Eeden, C. (Constance) van

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Eeden, Constance

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Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous

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Centrum voor Wiskunde en lnformatica Centre for Mathematics and Computer Science C. van Eeden Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous Department of Mathematical Statistics Report MS-R8414 November The Centre for Mathematics and Computer Science is a research institute of the Stichting Mathematisch Centrum, which was founded on February 11, 1946, as a nonprofit institution aim-ing at the promotion of mathematics, computer science, and their applications. It is sponsored by the Dutch Government through the Netherlands Organization for the Advancement of Pure Research (Z.W.O.). Copyright t Stichting Mathematisch Centrum, Amsterdam MEAN INTEGRATED SQUARED ER.ROR OF KERNEL ESTIMATORS WHEN THE DENSITY AND ITS DERIVATE ARE NOT NECESSARILY CONTINUOUS C. VAN EEDEN Universite de Montreal_, Depqrtement de Mathematiques et de Statistique_, Montrea?. Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sample XI> ... , Xn, were obtained by Rosenblatt [4] and Epanechnikov [l] for the case when the density f and its derivative f' are continuous. They found, under certain additional regu-larity conditions, that the optimal choice hno for the scale factor hn =Kn-a is given by hno=Kon- 115 with K0 depending on/ and the kernel; they also showed that MISE(hno)=O(n- 415) and Epanechnikov [ 1] found the optimal kernel. In this paper we investigate the robustness of these results to departures from the assumptions con-cerning the smoothness of the density function. In particular it is shown, under certain regularity condi-tions, that when f is continuous but its derivative f' is not, the optimal value of a in the scale factor becomes II 4 and MISE(hno)=O(n- 314); for the case when f is not continuous the optimal value of a becomes 112 and MISE(hno)=O(n- 112). For this lastcase the optimal kernel is shown to be the double exponential density. 1980 MATHEMATICS SUBJECT.CLASSIFICATION: 62G05. KEY WORDS & PHRASES: density estimation, mean integrated squared error, optimal kernel. NOTE: This report will be submitted for publication elsewhere. Supported by the Natural Sciences and Engtneering Research Council of Canada under Grant Nr.A 3114 and by the Gouverment du Quebec, Progrannne de formation de chercheurs et d'action concertee. Report MS-R8414 Centre for Mathematics and Computer Science P.O. Box 4079, 1009 AB Amsterdam, The Netherlands 1 1. Introduction Let Xi. ... , Xn be independent, identically distributed random variables with density function f and let I n [x-X;l fn(x) = -h ~w h , -oo<x<oo ~ n i=.1 n (1.1) be a kernel estimator off. Rosenblatt [4] and Epanecbnikov [1] (see also Rosenblatt [5]) studied ~e asymptotic behaviour of the mean integrateg squared error (MISE) of fn; they