Unified estimation of densities on bounded and unbounded domains

Transcription

1 Unified estimation of densities on bounded and unbounded domains Kairat Mynbaev International Scool of Economics Kazak-Britis Tecnical University Tolebi 59 Almaty 5, Kazakstan kairat and Carlos Martins-Filo Department of Economics IFPI University of Colorado 233 K Street NW Boulder, CO , USA & Wasington, DC 26-2, USA carlos.martins@colorado.edu c.martins-filo@cgiar.org Marc, 27 Abstract. Kernel density estimation in domains wit boundaries is known to suffer from undesirable boundary effects. We sow tat in te case of smoot densities, a general and elegant approac is to extend te density and estimate its extension. Te resulting estimators in domains wit boundaries ave biases and variances expressed in terms of density extensions. Te result is tat tey ave te same rates at boundary and interior points of te domain. Contrary to te extant literature, our estimators require no kernel modification near te boundary and kernels commonly used for estimation on te real line can be applied. Densities defined on te alf-axis and in a unit interval are considered. Te results are applied to estimation of densities tat are discontinuous or ave discontinuous derivatives, were tey yield te same rates of convergence as for smoot densities on. Keywords and prases. Nonparametric estimation, Hestenes extension, estimation in bounded domains, estimation of discontinuous densities. AMS-MS Classification. 62F2, 62G7, 62G2.

2 Introduction Kernel estimation of densities on te real line is a well-developed area. Te core of te teory is a series of results covering smoot densities tat do not exibit extreme curvature. Let K denote a kernel, an integrable function on, wic satisfies K(tdt =, > be a bandwidt and f be a density on. Assuming tat {X i } n i= is an independent and identically distributed (IID sample from f, te traditional osenblatt-parzen kernel estimator of f(x is defined by ˆf (x = n n i= K ( x X i. Tis estimator as tree desirable caracteristics: tere exists a great profusion of kernels tat can be used to construct te estimator (usually, tey are from te Gaussian or Epanecnikov families; tey are usually symmetric and do not depend on te point (x of estimation, or on te class of densities being estimated; 2 tere is a simple link between te degree of smootness of te density and te order of estimator s bias: if f Cb s (Ω and te kernel is of order s, ten E ˆf(x f(x = O( s. Te use of iger order kernels in te case of smoot densities is also a standard feature; 3 te optimal bandwidt is of order n /(2s for all estimation points, unless tere are areas of extreme curvature or discontinuities. In cases were te domain of f as a boundary, te main problem is bad estimator beavior in te vicinity of te boundary. Tis problem called into being a range of estimation metods. Among te widely used ones are te reflection metod, te boundary kernel metod, te transformation metod and te local linear metod (see, inter alia, Scuster (985, Karunamuni and Alberts (25, Malec and Scienle (24, Wen and Wu (25 and teir references. Oter metods ave proposed te use of asymmetric kernels and kernel adjustments near te boundary. Suc tecniques necessarily require variable bandwidts, separation of densities into subclasses tat vanis or not at te boundary, densities tat ave derivatives of a certain sign at te boundary, etc. Te difficulties of estimation near te boundary precluded researcers from identifying a core class for wic analogs of te standard results mentioned above would be true. In particular, we ave not seen in te literature results tat would guarantee a better bias rate for densities of iger smootness. In tis paper we propose estimation procedures tat permit a unified teoretical study of teir properties Let s N and Ω. Te class of functions f : Ω wic are s-times differentiable wit f (s (x C for some < C < is denoted by C s b (Ω. We say tat te kernel K is of order s 2 if t j K(tdt = for j =,, s and t s K(tdt.

3 under bounded and unbounded domains. We sow tat smootness is all one needs to ave a good bias rate, and for smoot densities te beavior at te boundary is irrelevant (derivatives at endpoints are one-sided derivatives. For densities on te alf-axis [, and on te unit interval (, we introduce new estimators for wic all standard facts old. Usual symmetric kernels and constant bandwidts can be used across te domain and for f Cb s(ω te biases of our estimators are of order O(s. Te bandwidt depends on te sample size in te same way as in case of estimation on te wole line. In te case of estimation of piece-wise continuous densities, wit known discontinuity points, our estimators supply te required jumps at tose points. As in, te estimator for densities in classes were s > 2 is not necessarily nonnegative, because te estimation involves iger-order kernels. Our metod does not cover densities wit poles at endpoints. Our estimation metod is based on Hestenes extension (Hestenes (94. Let D f be te domain of te density f and denote by g its Hestenes extension (te definitions for te alf-axis and interval are given below in te respective sections. Te key observation is tat g can be viewed as a linear combination of densities. Te sample generated from f is used to estimate eac of tese densities and te linear combination of te estimators estimates g. Te restriction of te estimator of g to D f estimates f. We sow tat te teory of estimation on a domain wit boundaries for smoot densities in effect becomes a capter in estimation on te wole line. Te essential link between te proposed estimator ˆf(x of f(x and te properties of g is of type E ˆf(x f(x = K(t (g(x t g(x dt, x D f. Tis representation as eluded previous work, and can be used for evaluating bias. Our estimation procedure does not require knowledge of g. Tere seems to be a sligt loss in te speed of convergence as compared to convergence on te line because te same data is exploited more tan once to estimate different parts of g. However, tis loss does not affect te rate in E ˆf(x f(x = c s o( s ; it affects only te constant c, in comparison wit te classical estimator for densities on te line. In section 2, we start wit estimation of a density on [,. Section 3 treats densities on a bounded interval. In section 4, te approac is extended to estimation of discontinuous densities. Section 5 provides two metods to satisfy zero boundary conditions. 2

