Improved Asymptotics for Zeros of Kernel Estimates via a Reformulation of the Leadbetter-Cryer Integral

Impro v ed Asymptotics for Zeros of Kernel Estimates via a Reform ulation of the Leadb etter-Cry er In tegral ∗ Kurt S. Riedel Coura n t Institute of Mathematica l Sciences New Y ork Universit y New Y ork, New Y ork 100 12-11 85 Abstract The exp ected num b er of false inflection p oin ts of k ernel sm o others is ev aluated. T o obtain the small noise limit, we use a reform ulation of the L eadb etter-Cry er integ ral for the exp ected n umb er of zero crossings of a different iable Gaussian pr o cess. Keyw ords: Kernel smo ot hers, Deriv ativ e estimation, Change p oin ts, Zero-cro ssings 1 Con v erge n ce of Kerne l Smo othe rs F or man y applications of nonparametric function estimation, obtaining the correct shap e of the unkno wn function is of imp ortance. A consequence of Mammen et al. (1992 , 1995) is that k ernel smo others hav e a nonv anishing probability of ha ving spurious inflection p oints if the smo othing lev el is c hosen to minimize the mean in tegrated square error (MISE). In Riedel (1996 ), w e prop ose a tw o-stage estimator where the num b er and lo cation of the c hange p o in ts is estimated using strong smo othing. In this letter, w e ev aluate the probabilit y of obta ining spurious inflection p oin ts for ker- nel smo others in the small noise/hea vy smo othing limit. The pro o f s are based on p ow erful and seldom used techniq ues: Ko ksma’s theorem and the Leadb etter-Cry er in tegral for the exp ected n umber of zeros o f a differen tiable Gaussian pro cess. W e consider a sequence of ke rnel smo o t her estimates, ˆ f N ( t ), of f ( t ), and examine the con v ergence of the estimate as t he n umber of measuremen ts, N , increases . W e b eliev e that our results are sligh tly stronger tha n previous theorems on k ernel smo others (G asser & ∗ W e thank the referee for us eful comments. Resea rch f unded by the U.S. Depa rtment of Energy . 1 M ¨ uller 1984). F or eac h N , the measuremen ts o ccur at { t N i , i = 1 . . . N } . W e suppress the sup erscript, N , on the measuremen t lo cations t i ≡ t N i . W e define the empirical distribution of measuremen ts, F N ( t ) = P t i ≤ t 1 / N , and let F ( t ) b e it s limiting distribution. Assumption A Consider the se quenc e of estimation pr oblems: y N i = f ( t N i ) + ǫ N i , wher e the ǫ N i ar e zer o me an r andom variabl e s and Cov [ ǫ N i , ǫ N j ] = σ 2 δ i,j . Assume that the dis tribution of me asur ement lo c ations c onver ges i n the sup norm: D ∗ N ≡ sup t {| F N ( t ) − F ( t ) |} → 0 , wher e 0 < c F < F ′ ( t ) < C F . The star-discrepancy , D ∗ N ≡ sup t { F N ( t ) − F ( t ) } , is useful b ecause it measures how closely a discre te sum o ve r an arbitrarily placed set o f p oin ts appro ximates an in tegral. (See Theorem 2.) F or regularly spaced p oints, F ( t i ) = ( i + . 