found the optimal value of hn, the asymptotically minimum value of MISE and the optimal kernel, assuming, among other regu-larity conditions, that w is a density symmetric around zero and that f has two continuous derivatives. Nadaraya [3] extended the results of Rosenblatt and Epanechnikov to the case where f has s (s ;;;;.2, s even) derivatives with J<s> bounded. In this paper the results of Rosenblatt and Epanechnikov are extended to the case where the density f and its derivative/' are not neeessarily continuous; a bounded density, symmetric around zero, is used for the kernel and it will be shown, under certain additional regularity conditions, that, with hno the asymptotically optimum value of hn, MISE(hno)=O(n- 314) if f is continuous and f' is not continuous, whereas MISE(hno)=O(n- 112) if f is not continuous. Expressions for limn-.oo n 314MISE(hno), respec-tively limn--+oo n 112 MISE(hno), in terms of wand/ will be given; further it will be shown that the dou-ble exponential density is the optimal kernel for the case when f is not continuous. · Section 2 contains the conditions on w and f, as well as some properties of, and some examples of, densitiesf satisfying these conditions. The main results are given in Section 3; a sketch of the proofs of these results is given Section 4. Full details of these proofs can be found in the technical report by van Eeden [6]; this report is available from the author on request. 2. The conditions on w and f The kernel w will be assumed to satisfy Condition A. w is a bounded density, symmetric around zero with +oo 0 < J t 2w(t)dt <oo. -oo The density f will be assumed to satisfy Condition B. 1. f has k (k ;;;;.O) points of discontinuity a 1 < ... <ak ; at each of these points f has a left-hand and a right-hand limit and f (a;-}=l=f (a/), i = 1, ... , k. 2. For each i = 1, ... , k + 1, the function g; defined on [a; -1>a;] by {/(a;:..i) if x =a;-1 g;(x) = f(x~ ~ a;- 1<x <a; f (a; ) if x =a;, where a 0 = - oo, ak + 1 = oo, has, except at the points b; 1 < ... <b; 1, (/; ;;;;.o, a;- I <b;" b; 1, <a;), a deriva-tive g/. At each of the points b;" ... ,bu,, g; has a left-hand and a right-hand derivative; g/(bij-)=l=g/(b;/), j = 1, ... , I; and )' ' g;(Y)-g;(x) = J g;(u)du, a;_ 1 ~x <y ~a;. x Further +oo J lf'(x) ldx <oo. -oo 2 3. For each pa1r (i ,j), i = 1, ... , k + 1, j =: 1, ... ,I;+ 1, the function g;j defined on [b;j-1> bij] by {g/(b;f-1) if x =bij-1 gij(x) = g;:(x~ ~ bij_ 1<x <b;j g;(bij ) if x =bij, where b;0=a; _ 1 and b; H 1 =a;, is absolutely continuous and g;j is continuous almost everywhere. Further f ~~lf"(x) ldx <oo and g;j is bounded. · The following Lemmas 2.2, 2.3, 2.4 and ~.5, needed for the proofs of the main results, give some pro-perties of densities satisfying Condition B. Lemma 2.1 is needed for the proofs of the Lemmas 2.3, 2.4 and 2.5. Lemma 2.1. If G (x ), - co <x <co, is absolutely continuous and +oo +oo J jG(x)ldx<co, J IG'(x)jdx <co (2.1) -oo -oo then x (a) G(x) = J G'(y)dy -co<x <co -oo (2.2) (b)G(x) is bounded Proof. (a) The absolute continuit of G (x) implies that x G(x)-G(y) = fG'(u)du -co<y<x<co. (2.3) y Further, (2.I) and the absolute continuity of G imply (see the proof of Lemma I.2.4.a of Hajek and Sidak [2]) (2.4) The result then follows from (2.3) and (2.4). (b) The boundedness of G follows from (2.4) and the fact that G is absolutely continuous, and hence of bounded variation, on finite intervals. D Lemma 2.2, For each i = l, ... , k + I the function g; is absolutely continuous on [a;_ 1' a;]. Proof. See Hajek and Siditk [2], Theorem I.2.1 ). D Lemma 2.3. For each i = l, ... , k + 1 the function g; is bounded. Proof. For i =2, ... , k the fact that g; is bounded follows from its definition. For i =I and for i =k + 1 the boundedness of g; follows from Condition B.2 and Lemma 2.1. D Lemma 2.4. For each pair (i ,j), i = 1, ... , k +I, j = 1, ... , I;+ I, the function g;j is bounded. Proof. For all but the pairs (i = l, j = 1) and (i =k +I, j =lk +I+ I) the fact that g;j is bounded follows from its definition. That gu and gk + 1,lk + 1 +I are bounded follows from Lemma 2.1 and Condition B.3. D Note that Lemma 2.3 implies that f is bounded, which implies that f is square integrable. Further, 3 Lemma 2.4 implies :tha,t f' is bounded and Condition B.3 implies that f" is bounded and square integr-able. Also, each of the functions f, f' and f" is continuous almost everywhere. If k = 0 then f (x ), -oo<x<oo, is, by Lemma 2.2, absolutely continuous and if k=O, 11=0 thenf'(x), -oo<x<oo, is, by Condition B.3, absolutely continuous. Now let {8; = f (a;-)-/ (a/) i = 1, ... , k aij = f'(bij_)_ f'(b;/) i = 1, ... , k + 1, J = 1, ... , Z; + 1 (2.5) and k k+1 f,+1 8 = ~8l, a=~· ~a;J (2.6) i=l i=I j=1 then Lemma 2.5. If the density f satisfies Condition B then +oo 8>0 or a>O or j {f"(x)}2dx>O. (2.7) -oo Proof. It is sufficient to prove that +oo {8=0, Ll=O} ~ j {f"(x)}2dx >0. (2.8) -oo To prove (2.8) first note that {8=0, a=O}~{k =O, 11=0} ~ {f(x), -oo<x <oo, and (2.9) f '(x ), - oo <x < oo, are absolutely continuous}. Then, by Lemma 2.1 and the fact that f, f' and f" are integrable, one obtains that { 8 = 0, a= 0} implies x y . f(x) = J<ff"(z)dz)dy, -oo<x<oo. (2.10) -00-00 Finally, f ~: {/"(x )}2dx =O contradicts (2.10) because f is a density. D The following are some examples of densities satisfying Condition B. 1. The normal, logistic and Cauchy densities satisfy ConditioiJ. B with k = 11 = 0, 8 =a= 0 and J~:{f"(x)}2dx >0, 2. the double exponential density /(x) = +e-lxl, -oo<x<oo, satisfies Condition B with k =O, 11=1, 8=0, Ll>O, f ~:{f"(x)}2dx >0, · 3. the density {l+x f(x) = ~-x -1.;;;;;x.;;;;;O O.;;;;;x.;;;;;l else satisfies Condition B with k =O, 11 =3, 8= f ~:{f"(x)}2dx =O and a>O, 4. the uniform density 4 {.l -1.s;;;x ..;;I j(x) = ~ else satisfies Condition B with k =2, / 1 =/2=/3=0, 8>0 and a= J~:{f"(x)}2dx =O, 5. the density a+x -1.s;;;xo;;;;O ---2a-l f(x) =: a-x Oo;;;;xo;;;;I 2a-l 0 else, where a> l, satisfies Condition B with k = , 11=13=0,12= l, 8>0, /:,.