4 2 Estimation of densities defined on [, Let w,..., w s be pairwise different positive numbers for s =,,. Of special interest are te decreasing sequence w i = /i, i =,..., s (used by Hestenes (94 and te increasing sequence w i = i. Let te numbers k,..., k s be defined from te following system s ( w i j k i =, j =,..., s. (2. Since tis system as te Van-der-Monde determinant... w w 2... w s , ( w s ( w 2 s... ( w s s i= k,..., k s are uniquely defined. If f as [, as its domain, its Hestenes extension for x < is defined by s φ s (x = k j f( w j x, x <. (2.2 j= φ s is not a density, but it is a linear combination of densities w j f( w j x wit coefficients k j /w j. Assuming tat f as s rigt-and derivatives f(,..., f (s ( (s = means continuity, we see tat te following sewing conditions at zero are satisfied due to (2.: s φ (m s ( = ( w j m k j f (m ( = f (m (, m =,, s. j= Now define g on by g(x = { f(x, x φ s (x, x <, (2.3 wit g being s times differentiable. Moreover, if, for example, f belongs to te Sobolev space W s p ([,, ten g belongs to W s p (, were p < (see Burenkov (998. Suppose f is s times differentiable, m =,,, s, te kernel K is m times differentiable, and let {X i } n i= be an IID sample from f. Te estimator of f (m (x, for x >, is defined by ˆf (m n ( x (x = K (m Xi s ( k j x n m K (m Xi /w j. (2.4 w j i= j= 3

5 Wen te kernel K is an even function and m = s =, ˆf ( (x ˆf S (x is te reflection estimator from Scuster (985, i.e., ˆf S (x = n n i= [ K ( x Xi K ( x Xi ]. Te next assumption is used only for m, wen integration by parts is needed. Assumption 2.. a K is even, m times differentiable and max f (j (x = O(x as x. j m max j m K(j (t t = o( as t ; b Te estimator in (2.4 can be constructed using kernels in te class {M k (x} k N proposed by Mynbaev and Martins-Filo (2, were M k (x = C k 2k k l = ( l C lk ( 2k x K l l wit C l 2k = 2k! (2k l!l! for l =,, 2k. In tis context, K is called te seed of M k. Tese kernel are used togeter wit an order 2k finite difference 2k g(x = k ( lk C lk g(x l l = 2k wen Besov type norms are employed to measure smootness (see Mynbaev and Martins-Filo (2. We let (m ˆf k (x denote te estimator defined in (2.4 wit K replaced by M k. Note tat if K is even, ten M (x = K(x and ˆf (m (x = ˆf (m (x. Teorem 2.. Suppose f is s-times differentiable on D f = [, and te kernel K is m times differentiable on wit m =,,, s. In case m suppose tat Assumption 2. olds. Ten, Te bias of ˆf (m (x as te representation E ˆf (m (x f (m (x = [ ] K(t g (m (x t g (m (x dt, x D f. (2.5 2 If in (2.4 a kernel M k wit seed K is used, ten E (m ˆf k (x f (m (x = ( k C2k k K(t 2k tg (m (xdt, x D f. (2.6 4

6 Proof. By te IID assumption ( E ˆf (m x (x = m E K (m X ( = x t m K (m In te first integral let u = x t E ˆf (m (x = m = m s j= k j w j K (m s f(tdt w j= j, in te oters u = xt/wj. Ten x/ x/ ( x X /w j k j s K (m (u f(x udu k j j= K (m (u f(x udu x/ x/ s K (m (u ( x K (m t/wj f(tdt. (2.7 K (m (uf ( w j (x u du j= k j f ( w j (x u du. In te first integral we ave x u > and f(x u = g(x u; in te second one x u <, so s j= k jf ( w j (x u = g(x u. Hence, E ˆf (m (x = m K (m (u g(x udu. (2.8 By Assumption 2. K (j (ug (m j (x u = o(, as u for j =,..., m, >. Terefore, integration by parts gives te following expression for (2.8 E ˆf (m (x = = m j= m j K(m j (ug (j (x u K(ug (m (x udu K (u g (m (x udu. (2.9 Since K(tdt =, tis implies ( Plug te definition of M k in (2.7 to get (m E ˆf k (x = k ( l C lk 2k C2k k l l m m l = ( x t K (m l s f(tdt k j w j= j ( x K (m t/wj f(tdt. l (2. 5