5) / N and D ∗ N ∼ 1 / N , while for randomly spaced p oin ts, D ∗ N ∼ q ln[ln[ N ]] / N by the Gliv enk o-Cantelli Theorem. W e consider k ernel estimates o f the form: ˆ f ( ℓ ) ( t ) = 1 N h ℓ +1 N N X i y i w i F ′ ( t i ) κ ( ℓ ) ( t − t i h N ) , (1) where h N is the k ernel halfwidth and { w i } are w eigh ts. W e need conv ergence results for k ernel estimators, ˆ f ( ℓ ) N ( t ), of f ( ℓ ) ( t ). Our h yp o theses are stated in terms of the star dis- crepancy while previous results imp ose stronger/redundan t conditio ns. W e define σ 2 N ( t ) = V ar [ ˆ f ( ℓ ) N ( t )], ξ 2 N ( t ) = V ar [ ˆ f ( ℓ +1) N ( t )], µ 2 N ( t ) = Cor r [ ˆ f ( ℓ ) N ( t ) , ˆ f ( ℓ +1) N ( t )]. W e now ev aluate the limiting quan tities for a class of kerne l smoo t hers. W e use the notation O R ( · ) to denote a size o f O ( · ) r elat ive to the main term: O R ( · ) = × [1 + O ( · )]. W e denote C ℓ as the set of ℓ times contin uously differen tiable f unction, T V [0 , 1] as the function of b o und v ariation with the total v ariation norm, k · k T V . W e define k f k bv to b e the sum of the L ∞ and total v ariation nor ms of f and define k f k to b e the L 2 norm. Theorem 1 (Generalized Gasser-M¨ uller ( 1984) ) L et f ( t ) ∈ C ℓ +1 [0 , 1] ∩ T V [0 , 1] and c onsider a se quenc e of estimation pr oblems satisfying Assumption A. L et ˆ f ( ℓ ) N ( t ) b e a kernel smo other estimate as give n in (1), whe r e the hal f width, h N , an d the weights, { w i } , sa tisfy | w i − 1 | ∼ O ( D ∗ N /h N ) . L et the kerne l, κ ( ℓ +1) ∈ T V [ − 1 , 1] ∩ C [ − 1 , 1 ] , sa tisfy the moment c ondition: R 1 − 1 κ ( s ) ds = 1 , and the b oundary c onditions: κ ( j ) ( − 1) = κ ( j ) (1) = 0 for 0 ≤ j ≤ ℓ . Cho ose the ke rnel halfwidths such that h N → 0 , an d D ∗ N /h ℓ +2 N → 0 ; then i) E [ ˆ f ( ℓ ) N ]( t ) → f ( ℓ ) ( t ) + O R ( h N + D ∗ N /h ℓ +1 N ) , ii) E [ ˆ f ( ℓ +1) N ]( t ) = R 1 − 1 f ( ℓ +1) ( t + hs ) κ ( − s ) ds + O ( k f κ ( ℓ +1) k bv D ∗ N /h ℓ +2 N ) , iii) σ 2 N ( t ) → σ 2 k κ ( ℓ ) k 2 / ( N F ′ ( t ) h 2 ℓ +1 N ) + O R ( h N + D ∗ N /h N ) , iv) ξ 2 N ( s ) → σ 2 k κ ( ℓ +1) k 2 / ( N F ′ ( s ) h 2 ℓ +3 N ) + O R ( h N + D ∗ N /h N ) , a nd v) µ 2 N ( t ) → O ( h N + D ∗ N /h N ) uniformly in the interval, [ h N , 1 − h N ] . Our pro of of Theorem 1 is based on Koksma’s Theorem whic h bounds t he difference b et we en integrals a nd discrete sum approximates : 2 Theorem 2 (Generalized Koksma Niederieter (1992) ) L et g b e a b ounde d function of b ounde d variation, k g k T V , on [0 , 1] : g ∈ T V [0 , 1 ] ∩ L ∞ [0 , 1] . L et the star dis c r ep a ncy b e me asur e d b y a distribution, F ( t ) ∈ C 1 [0 , 1] with 0 < c F < F ′ ( t ) < C F . If the discr ete sum weights, { w i , i = 1 , . . . N } , satisfy | w i − 1 | ≤ C D ∗ N , then      Z 1 0 g ( t ) dF ( t ) − 1 N N X i =1 g ( t i ) w i      ≤ [ k g k T V + C k g k ∞ ] D ∗ N . (2) In our v ersion of Koksma’s Theorem, w e hav e added tw o new effects: a nonun ifo rm w eigh ting, { w i , i = 1 , . . . N } , and a non uniform distribution of p o ints, dF . The total v ariation of g ( t ( F )) with resp ect t o d F is equal to the total v ariation of g ( t ) with r esp ect to dt . Theorem 2 follows from Koksma’s Theorem b y a c hange o f v ariables. Pr o of of The or em 1. W e rescale: s i = ( t i − t ) /h N and apply Koksma’s theorem to f ( t + hs ) κ ( ℓ ) ( − s ) ∈ T V s [ − 1 , 1]. The contribution of the w eigh ts, w i , is O R ( D ∗ N / N h ℓ +1 N ). T hus