>0 and /~:{f"(x)}2d.x =O, 6. an example of a density for which 8>0, ~=O and f ~: {f "(x )}2dx >0 can be constructed as follows. Letf be a density satisfying Condition B. with k=/1=0. Then /~:{f"(x)}2dx>O. Let c>O and let f * be defined by f*{x) = .fl& 1+2c f(x)+c 1+2c for lxl>l for lxlo;;;;I. Thenf°(x), -oo<x <oo, is a density; further ..!i...1. ( ) = .t:Q)_ dx x 1+2c and f* satisfies Condition B with k =2, 11 =/2=/3 =0, 8>0, /:,.=O and +oo l +oo J {f*"(x)}2dx = J {f"(x)}2dx>O -oo (1+2c)2 -oo , 7. an example where 8>0, /:,.>0 and J~:{f"(x)}2dx >0 is the density 3. The main results {f[l+(x.f-1)2] -1.s;;;xo;;;;O f(x) = 0f[l+(x-1)2] Oo;;;;xo;;;;l else, The main results of this paper are given in the following Theorems 3.1, 3.2 and 3.3; the proofs of these theorems are sketched in Section 4. Theorem 3.1. If the conditions A and Bare satisfied then, for n ~oo and hn ~o, 1 +oo MISE(hn) = nh f w2(t)dt + n -oo thn4{ +[t2w(t)dt}2 +[lf"(x)}2dx+o(n! +hn4) if ~=A=O -oo -oo n 0 y ·1 + 2hjA J { J (y-t)w(t)dt }2dy +o( nh +hj) if 6=0, A>O -oo -oo · n 00 00 1 2hn6 J { J w(x)dx }2dy +o(-h +hn) if 6>0. o Y n n When f satisfies Condition B with k =11 =O, Theorem 3.1 reduces to the results of Rosenblatt [4], Epanechnikov [1] and to Nadaraya's [3] result for s =2. The following Theorem 3.2 gives, for hn =Kn-a, a>O, the asymptotically optimum values ao and K0 of a and K, as well as the value of M 0 = 1im n I -a.iMISE (K0n -a.i). n->oo Theorem 3.2. If the conditions A and B are satisfied and hn =Kn -a, a>O, then (a) the asymptotically optimum value ao of a is given by .l if 6=A=O 5 ao = I 4 if 6=0, A>O .l if 6>0 2 (b) the asymptotically optimum value K0 of K is given by Ko= +oo J w2(t)dt +oo +oo { J t 2w(t)dt}2 J lf"(x)}2dx -oo -oo .l +oo 4 J w2(t)dt 0 y 6A j[ f (y-t)w(t)dtfdy -00-00 +oo J w2(t)dt .l 2 .l 5 if 6=A=O if 6=0, A>O 00 00 if 6>0 26 j[j w(t)dtfdy 0 y 6 (c) the asymptotically minimum value of MISE, MISE(Kon-a.), satisfies lim n l-a.MISE (K0n -a.) = n--+OO [ +oo ]4[+00 ]t[+oo lt f _£ w2(t)dt _£ t 2w(t)dt -L {f"(x)}2~ if 8=11=0 [ +oo ]* [ 0 y ]± 4 _£ w2(t)dt 611_£ ~£ (y-t)w(t)dt )2dy if 8=0, 11>0 As for Theorem 3.1, Rosenblatt's [4] result, Epanechnikov's [I] result and Nadaraya's [3] result for s = 2 are a special case of Theorem 3.2. If it is known that the density to be estimated satisfies Condition B, but it is not known whether f, nor whether f' is continuous, then the optimal choice for hn =Kn -a is unknown; in this case, for any choice of a among the values t, ± and ± that is not the optimal choice, the asymptotic efficiency relative to the optimal choice is zero; that is lim MISE (Kon -a.) = 0 1 1 1 K>O, aE{5;4,z-}, a=foao. n-->oo MISE (Kri -a) On the other hand, if ao is known and one uses hn =Kn -a. then the asymptotic efficiency relative to the optimal choice is positive for all K>O and is equal to one if a consistent estimator of Ko is used for K. Nadaraya [3] gives such an estimator