7 For l <, ( x t s K (m f(tdt l (replacing x t l = l x/(l k j w j= j = u and x t/w j l = u s K (m (u f(x ludu = l K (m (u g(x ludu. Similarly, we ave for l > ( x t s K (m f(tdt l Terefore, (2. gives Finally, wic is (2.6. E k j w j= j ˆf (m k (x = C k 2k j= ( x K (m t/wj f(tdt l x/(l k j K (m (u f( w j (x ludu ( x K (m t/wj f(tdt = l K (m (u g(x ludu. l k l = ( l C lk 2k (l m K (m (u g(x ludu (integrating by parts as above = k C2k k ( l C lk 2k K (u g (m (x ludu. l = (m E ˆf k (x f (m k (x = ( k C2k k ( lk C lk 2k K (u g (m (x ludu l = ( k C2k k ( k C2k k K (u g (m (xdu = ( k C2k k K(u 2k ug (m (xdu Te integral representations for biases obtained in Teorem 2. depend on te extension g, not te density f. Consequently, existing results for smoot functions (not densities on allow us to easily obtain bias estimates. If classical smootness caracteristics in terms of derivatives and Taylor expansions are used, ten part of Teorem 2. is relevant. Tis approac can be used for derivatives of orders m s wen te bias order is O( s m and guaranteed to tend to zero as. If, on te oter and, smootness is caracterized in terms of finite differences and Besov spaces, ten te second representation sould be 6

8 applied. It is appropriate for m = s or m = s wen te derivative of order s may ave a residual fractional smootness of order < r <. For p, q and Ω an open subset of put 2k,Ωf(x = 2k f(x if [x k, x k] Ω and 2k,Ωf(x = oterwise and let f b r p,q (Ω = ( Ω,Ω f(x p /p q dx d r 2k were k is any integer satisfying 2k > r, and in case p = and/or q = te integral(s is (are replaced by sup. Furter, f B r p,q (Ω = f b r p,q (Ω f L p(ω. Te Hestenes extensionis known to be bounded from B r p,q(ω to B r p,q(. /q Assumption 2.2. For m s, f (m b r < wit some r > and q and,q (, ( q K(t q t (r/qq dt < were /q /q =. Teorem 2.2. Let Assumption 2. old and assume tat K(ttj dt =, for j =,..., s m, K(tts m dt < and f (s (x < C for all x >, ten E ˆf (m (x f (m (x = O( s m for all x D f. (2. 2 Let f and K satisfy Assumption 2.2, ten E ˆf (m k (x f (m (x = O( r for all x D f. (2.2 Proof. For part, we note tat since K(tdt = we ave from Teorem 2. ( E ˆf (m (x f (m (x = K(t(g m (x t g (m (xdt = K(t g (m (x( t 2! g(m2 (x( t 2 (s m! g(s (x tτ( t s m dt for some τ (,. If K(ttj =, for j =,..., s m ten E ˆf (m (x f (m (x s m (s m! t s m K(t g (s (x tτ dt C s m, 7

9 were te last inequality follows from te assumptions tat K(tts m dt <, f (s (x < C for all x > and te structure of g (s. For part 2, using (2.6 and Hölder s inequality we ave (m E ˆf k (x f (m (x = c K(t t r/q 2k t g(m (x dt t r/q ( /q c K(t q t (r/qq dt sup 2k x t r t g(m (x (canging variables on te second integral ( /q c r g K(t q t (r/qq dt (m = O( r. b r,q ( q dt t /q In te last line we used te bound g (m B r p,q ( c f (m B r p,q (,. From now on we give just expressions for bias and variance leaving consequences of type (2. and (2.2 to te reader. Teorem 2.3. Suppose tat f is continuous, sup f(x <, sup K (m (x <, x x K (m (t 2 dt and K (m (t (m dt <. In case m > also let Assumption 2. old. Denote F (t = M k (t. Ten ( { } (m V ˆf k (x = n 2m f(x F 2 (tdt o(, x >. (2.3 Te estimator ˆf (m (x as a similar property wit F (t = K (m (t in place of M (m k (t. Proof. Denoting u i = m [ F ( x X i s k j j= w j F ( ] xxi/w j we ave ˆf (m k (x = n n i= u i and given te IID assumption V ( (m ˆf k (x = [ Eu 2 n (Eu 2]. (2.4 Let w = 2, k =, w =, k = (te value of w does not matter. Ten Eu 2 = E s ( k j x X /w j m F 2 w j = 2m2 j= s i,j= k i w i k j w j ( x t/wi F F ( x t/wj f(tdt. (2.5 Denote by B(x, r = {t : x t r} te ball centered at x wit radius r. For any positive ε tere is a > suc tat F (u du < ε. Note tat u >a B i { t : x t/w i } a = {t : w i x t w i a} = B( w i x, w i a. 8

10 Since all w i are different and x >, te balls B i do not overlap for small. Let be tat small and denote F i (t = F ((x t/w i /. Ten using = B i B c i (ere c stands for te complement we ave F i (t F j (t f(tdt sup f B i B c i F i (t F j (t dt [ = c B i (B j Bj F i (t F j (t dt c (using B i B j =, B i Bj c Bj c ( c 2 F j (t dt F i (t dt. Bj c Bi c B c i F i (t F j (t dt ] Furter, B c i F i (t dt = w ixt > w i a It follows tat F i (tf j (tf(tdt = o( and from (2.5 Eu 2 = = = 2m2 2m2 2m [ s i= [ ( ki w i ( x F t/wi dt = w i F (u du = O(ε. u >a 2 ] Fi 2 (t f(tdt o( F 2 ( x t s f(tdt [ x/ F 2 (u f(x udu ( ki w i= i s ( ki i= 2 w i 2 ( ] x F 2 t/wi f(tdt o( F 2 (u f( w i (x udt o( x/ ]. Defining g (x = { f(x, x > ( 2 s k i i= w f( wj i x, x < (2.6 we can write Eu 2 = [ ] 2m F 2 (u g (x udu o(. (2.7 g is not a smoot extension of f but it is integrable on and continuous at x >. Terefore F 2 (u g (x udu g (x F 2 (u du = f(x F 2 (u du. Teorem 2.2, (2.4 and (2.7 finis te proof. 9

