel ad Fa (990)). Prf f Therem 3. We eed ly t prve the result fr the decreasig desity case the uimdal case fllws frm the result f estimatig a decreasig desity ad the prf f Therem. By Lemma 3, ad the Hugaria embeddig f Kmls et al. (973), the prcess F(t) has the fllwig decmpsiti: =2 (F (t) ; F (t)) = =2 ^F (t) ; F (t) = B (F (t)) + p () + =2 F (t) ; ^F (t) where fb g is a sequece f Brwia bridges, cstructed the same space as the ^F (t), the empirical prcess. The cclusi fllws frm the prf f Therem 2. f Greebm (985). Prf f Therem 4. The result fllws frm the prf f Therem 6. Prf f Therem 5. Let l(x) =jf(x)f 0 (x)=2j ;=3. By the prf f Lemma 2, with prbability tedig t e, the maximum spacig fr the data set fx i : X i 2 x "g is f rder O( ; lg ), where " is small eugh s that if y2x" f(y) > 0. Thus, with prbability tedig t, the pits x ; ", x, x + " are i dieret itervals f (z j z j+ ), where " = ;2=5, ad z j was deed i (2.6). Thus, by (2.7), we have with prbability tedig t e that ^f (x + " ) ^f L (x ) ^f (x ; " ) (5:7) where ^f (x) was deed after (2.6). Nte that the mdied empirical distributi deed by (2.6) satises 0 ^F (x) ; ^F (x) =, where ^F () is the usual empirical cdf. Thus, by the same argumet as i the prf f Therem 3, we have P =3 l(x)( ^f (x + " ) ; f(x)) t ;! P f2z tg 8t 2 (; ):

21 Csequetly, by (5.7), UNIMODAL DENSITY ESTIMATION 43 lim sup P =3 l(x)( ^f L (x ); f(x)) t =3 l(x)( ^f (x + " ) ; f(x)) t lim sup P = P f2z tg 8t 2 (; ): The cclusi fllws frm a similar iequality: lim if P =3 l(x)( ^f L (x ); f(x)) t P f2z tg 8t 2 (; ): We eed the fllwig lemma (Therem.5. f Rberts et al: (988)) t prve Therem 6. Lemma 4. Suppse that () is dieretiable, ad cvex a iterval I. Let (u v) = (u) ; (v) ; (u ; v) 0 (v). If f j is a sluti f prblem (5:2), the f miimizes P j (g j f j )w j i the class f istic fuctis f. Prf f Therem 6. We eed ly prve that ^f (x a) is the sluti t the prblem (2.5) with a additial cstrait that the lcati f the mde is X a. Let f j = f(x j ). The the prblem is equivalet t max X j lg f j subject t :(uimdality) f f 2 f a f a+ f (5:8) (Area e) ; X j= f j+ + f j (X j+ ; X j )=: (5:9) 2 Write c j =(X j+ ;X j; )=2 with X 0 =X,adX + =X. The the equality cstrait (5.9) ca be rewritte as j= c j f j =: (5:20) Dete g j ==(c j ) ad w j = c j. The, the ptimizati prblem is equivalet t maximizig P lg f j subject t (5.8) ad P (g j ; f j )w j =0. Csider the prblem f istic regressi mi f (f j ; g j ) 2 w j (5:2)

22 44 PETER J. BICKEL AND JIANQING FAN with a partial rder 2 a a + a +2. The, the sluti t the prblem (5.2) is give by (2.4) (see page 23 f Rberts et al. (988)). The sluti als satises (Therem.3.6 f Rberts et al. (988)) ( ^f aj ; g j )w j =0 i.e. (5.9). Nw, let us apply Lemma 4. Take acvex fucti (u) =u lg u. The, ^f a als miimizes (g j lg g j ; g j lg f j ; g j + f j )w j = c ; lg f j + c j f j uder the istic cstraits, where c = P lg g j ;. Sice we are iterested ly i the class f istic regressi satisfyig (5.20), ^fa maximizes P lg f j uder the cstraits (5.8) ad (5.9). The desired cclusi fllws. Ackwledgmet The research f the rst authr was supprted by ONR N J-563. The research f the secd authr was supprted i part by NSF Grat DMS ad NSF Grat DMS We thak the assciate editr ad referees fr the helpful suggestis ad cmmets. I particular, we appreciate the prle likelihd heuristic raised i the review prcess. Refereces Barlw, R. E., Barthlmew, D. J., Bremer, J. M. ad Bruk, H. D. (972). Statistical Iferece uder Order Restrictis. Jh Wiley, Ld. Barlw, R. E. ad va Zwet, W. R. (970). Asympttic prperties f istic estimatrs fr the geeralized failure rate fucti, part I: strg csistecy. I Nparametric Techiques i Statistical Iferece, (Edited by M. L. Puri), 59-73, Cambridge Uiversity Press. Bickel, P. J. ad Fa, J. (990). Sme prblems the estimati f desities uder shape restrictis. Techical Reprt 258, Dept. f Statist., Uiv. f Califria, Berkeley. Breima, L., Friedma, J. H., Olshe, R. A. ad Ste, C. J. (983). CART: Classicati ad Regressi Trees. Wadswrth, Belmt, CA. Birge, L. (987a). Estimatig a desity uder rder restrictis: Nasympttic miimax risk. A. Statist. 5, Birge, L. (987b). O the risk f histgrams fr estimatig decreasig desities. A. Statist. 5, Birge, L. (987c). Rbust estimati f uimdal desities. Upublished mauscript. Birge, L. (989). The Greader estimatr: A asympttic apprach. A. Statist. 7, Cher, H. (964). Estimati f the mde. A. Ist. Statist. Math. 6, 3-4. Eddy, W. F.(980). Optimum kerel estimatrs f the mde. A. Statist. 8,

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