E [ ˆ f ( ℓ ) N ]( t ) = R 1 − 1 f ( ℓ ) ( t + hs ) κ ( − s ) ds + O ( k f κ ( ℓ ) k bv D ∗ N /h ℓ +1 N ). Since | κ ( ℓ +1) ( − s ) | 2 /F ′ ( t + h N s ) is in T V [ − 1 , 1], the v ariance satisfies ξ 2 N ( t ) = σ 2 N h 2 ℓ +1 N Z | κ ( ℓ ) ( − s ) | 2 F ′ ( t + h N s ) ds + O R ( D ∗ N /h N ) . The result follows from expanding F ′ ( t ) in h N . ✷ Theorem 1 is one of tw o ingredien ts whic h we need to b ound t he exp ected num b er of c hange p o in ts of ˆ f ( ℓ ) N ( t ). Section 2 presen ts t he second ingredien t. 2 Asymptotics of Zero Cross ings The Leadb etter- Cry er (L-C) expression ev aluates the exp ected num ber of zeros of a differ- en tiable Gaussian pro cess, Z ( t ), in terms of a time history integral in volving the first and second momen ts of Z ( t ) (Leadb etter and Cry er 19 65). W e reexpress this in tegral in terms of the zeros of E [ Z ( t )] and a remainder term. This alternativ e expression is particularly useful in the small noise limit when o ne desires an asymptotic ev aluation of the num b er of noise induced zero crossings. Theorem 3 (Leadb etter & Cryer (1965) , Cram´ er & Leadbet ter, 1967, Sec. 13.2) L et Z ( t ) b e a p a thwise c ontinuously d iffer entiable Gaussian pr o c e ss in the t ime interval [0,T]. Denote m ( s ) = E [ Z ( s )] , Γ( s, t ) = Co v [ Z ( s ) , Z ( t )] , σ 2 ( s ) = V ar [ Z ( s )] = Γ( s, s ) , ξ 2 ( s ) = V ar [ Z ′ ( s )] , µ ( s ) = Corr [ Z ( s ) Z ′ ( s )] . L et N z b e the numb er of zer o cr ossings of Z ( t ) . If m ( t ) is c ontinuously differ entiable, Γ( s, t ) has mixe d s e c ond deriva tive s that a r e c ontinuous at t = s and µ ( s ) 6 = 1 at any p oint s ∈ [0 , T ] , then E [ N z ] = Z T 0 ξ ( s ) γ ( s ) σ ( s ) φ m ( s ) σ ( s ) ! Q ( η ( s )) d s , (3) 3 wher e Q ( z ) ≡ 2 φ ( z ) + z [2Φ( z ) − 1] , γ ( s ) 2 = 1 − µ ( s ) 2 , η ( s ) = m ′ ( s ) − ξ ( s ) µ ( s ) m ( s ) /σ ( s ) ξ ( s ) γ ( s ) . By decomp osing (3) in to t w o pieces, we deriv e the follo wing b ounds: Theorem 4 (Alt er nate form) L et the hyp otheses of The or em 3 hold and define M ( t ) ≡ m ( t ) /σ ( t ) . L et | M ( t ) | have N o z zer os, L mx r elative maxima, M j , j = 1 . . . L mx and L mn nonzer o r elative minima, m j 6 = 0 , j = 1 . . . L mn , wher e M (0) and M ( T ) ar e c ounte d as r elative extr ema. L et ν j e qual 1 if m j o c cur s at 0 o r T and ν j = 2 otherwise. Define ˆ ν j similarly for the M j . Equation (3) c an b e r ewritten as E [ N z ] − N 0 z = L mn X j =1 ν j Φ( − m j ) − L mx X j =1 ˆ ν j Φ( − M j ) + Z T 0 ξ ( s ) γ ( s ) σ ( s ) φ m ( s ) σ ( s ) ! ˜ Q ( η ( s )) ds , (4 ) wher e ˜ Q ( z ) ≡ 2 R ∞ | z | φ ( s ′ )[ s ′ − | z | ] ds ′ . Pr o of . W rite Q ( z ) = | z | + ˜ Q ( z ). The first term in (3) equals − R Φ ′ ( M ) | M ′ ( t ) | dt . In tegrat- ing this term yields the w eigh ted sum of the relativ e extrema of Φ( −| M | ( t )). W e decomp ose this sum into N 0 z zeros of | M | ( t ) plus the additional relative extrema: P L mn j =1 ν j Φ( − m j ) − P L mx j =1 ˆ ν j Φ( M j ) . ✷ W e a r e una w are of an y previous deriv a tion of Theorem 4. The second term on the righ t hand side of (4) corresp o nds to the probabilit y that Z ( t ) lac ks a zero of m ( t ) while the the first and third terms corresp ond to extra zeros. Note that ˜ Q ( z ) ≤ φ ( z ) ≤ 1 / √ 2 π . Corollary 5 Under the hyp otheses of The or ems 3 & 4, let { ( x k , w k ) , k = 1 . . . K } b e chosen such that | t − x k | ≤ w k implies that m ′ ( t ) > 0 and | M ( t ) | ≥ cm ′ ( x k ) | t − x k | /σ ( x k ) , wher e c is a fixe d numb er, 0 < c < 1 . Define Ψ k ≡ sup | s − x k |≤ w k { ˜ Q ( s ) ξ ( s ) γ ( s ) σ ( x k ) /σ ( s ) } , C = su p t { ξ ( t ) /σ ( t ) } , and m o ≡ inf { | M ( s ) | for s suc h that | s − x k | ≥ w k , k = 1 . . . K } . The exp e cte d numb er of zer os of the Gaussian pr o c ess, Z ( t ) , satisfies E [ N z ] − N o z ≤ K X k =1 Ψ k cm ′ ( x k ) + O (( C T + 2 L mn ) φ ( m o )) . (5) Pr o of . The first term in (5) a rises from replacing Ψ R x k + w k x k − w k φ  m ( s ) σ ( s )  ds by Ψ R + ∞ −∞ φ  cm ′ ( x k ) s σ ( x k )  ds and in tegra t ing. ✷ A sufficien t additional condition f o r the existenc e of a set of ( x k , w k ) satisfying Corollary 5 is that m ( s ) v anishes only at a finite n umber o f p oin ts, { x k } , and at these p oin ts, m ′ ( x k ) 6 = 0. Let δ b e a small parameter related to the w eakness of the noise amplitude. In man y cases, the { w k } can b e chose n to b e p ow ers of δ and the upp er b ound of (5) r educes to E [ N z ] − N o z ≤ K X k =1 ˜ Q ( x k ) ξ ( x k ) γ ( x k ) cm ′ ( x k ) [1 + o (1) ] . (6) 4 In contrast, a similar naiv e expansion of the original in tegral (3) yields the asymptotic expression: E [ N z ] − N o z ≤ N o z o (1) + K X k =1 ˜ Q ( x k ) ξ ( x k ) γ ( x k ) cm ′ ( x k ) [1 + o (1) ] . (7) The adv an tage of ( 6) ov er (7) is that the remainder term, N o z o (1), ha s b een integrated a w ay . 3 Num b e r of false c hang e p oin ts W e no w consider sequences of k ernel estimates of f ( ℓ ) ( t ), and examine the num b er o f false ℓ -c hange p oints. W e r estrict to indep enden t Gau s s ian errors: ǫ i ∼ N (0 , σ 2 ). Th us, ˆ f ( ℓ ) N ( t ) is a Gaussian pro cess. Mammen et al. (1992,19 95) consider the statistics of c hange p oin t estimation for k ernel estimation of a probability density . W e presen t the analog ous result for regression function estimation. In b o th cases, the ana lysis is based on the Leadb etter- Cry er fo rm ula fo r zero crossings. The f o llo wing a ssumption rules out nong eneric cases: Assumption B L et f ( t ) ∈ C ℓ +1 [0 , 1] have K ℓ - c hange p oints, { x 1 , . . . x K } , with f ( ℓ ) ( x k ) = 0 , f ( ℓ +1) ( x k ) 6 = 0 , f ( ℓ ) (0) 6 = 0 and f ( ℓ ) (1) 6 = 0 . Consi d er a se quenc e of estima tion pr oblems with indep endent, normal ly distribute d me asur ement e rr ors, ǫ N i , with varianc e σ 2 . L et ˆ f ( ℓ ) N ( t ) b e a se quenc e of kernel estimates of f ( ℓ ) , o n the se quenc e of in tervals, [ δ N , 1 − δ N ] . Gasser and M ¨ uller (1 984) ev aluate the v ariance of a c hange p oint estimate: V ar [ ˆ x k − x k ] ≈ σ 2 if ( x k ) ≡ V ar [ ˆ f ( ℓ ) N ( x k )] / | f ( ℓ +1) ( x k ) | 2 . The following theorem b ounds the tail of the empirical c hange p oin t distribution | ˆ x k − x k | > > σ if . By using the L-C in tegral, w e require w eak er conditions than the h yp o theses of G a sser and M ¨ uller (1984). Theorem 6 L et Assumption B hold and c onsider a se quenc e of kernel estimators, ˆ f ( ℓ ) N ( t ) , that s atisfy the hyp otheses of L emma 1. Cho ose kerne l halfw idths, h N , and unc ertainty intervals, w N , such that h N /w N → 0 , w N → 0 , w 2 N ,k N h 2 ℓ +1 N ≥ 1 . Th e p r ob ability, p N ( w N ) , that ˆ f ( ℓ ) N has a false change p oint outside of a width of w N fr om the actual ( ℓ + 1) -change p oints satisfies p N ( w N ) ≤ K X k =1 O σ if ( x k ) h N exp − w 2 N 2 σ 2 if ( x k ) ! ! , (8) wher e σ 2 if ( x k ) → σ 2 k κ ( ℓ ) k 2 . | f ( ℓ +1) ( x k ) | 2 N F ′ ( x k ) h 2 ℓ +1 N on the interval [ h N , 1 − h N ] . Pr o of . Lemma 1 sho ws that ξ N ( t ) /σ N ( t ) → O ( h − 1 N ). Within a neigh b or ho o d of √ w N of x k , E [ ˆ f ( ℓ ) N ( t )] = f ( ℓ +1) ( x k )( t − x k ) + O ( √ w N + D ∗ N /h ℓ +1 N ). D efine b N = inf {| f ( ℓ ) ( t ) | such that t / ∈ ∪ K k =1 ( x k − √ w N , x k + √ w N ) } . Note that b N ≥ C √ w N asymptotically and the in tegral of (3 ) 5 outside of ∪ K k =1 ( x k − √ w N , x k + √ w N ) is b ound by exp( − cw N /σ 2 N ) << exp( − w 2 N / 2 σ 2 if ( x k )). In tegrating the O (1) in tegrand b ound, exp  −| f ( ℓ +1) ( x k ) | 2 | t − x k | 2 / 2 σ 2 N ( x k )  /h N , o ve r the in terv als [ x k ± √ w N , x k ± w N ] yields (8). ✷ Mammen et a l. (19 92,1995) deriv ed the n umber of false c hang e p oin ts for k ernel estima- tion of a probability densit y for nonvanish ing error pro babilities. W e now sho w t hat there expression remains v alid as the error probabilit y go es to zero. Giv en G a ussian measuremen t errors, the sophisticated pro of in Mammen (1995) can b e simplified in our case. Theorem 7 (Analog of Mammen et al. (1992,1995)) L et Assumption B hold. Con- sider a se quenc e of ke rn e l sm o other estimates ˆ f N which satisfy the hyp otheses of L emma 1 with R 1 − 1 sκ ( s ) ds = 0 . L et the se quenc e of kernel halfwidths, h N , satisfy D ∗ N N 1 / 2 h 1 2 N → 0 and 0 < liminf N h N N 1 / (2 ℓ +3) ≤ limsup N h N N 1 / (2 ℓ +3) < ∞ . The exp e cte d numb er of ℓ -change p oints of ˆ f N in the estimation r e gion, [ h N , 1 − h N ] , is asymptotic al ly E [ ˆ K ] − K = 2 K X k =1 H   v u u t | f ( ℓ +1) ( x k ) | 2 N F ′ ( x k ) h 2 ℓ +3 σ 2 k κ ( ℓ +1) k 2   + o R (1) , (9) wher e H ( z ) ≡ φ ( z ) / z + Φ( z ) − 1 wi th φ and Φ b eing the Gaussian density. If f ( ℓ +1) ( t ) h a s H¨ older smo othness of or der ν for some 0 < ν < 1 , and h N N 1 / (2 ℓ +3) → 0 , then (9) r emains valid pr ovide d that h N N 1 / (2 ℓ +3+2 ν ) → 0 . In Mammen (1992,19 9 5), the correction in (9) is sho wn to b e o (1) if limsup N h N N 1 / (2 ℓ +3) < ∞ . W e strengthen t his result b y showing tha t (9) contin ues to represen t the leading order asymptotics ev en when h N N 1 / (2 ℓ +3) → ∞ . Our secret is to use (4) instead of (3) b ecause (4) has in tegra t ed out the t erm equal to K . Pr o of of The or e m 7. Theorem 6 sho ws tha t the contribution aw a y from the ℓ -c hange p o in ts is exp o nen tially small for | s − x k | > > σ N ( s ). Lemma 1 sho ws that ξ N ( s ) γ N ( s ) σ N ( s ) → k κ ( ℓ +1) k h N k κ ( ℓ ) k and that for | s − x k | << 1, η N ( s ) → f ( ℓ +1) ( s ) /σ N ( s ). Equation (9) is an appro ximation of ( 4 ) using La pla ce’s metho d. T o prov e (9) , w e must sho w t ha t E [ ˆ f ( ℓ ) N ( t )] = f ( ℓ ) ( t ) + o R ( σ N ) for | t − x k | ∼ σ N . Near the change p oint, x k , E [ ˆ f ( ℓ ) N ( t )] = f ( ℓ ) ( t ) + Z 1 − 1 κ ( s ) h f ( ℓ ) ( t + h N s ) − f ( ℓ ) ( t ) i ds + O R ( D ∗ N /h ℓ +1 N ) = f ( ℓ ) ( t ) + h N Z 1 − 1 sκ ( s ) h f ( ℓ +1) ( t + h N τ N ( s )) − f ( ℓ +1) ( t ) i ds , (10) where τ N ( s ) lies in [0 , s ] by the mean v alue theorem. Since f ( ℓ +1) ( t ) is con tinuous at x k , for eac h δ , there is a ˜ h N ( δ ) suc h t ha t | f ( ℓ +1) ( t + h N τ N ( s )) − f ( ℓ +1) ( t ) | < δ for all t , t + h N τ N ∈ [ x k − ˜ h N ( δ ) , x k − ˜ h N ( δ )]. Th us E [ ˆ f ( ℓ ) N ( t )] = f ( ℓ ) ( t ) + O R ( δ h N + D ∗ N /h ℓ +1 N ). Here δ may b e ta ken arbitrarily small. Applying the Laplace’s metho d yields (9) with 6 corrections of O R (exp( − δ h N /σ if ) − 1) + O R  exp( − D ∗ N /h ℓ +2 N σ if ) − 1  . The scaling, h N ∼ N − 1 / (2 ℓ +3) , implies that the first t erm is O R ( δ ). The discrete sampling effect (the second term) requires the h yp othesis that D N √ h N N → 0 to b e o R (1). When f ( ℓ +1) ( t ) is H¨ older of order ν , w e hav e the stronger b ound: | f ( ℓ +1) ( t + h N τ N ( s )) − f ( ℓ +1) ( t ) | < C t h ν N , and E [ ˆ f ( ℓ ) N ( t )] = f ( ℓ ) ( t ) + O R ( h 1+ ν N + D ∗ N /h ℓ +1 N ). The next order correction in Laplace’s metho d is O R  exp( h 1+ ν N /σ if )  . This term is o R (1) when h N N 1 / (2 ℓ +3+2 ν ) → 0. ✷ In R iedel (1996), we prop ose a tw o-stage nonpara metric function estimator whic h ac hiev es the correct shap e with high probability . In the first stag e, w e estimate the num ber and appro ximate lo cations of the ℓ -c hange p oint using a pilot estimate with la rge smo o t h- ing. In the second stage, the smo othing is r educed, but w e imp ose the shap e restrictions obtained from the pilo t estimate. Theorems 6 and 7 imply t hat if the kernel half width of the pilot estimator satisfies h N >> ln[ N ] N − 1 / (2 ℓ +3) , then spurious inflection p oin ts will o c- cur with a probability smaller than N c for an y c . T o achie ve this result, w e use an a lternate form of the Leadb etter-Cryer in tegra l to remo ve the N z o (1) from (7). References [1] Cram ´ er, H. and Leadb etter, M. R. (1 967), S tationary a n d r elate d pr o c ess e s , John Wiley , New Y ork. [2] Gasser, Th. and M ¨ uller, H., Estimating regression functions and their deriv ativ es by the k ernel metho d, Sc and. J. of Stat. 11 (19 84), 171–18 5. [3] Leadb etter, M. R. and Cry er J. D. (1965 ) Curv e crossings by normal pro cesses, A nn. Math. Stat. 36 , 509-516 . [4] Mammen, E., Marron, J. S. and Fisher, N. J. ( 1992), Some asymptotics fo r m ultimo da l tests based on k ernel densit y estimates, Pr ob. Th. R el. Fields 91 11 5-132. [5] Mammen, E. (1995) On qualitativ e smo othness of k ernel densit y estimates, S tatistics , 26 253-267. [6] Niederrieter, H. (1992 ) R andom Numb er Gener ators and Quasi-Monte Carlo Metho ds , SIAM, Philadelphia, P A. [7] Riedel, K. S. (1996), Piecewise con v ex function estimation I: pilot estimators. Submit- ted for publication. 7