11 3 Estimation on a bounded interval Let f be defined on D f = (, and let te vectors w, k be as before. We would like to extend f to te left of zero using (2.2. To obtain a common domain for te components of φ, we put a = min i (/w i and let s φ (x = k j f( w j x, a < x <. Te sewing conditions at are satisfied as before. Put s φ 2 (x = k j f( w j (x, < x < a, j= j= and define te extension by Te sewing condition olds at x = : φ (x, a < x < g(x = f(x, < x < φ 2 (x, < x < a (3. s 2 ( = ( w m j k m f (j ( = f (j (, j =,..., s. φ (j m= Suppose f is s times differentiable, m is an integer, m s, te kernel K is m times differentiable, and let X,..., X n be an IID sample from f. Te estimator of f (m (x, x (,, is defined by ˆf (m n ( x (x = n m K (m Xi s k j ( x K (m Xi /w j w i= j= j X i<aw j ( x (Xi /w j. (3.2 X i> aw j K (m Teorem 3.. Let f be s times differentiable on D f = (, and let K be an m times differentiable kernel wit finite support, m s. Let > be small (specifically, it sould satisfy te condition suppk ( a/, a/. Ten, te following statements old: For a classical kernel K (2.5 is true. 2 If in (3.2 K is replaced by M k, ten (2.6 is true. Proof. Let I A denote te indicator of a set A. Ten, for an arbitrary function g, X i<aw j g(x i =

12 n i= I {X i<aw j}g(x i. Using indicators in (3.2 and te fact tat {X i } n i= is IID, we ave E ˆf (m (x = m Canging variables using x t E ˆf (m (x = m (x / aw j K (m xt/wj = u, (x / x/ (x a/ Applying (3., we ave E ˆf (m (x = m x/ (x / ( x t K (m s f(tdt ( x (t /wj = u, x (t /wj k j w j= j f(tdt [ awj ]} = u, we ave [ s (xa/ K (m (u f(x udu k j j= K (m (u f( w j (x u du (x / (x a/ K (m (u f(x udu ]}. (xa/ x/ ( x K (m t/wj f(tdt. (3.3 x/ s K (m (u k j f( w j (x u du j= K (m (u f( w j (x udu s K (m (u k j f( w j (x udu = (xa/ m K (m (u g(x udu. (3.4 (x a/ egardless of x (,, te interval ((x a/, (x a/ contains ( a/, a/ wic contains suppk j= for all small. Terefore, E ˆf (m (x = m K (m (u g(x udu. For tis to old formally, g sould be extended outside ( a, a smootly; te manner of extension does not affect te above integral. Finally, integration by parts and te condition K(tdt = prove te statement. 2 Since K is assumed to ave finite support, we do not need Assumption 2.. Calculations done in te proof of Teorem 2.2 afterequation (2. include cange of variables and integration by parts and can be easily repeated ere. emark. Instead of requiring K to ave compact support one can define te extension so tat it is sufficiently smoot and as compact support. Take a smoot function suc tat (x = on ( a/2, a/2 and (x = for x outside ( a, a. Instead of (3. consider te extension g (x = (xg(x and cange

13 (3.2 accordingly. Ten te statement of Teorem 3. will be true for g witout te assumption tat K as compact support. Wen m = and integration by parts is not necessary, te function does not ave to be smoot. g can be extended by zero outside ( a, a or, equivalently, one can take (x = on ( a, a and (x = for x outside ( a, a. Teorem 3.2. Under conditions of Teorem 3. te following is true. For a classical kernel K were F (x = K (m (x. ( { } V ˆf (x (m = n 2m f(x F 2 (tdt O(, x D f. 2 If M k is used in place of K, ten te same asymptotic expression is true wit F (x = M (m k (x. Proof. Define u i = m K(m ( x Xi s k j w j= j I {Xi> aw j}k (m ( x (Xi /w j [ ( x I {Xi<aw j}k (m Xi /w j ]}. Ten we can use equations similar to (2.4 and (2.5. We want to sow tat in te analog of (2.5 all cross-products vanis if is small. Suppose suppk ( b, b for some b >. In te cross-products we ave integrands tat include products of functions ( ( ( x t x f (t = K (m, f j (t = K (m t/wj x (t, g j (t = K (m /wj. It is easy to see tat suppf B(x, b, suppf j B( w j x, w j b, supp g j B( w j (x, w j b. Since te radii of te balls ere tend to zero, for small tey do not overlap if teir centers are different. It is easy to sow tat all te centers are indeed different using te conditions tat < x <, w j > for all j and all w j are different. 2