for the case when 8=11=0. Finally, the optimal kernel, that is the kernel w that minimizes M 0(w) = lim n 1-a.MISE(K0n-a.), n->OO depends on the unknown density orily through ao- For the case where 8=11=0, it is well-known (see Epanechnikov [I]) that the optimal kernel is given by t3 (l-t2) t2~1 w(t) = 12> 1. The following Th rem 3.3 gives the optimal kernel for the case when 8>0. Theorem 3.3. If the conditions A and B are satisfied and 8>0 then the optimal kernel is given by w(t) = ±e- 1' 1 -oo<t<oo. We have been unable to find an optimal kernel for the case when 8=0, 11>0. The following Table I gives, for several kernels, the asymptotic efficiency, e(w), of the kernel w as defined by · M 0(w) e(w) = M~(wo)' 7 where w0 is the optimal kernel. The kernels used in Table 1 are the same as those used by Epanechnikov [l]; he gives, in his Table l, the values of [e(w)r5/ 4 for the case where 8=.::l=O. w It I :s;;; 1 1 .940 0 It l>l tcost I.OOO .945 0 ltl>; 1-lt I .989 .968 0 It l>l 1 I 2 -21 -oo<t<oo .961 .974 --e V2ir I It I :s;;; 1 2 .943 .866 0 It l>l .!e-111 2 -oo<t<oo .802 1 Table 1. Asymptotic efficiencies e ( w) of the kernel w. Proofs of the theorems in section 3 In order to simplify the notation the index n on hn will be omitted. For the proofs of the theorems in Section 3 the following lemmas are needed. The proofs of these lemmas, which are straightforward but lengthy and tedious, are omitted; they can be found in van Eeden [6]. Lemma 4.1. If the conditions A and B are satisfied then +oo J fi>{fAx)-f(x)} 2dx = (4.1) -oo l +oo 2 +oo +oo nh J w2(t)dt--;; J f(x) J w(t)f(x -th)dtdx -oo -oo -oo 8 1 ~ 00 · 1 +oo [ +oo ] 2 +; _£ f 2(x)dx + n ~ _£ _£ w(t)(f(x -th)-f(x)}dt dx. Lemma 4.2. If the conditions A and B are satisfied then +oo +oo +oo Eo J f(x) J w(t)f(x -th)dtdx = J f 2(x)dx. (4.2) -oo -oo -oo The following three lemmas are needed to obtain the behaviour of (see (4.1)) [ ]2 +oo +oo _£ _£ w(t)(f(x -th)-f(x)}dt dx as h ~o; the first two of these lemmas give Taylor series-like expansions for functions f satisfying Con-dition B; if k = 11 = 0 these expansions are Taylor series expansions. Lemma 4.3. If f satisfies Condition B then, for bij1-1<x<b;j,, b;j2-1<x-th<bij» l,.,;;;j1~l;+l, l~h ~/; +l, i =l, ... ,k +l t f (x -th )-f (x)+thf'(x)-h 2 J (t -s)f"(x -sh )ds 0 (i) 0 ifJi=h j?-1 (ii) ~ (b; 1 -x +th)a;1 if Ji<h y=j1 j1-I (iii) - L (bi 1 -x +th)a;1 ifJ1>Ji. y=j2 (4.3) Lemma 4.4. If f satisfies Condition B then, for b;,j,-1 <x <b;,j,, b;,j2-1<x -th <b;2j,, l~}I ~/;, + 1, l~ji~/;2 +1, l~i1~k+l, l~i2~k+I, i1::f=ii. Now let t f (x -th )-f (x)+thf'(x)-h 2 J (t -s)f"(x -sh )ds = 0 /;,+I i2-I l,+I (i) L (b;,1 -x +th)a;,1 + L L (byy,-x +th)~, y=j1 y=i1+I y'=I fa-I ii-I + L (b;,1 -x +th)a;,1 - L 8,. if i 1<i2 y= I p.=i1 /,,+J i1-I l,+I (ii) - L (b;,1 -x +th)A;21 - L L (bn,-x +th)firr y=j2 y=ii+I y'=I ji-1 ;,-) - L (b;,1 -x +th)A;,1 + L 8" if i 1>i2• y=I p.