14 Tus, for small Eu 2 = { ( x t 2m2 F 2 f(tdt F 2 aw j ( x (t /wj s ( kj j= w j f(tdt 2 [ awj ]}. ( x F 2 t/wj f(tdt Define ( 2 s k i i= w f( wi i x, a < x < g (x = f(x, x (, ( 2 s k i i= w i f( wi (x, < x < a epeating te canges of variables tat led from (4.2 to (5. and recalling tat suppk ( a/, a/ (3.5 we obtain Eu 2 = (xa/ 2m F 2 (u g(x udu = (x a/ 2m F 2 (u g(x udu. Overall, using Teorem 3.. ( V ˆf (x (m = n = [ 2m n 2m Te proof for M k is te same. ( f(x ( F 2 (u g(x udu F 2 (u du O(. 4 Estimation of smoot pieces of densities m ] 2 K (m (u g(x udu Ideas developed in te previous sections can be applied to estimation of densities wit discontinuities or wit discontinuous derivatives. Here we provide two results. Cline and Hart (99 used Scuster s symmetrization device to improve bias around a discontinuity point. Te first result in tis section applies, for example, to te Laplace distribution wic is continuous everywere but as a discontinuous derivative at zero. Te usual kernel density estimator on te wole line will inevitably ave a large bias at zero. Te suggestion is to estimate its smoot restrictions f and f on te rigt alf-axis (, and left alf-axis (,. As a second example, consider a piece-wise constant density on te interval (,. Te restriction of te density on eac interval were it is constant is smoot 3

15 and can be estimated using our approac. Obviously, te jumps of te estimators will estimate te jumps of te density. f and f do not need to ave te same degree of smootness. Suppose tat te rigt part f is s times differentiable and m s. Te estimator of f (m (x, x >, is defined by ˆf (m ( x (x = K (m Xi s ( k j x n m K (m Xi /w j. w j X i> Teorem 4.. In Teorem 2. and in definition (2.3 let f = f and D f = (,. If te conditions of j= Teorem 2. are satisfied for f and K, ten (2.5 and (2.6 are true and 2 for te variance of ˆf (m (x one as (2.3. Proof. Instead of (2.7 we ave (m ( x E ˆf (x = m E I {X >} K (m X ( = x t m K (m s j= s f (tdt epeating calculations tat led from (2.7 to (2.9 we get E k j w j= j k j w j K (m (m ˆf (x = K (s g (m (x sds (tose calculations did not use te fact tat f was a density. ( x X /w j ( x K (m t/wj f (tdt. 2 Similarly, replacing in (4.3 f by f and repeating te argument of Teorem 2.3 we see tat were F (t = K (m (t. V ( { } (m ˆf (x = n 2m f (x F 2 (tdt o(, x >, Now suppose tat te domain D f of a density f contains a finite segment (c, d suc tat te restriction f r of f onto (c, d is smoot. Denote s φ (x = k j f r (c w j (x c, c a < x < c, j= 4

16 te extension of f r to te left of c and s φ 2 (x = k j f r (d w j (x d, d < x < d a, j= te extension of f r to te rigt of d. Here we coose a = a(d c, to make sure tat c w j (x c and d w j (x d belong to (c, d. Te extended restriction ten is defined by φ (x, c a < x < c, g r (x = f r (x, c < x < d, φ 2 (x, d < x < d a. (4. Definition (3.2 guides us to define ˆf (m (x = n m { s k j w j= j c<x i<d d a w j<x i<d ( x K (m Xi c<x i<ca w j K (m ( x c (Xi c/w j K (m ( x d (Xi d/w j, x (c, d. Teorem 4.2. Let f r be s times differentiable, m s, and let K ave compact support. For sufficiently small (suc tat suppk ( a /, a / we ave E ˆf [ (m (x f r (m (x = K(u g (m r ] (x u g r (m (x du, c < x < d. (4.2 Furter, ( { } V ˆf (x (m = n 2m f(x F 2 (tdt O(, c < x < d, were F (t = K (m (t or F (t = M (m k (t depending on wic kernel is used in te definition of ˆf (m (x. Proof. By te i.i.d. assumption E ˆf (m (x = { d ( x t m K (m f r (tdt c s k j w j= j d [ caw j c d a w j K (m K (m ( wj (x c (t c w j ( wj (x d (t d w j f r (tdt f r (tdt ]}. 5

17 Te obvious canges of variables are: x t = u, w j (x c (t c w j = u, w j (x d (t d w j = u. Te mean value becomes { E ˆf (m (x = m s (x d/ (x c/ K (m (u f r (x udu [ (x ca/ k j K (m (u f r (c w j (x c w j udu j= (x c/ (x d/ (x d a / Applying (4. we see tat tis is te same as K (m (u f r (d w j (x d w j udu { E ˆf (m (x = (x c/ m K (m (u f r (x udu (x d/ = m (x ca/ (x c/ (x d/ (x d a / (x ca/ (x d a / ]} s K (m (u k j f r (c w j (x u cdu j= s K (m (u j= k j f r (d w j (x u ddu K (m (u g r (x udu. egardless of x (c, d, te interval ((x d al/, (x c a/ contains ( a/, a/ wic contains suppk for all small. Terefore, also integrating by parts, E ˆf (m (x = m K (m (u g r (x udu = K (u g r (m (x udu. Te derivation of te expression for variance largely repeats tat from Teorem 3.2. Define. u i = { ( x m I {c<xi<d}k (m Xi s k j w j= j [ ( x c I {c<xi<ca w j}k (m (Xi c/w j I {d aw j<x i<d}k (m ( x d (Xi d/w j ]}. Ten we can use equations similar to (2.4 and (2.5. 6

18 We want to sow tat in te analog of (2.5 all cross-products vanis if is small. Suppose suppk ( b, b for some b >. In te cross-products we ave integrands tat include products of functions It is easy to see tat ( x t f (t = K (m g j (t = K (m ( x d (Xi d/w j ( x c, f j (t = K (m (Xi c/w j., suppf B(x, b, suppf j B(c w j (x c, w j b, supp g j B(d w j (x d, w j b. Since te radii of te balls ere tend to zero, for small tey do not overlap if teir centers are different. It is easy to sow tat all te centers are indeed different using te conditions tat c < x < d, w j > for all j and all w j are different. Tus, for small Eu 2 = { d ( x t 2m2 F 2 f r (tdt c s ( 2 [ caw kj j j= d w j d a w j F 2 ( x c (t F 2 c/wj c ( ]} x d (t d/wj f r (tdt. f r (tdt Define ( 2 s k i i= w i f(c wi (x c, c a < x < c gr(x = f(x, x (c, d ( 2 s k i i= w i f(d wi (x d, d < x < d a epeating te canges of variables tat led from (4.2 to (5. and recalling tat suppk ( a /, a / (4.3 we obtain Eu 2 = 2m (x ca/ (x d a / F 2 (u gr(x udu = 2m F 2 (u gr(x udu. Overall, combining tis wit te bias estimate 4.2 we obtain [ ( V ˆf (x (m = n 2m ( = n 2m f r (x ( F 2 (u gr(x udu F 2 (u du O(. m ] 2 K (m (u gr(x udu 7

19 5 Estimators satisfying zero boundary conditions For simplicity we consider only densities on D f = (,. For estimator (2.4 we provide two modifications designed to satisfy zero boundary conditions for te estimator itself and/or its derivatives. In bot cases te bias rate is retained. Te first estimator also reduces te variance near zero. Te main difference between te estimators is in te number of derivatives tat are guaranteed to vanis. Everywere it is assumed tat f is s times differentiable, m s and te purpose is to estimate f (m (x. Let l be an integer between m and s and let ψ be a function on D f wit properties ψ( =... = ψ (l m ( =, ψ (l m (, (5. ψ(x = for x, ψ(x everywere. If f (m ( =... = f (s ( =, (5.2 put α =. Oterwise, let k be te least integer suc tat m k s, f (k ( and put α = s m k m. For any estimator ˆf (m (x of f (m (x define anoter estimator f (m (x = ψ(x α ˆf (m (x. Teorem 5.. Let te estimator ˆf (m (x of f (m (x satisfy (2.. Ten f (m (x satisfies f (m ( =... = f (l ( =, (5.3 E f (m (x f (m (x = O( s m for all x D f. (5.4 and ( ( V ˆf (x V f (x (m, x α (m = ( V ˆf (x (m (x α 2(l m, x < α. (5.5 Proof. (5. implies ( j d f (m (x x= = dx for j =,..., l m, so (5.3 is satisfied. To prove (5.4, consider two cases. j i= C i j [ ( ] [ i ( j i d d ψ(x α ˆf (x] (m x= =, dx dx Case x α. (5.4 follows trivially from (2. because f (m (x = ˆf (m (x. 8

20 Case x α. Obviously, in te equation E f [ (m (x f (m (x = ψ(x α E ˆf ] (m (x f (m (x [ ψ(x α ] f (m (x te first term on te rigt is O( s m, and it remains to prove tat [ψ(x α ] f (m (x = O( s m. Suppose (5.2 is true, so tat α =. Ten f (m (x = f (m (... f (s ( xs m (s m! o(xs m = o ( ( α s m = o( s m. (5.6 Suppose (5.2 is wrong. Ten α = s m k m and f (m (x = f (m (... f (s ( xk m (k m! o(xk m = O ( x k m = O( s m. (5.7 (5.6 and (5.7 prove wat we need. To prove (5.5, consider two cases. Case x α. Te first part of (5.5 is obvious because f (m (x = ˆf (m (x. Case x α. From (5. it follows tat ψ(x α = ψ(... ψ (l m ( (x α (l m (l m! ( x α (l m wic proves te second part of (5.5. Let ψ be a function wit properties: ψ is m times differentiable on D f, ψ(x = for x (, 2], ψ(x = for x 3, ψ(x everywere. Define ˆf (m n ( x (x = n m ψ(x i /x K (m Xi s ( k j x K (m Xi /w j. w j i= Teorem 5.2. All derivatives of ˆf (m (x tat exist vanis at zero and E ˆf (m (x f (m (x = [ K(t g (m x j= ] (x t g x (m (x dt, x D f, (5.8 ( were g x is te Hestenes extension of f x (t = f(tψ (t/x. Besides, V ˆf (x (m satisfies (2.3. 9

21 Proof. For almost all samples min i X i > and for < x < 3 min i X i one as ψ(x i /x =, i =,..., n. Hence ˆf (m (x vanises, togeter wit all its derivatives, in te neigborood of zero for almost all samples. Following (2.7 we see tat te mean is E ˆf (m (x = ( x t s ( K (m k j x m K (m t/wj ψ w j j= ( t f(tdt. x Here te function f x (t = f(tψ (t/x as support suppf x [, 3]. Implementing canges applied after (2.8, including integration by parts, we obtain an analog of (2.9 wit g x instead of g. g x is obtained by replacing f in (2.2-(2.3 by f x. Te rest is familiar. epeating te calculations from te proof of Teorem 2.3 wit f x in place of f and using f x (x = f(x we obtain ( Simulations We conducted a series of simulations to provide some evidence on te finite sample performances of our estimators and to contrast tem wit tat of some of te most commonly used estimators for densities wit supports tat are subsets of. We focus on two broad cases: first, we consider densities tat are defined on [, ; second we consider te case of a density wit a discontinuity at x =. In te first case we considered random variables wit te following densities:. Normal density left-truncated at x = : f T N (x = 2 2π exp( 2 x2, 2. Gamma density: f G (x = β α Γ(α xα exp( β x wit α = β =, 3. Ci-squared density: f χ (x = 2 v/2 Γ(v/2 xv/2 exp( 2x wit v = 5, 4. Exponential density: f E (x = λexp( λx wit λ =. For eac density we generated samples of size n = 25, 5 and calculated te following estimators: ˆf, ˆfS and ˆf k for k =, 2, 3, w i = i, i and s =, 2. In eac case we used a Gaussian kernel and a Gaussian seed kernel, as necessary. We also calculated te Gamma kernel estimator of Cen (2, wic we denote by ˆf C. For eac estimator we selected an optimal bandwidt by minimizing te integrated squared error over 2

22 a fixed grid on te interval (, 4 wit step 3. Figure gives a set of estimates for one of te generated samples of size n = 25 associated wit te Gamma density. For eac sample, and eac estimator, a root average squared error (ASE over te grid is calculated. Te average of ASE across all generated samples are reported on Table. As expected, te ASE, for eac estimator and across all densities, decreases as n increases from 25 to 5. Except for data generated from te f χ density, all estimators tat are based on Hestenes extension and constructed using te M k kernels (including te case were k = and M = K ave smaller ASE wen w i = i. Also, for estimators ˆf 2 and ˆf 3, coosing s = reduces ASE (compared wit s = 2 for all densities, except f G and f E wen n = 5. Except for te case of te truncated normal density - f T N - tere is always a Hestenes based estimator tat outperforms te estimator ˆf S proposed by Scuster (985. In fact, if we coose w i = i, all Hestenes based estimators ave smaller ASE tan tat of ˆf S. For all densities, tere is always a Hestenes based estimator tat outperforms te estimator ˆf C proposed by Cen (2. In addition, as in te comparison wit ˆf S, if we coose w i = i, all Hestenes based estimators ave smaller ASE tan tat of ˆf C for all densities. Lastly, as expected, te traditional osenblatt-parzen estimator as te poorest performance across all densities and all estimators. Te coice of kernel, or seed-kernel, does not qualitatively impact te relative performance described above. Tis preliminary experimental evidence seems to support te use of w i = i and te coice of s = relative to s = 2. esults for s = 3 and k > 3 (not reported suggest rapid deterioration of performance. Te evidence also suggests tat Hestenes-based estimators outperform te well known estimators proposed by Scuster (985 and Cen (2. In te second broad case, we consider a density tat as a point of discontinuity at x =. Specifically, f(x = { ( exp 2πσ 2 x 2 2 σ 2 if x < 2π exp ( 2 x2 if x were σ 2 controls te size of te jump. If σ 2 = te density is continuous everywere, and for < σ 2 < te jump at x = is given by J f ( = f( f( = 2π ( σ. Te left and rigt panels of Figure 2 provide graps of tis density for σ 2 =.5 and σ 2 =.25. Wit knowledge of te point of discontinuity (x = we evaluate two estimators for f. Te first, ˆfH, (6. 2

23 based on Hestenes extension, is composed of te estimator ˆf (x = ( x Xi s ( k j x Xi /w j K K. n w j X i> for te part f of f defined on [, and ˆf is te analog estimator for te f part of f defined on (, ]. Te jump J f ( is estimated by J ˆf ( = ˆf ( ˆf (. Te second estimator is composed of a osenblatt-parzen estimator ˆf (x = n j= ( x Xi K for f and ˆf is te analog estimator for f. J f ( is estimated by J ˆf ( = ˆf ( ˆf (. X i> For eac density (σ 2 =.25,.5 we generated samples of size n = 25, 5 and calculated te osenblatt- Parzen and Hestenes estimator for w i = i, i, i/(s and s =, 2 using a Gaussian kernel. For eac estimator we selected an optimal bandwidt by minimizing te integrated squared error over a fixed grid on te interval ( 2.5, 3 wit step 5 3 for te case were σ 2 =.5 and (.5, 3 wit step 5 3 for te case were σ 2 =.25. For eac sample, and eac estimator, a root average squared error (ASE over te grid is calculated. Te average of ASE across all generated samples are reported on Table 2 and 3. Average estimated jumps (across all generated samples for bot estimators, as well as teir deviation from te true jump, are also provide for bot estimators. We observe te following general regularities. First, all estimators ave smaller average ASE wen n = 5 compared to n = 25 for bot σ 2 =.5 and σ 2 =.25. All versions of ˆf H ave smaller average ASE tan ˆf and ˆf H performs better wen s = and w i = i for bot σ 2 =.5 and σ 2 =.25. For eac coice of s, te estimator ˆf H performs similarly wen w i = i or w i = i/(s. Te estimated jump is muc closer to te true jump under ˆf H tan under ˆf. J ˆf severely underestimates te true jump for bot σ 2 =.5 and σ 2 =.25. J ˆfH, calculated under all versions of te estimators, is muc closer to te true value of te jump, but for bot σ 2 =.5 and σ 2 =.25 tere seems to be some evidence of a sligt overestimation of te jump. Except for te case were s = and w i = i, jump estimators become more accurate wen n = 5 compared to n = 25. However, results are mixed regarding te impact of s and w i on te accuracy of jump estimates. For an arbitrary sample of size n = 5 and σ 2 =.25, Figure 3 22

24 provides a grap for ˆf and ˆf H for s = 2 and w i = i. 7 Summary and conclusions We provided a set of easily implementable kernel estimators for densities defined on subsets of tat ave boundaries. Te use of Hestenes extensions allows us to obtain teoretical representations for bias and variance of our proposed estimators tat preserve te orders of traditional kernel estimators for densities defined on. In effect, te insigts gained from using Hestenes extensions make te study of suitably defined kernel estimators in sets tat ave boundaries a special case of te teory developed for densities defined on. Preliminary simulations reveal very good finite sample performance relative to a number of commonly used alternative estimators. Furter work sould investigate te possible existence of optimal coices for s and w,, w s under a suitably defined criterion. If possible, tis would produce a best estimator in te class we ave defined. 23

25 .2 Kernels Density, Estmated density Values of x Figure : Density f G (x (blue, ˆf (yellow, ˆf C (green, ˆf S (red and ˆf k (black for k = 2, s =, w i = i Value of te density Value of te density Value of x Value of x Figure 2: Density f(x wit a discontinuity at x =. Left panel is for σ 2 =.5, rigt panel is for σ 2 =.25 24

26 .2 density, Estmated density Values of x Figure 3: True density f(x (blue, ˆf (red and ˆf H for s = 2 and w i = i (black for σ 2 =.25 25

27 Table. Average ASE 2 ˆf and ˆf constructed using K(x = 2π e 2 x2 ˆf ˆfC ˆfS ˆf ˆf2 ˆf3 s wi i i i i i i i i i i i i n = 25 ft N fg fχ fe n = 5 ft N fg fχ fe

28 Table 2. Average ASE and Jump Estimate ˆf and ˆf H for σ 2 =.5 Experiment (s,w i,n Average Squared Error Jump Estimate True Jump: J =.652 ˆf ˆfH J ˆf J ˆfH J ˆf J J ˆfH J (s=,w i = i, (s=2,w i = i, (s=,w i = i, (s=2,w i = i, (s=,w i = i/(s, (s=2,w i = i/(s, (s=,w i = i, (s=2,w i = i, (s=,w i = i, (s=2,w i = i, (s=,w i = i/(s, (s=2,w i = i/(s, Table 3. Average ASE and Jump Estimate ˆf and ˆf H for σ 2 =.25 Experiment (s,w i,n Average Squared Error Jump Estimate True Jump: J =.3989 ˆf ˆfH J ˆf J ˆfH J ˆf J J ˆfH J (s=,w i = i, (s=2,w i = i, (s=,w i = i, (s=2,w i = i, (s=,w i = i/(s, (s=2,w i = i/(s, (s=,w i = i, (s=2,w i = i, (s=,w i = i, (s=2,w i = i, (s=,w i = i/(s, (s=2,w i = i/(s,

29 eferences Burenkov, V. I., 998. Sobolev spaces on domains. B. G. Teubner, Leipzig. Cen, S. X., 2. Probability density function estimation using gamma kernels. Annals of te Institute of Statistical Matematics 52, Cline, D. B. H., Hart, J. D., 99. Kernel estimation of densities wit discontinuities or discontinuous derivatives. Statistics: A Journal of Teoretical and Applied Statistics 22, Hestenes, M.., 94. Extension of te range of a differentiable function. Duke Matematical Journal 8, Karunamuni,. J., Alberts, T., 25. A generalized reflection metod of boundary correction in kernel density estimation. Te Canadian Journal of Statistics 33, Malec, P., Scienle, M., 24. Nonparametric kernel density estimation near te boundary. Computational Statistics and Data Analysis 72, Mynbaev, K., Martins-Filo, C., 2. Bias reduction in kernel density estimation via Lipscitz condition. Journal of Nonparametric Statistics 22, Scuster, E. F., 985. Incorporating support constraints into nonparametric estimators of densities. Communications in Statistics - Teory and Metods, 4, Wen, K., Wu, X., 25. An improved transformation-based kernel density estimator od densities on te unit interval. Journal of te American Statistical Association,