=ii (4.4) where +oo l(x,h) = J w(t){j(x-th)-j(x)}dt, -oo<x<oo, -oo +oo t G(x,h) = h2 J w(t)j(t-s)j"(x -sh)ds dt, -oo<x <oo, -oo 0 x-b,, 1,+1 h H;j 1(x,h) = ~~iy J (b;y-x +th)w(t)dt y=j -oo x-brr' k+I /,+I h + ~ ~~ J (bn·-x+th)w(t)dt, y=i+I y'=I -oo bij-1<x <bij, j::::: I, ... ,I;+ I, i = 1, ... ,k + 1, j-1 00 Hij2(x,h) = - ~~iy J (b;y-x+th)w(t)dt y=I x-b,, h i-1 l,+I oo -~ ~~ J (bn·-x+th)w(t)dt, y=I y'=I x-brr' h bij-1<x <b.;j, j = 1, ... ,I;+ 1, i = 1, ... ,k + 1, i-1 x-a V; 1(x,h) = ~«5,.{1-W(-h ")}, a;- 1<x<a;, i=l, ... ,k+l, 1£=1 k x-a Vi2(x,h) = -~.«5,.W(T), a;-1<x<a;, i=l, ... ,k, 1£=1 y W(y) = J w(t)dt, -oo<y<oo. -oo Further let, for j = 1, ... ,I; + l, i = 1, ... ,k + 1, {H;j(x,h):: H;j 1(x,h)+Hij2(x,h) V;(x ,h) - Vf 1(x ,h )+ Vi2(x ,h) then it follows from Lemma 4.3 and Lemma 4.4 that Lemma 4.5. If the conditions A and B are satisfied, then +oo +oo J [ J w(t){f(x -th)-j(x)}dtfdx = -oo -oo k+I /,+I b,1 = ~ ~ J {G(x,h)+Hij(x,h)+ Vj(x,h)}2dx. i=I j=lblJ-• 9 (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) (4.12) (4.13) 10 The following four lemmas give the behaviour, as h ~o, of each of the six terms obtained by expand-ing the square in the right-hand side of (4.13). Lemma 4.6. If the conditions A and B are satisfied then . l +oo l +oo +oo Jim - J G2(x,h)dx = -{ J t 2w(t)dt}2 J {f"(x)}2dx. h->0 h4 4 -oo -oo -oo Lemma 4.7. If the conditions A and Bare satisfied then · I ..;;;.j ..;;;./; + 1, i = 1, ... ,k + 1. Lemma 4.8. If the conditions A and Bare satisfied then, for i = 1, ... ,k + 1, . lja, 2 h dx. - ~2 ~2 Joo 2 lim -h Vi (x, ) - (ui +u;-1) {1-W(Y)} dy. h->0 Q1-1 0 Lemma 4.9. If the conditions A and B are satisfied then . I b,; Jim - 3 J G(x,h)Hij(x,h)dx h->0 h blj-1 . 1 a, = Jim -h jG(x,h)V;(x,h)dx h->0 01-1 . 1 bi; {l ..;;;.j ..;;;_/; + 1 =lim-h jH;j(x,h)V;(x,h)dx=O . 1 k+l h->0 l = , ... , . b11-1 Proof of Theorem 3.1. From Lemma 4.1, 4.2 and 4.5 it follows that, for h ~o. +oo +oo MISE (h) = __l._h j w2(t )dt _ _!_ j f 2(x )dx + o (h) n n . -co -co n _ 1 k+11,+1 blj + -- ~ ~ J {G(x,h)+H;/x,h)+ V;(x,h)}2dx. n i=1 j=lby-1 The theorem then follows from the Lemmas 4.6, 4.7, 4.8 and 4.9 and from the fact that k+l 11+1 k+1 /,+1 ~ ~dG-1 = ~ ~d;J = d i=l j=I i=I j=I and k+I k+I ~Sl-1 = ~8? = 8. D i=I i=I (4.14) (4.15) (4.16) (4.17) (4.18) Proof of Theorem 3.2. For the case where 8=d=O, the proof can be found in Rosenblatt [4] for the uni-form kernel and in Epanechnikov [l] for the more general case of a kernel satisfying Condition A. The proofs for the other two cases are analogous to these proofs of Rosenblatt [4] and Epanechnikov [l]. D ·Proof of Theorem 3.3. The kernel that minimizes M0(w) = Jim n 1-a,,MISE(K0n-IXJ:J) n->co when 8>0, is the kernel that minimizes +oo oo oo oo J w2(t )dt J { 1-·w(y )}2dy = 2 J w2(t )dt J {I - W(y )}2dy. -oo 0 0 0 By Schwartz's inequality oo oo oo I J w2(t)dt J {1-W(y)}2dy ~ [ J w(()(l-W(t))dt]2 = 64 , 0 0 0 with equality if and only if or, equivalently, if and only if References W(t) = aw(t) O<t<oo for some a>O w(t) = .!!... e-altl -oo<t<oo, a>O. 2 II (4.19) (4.20) (4.21) (4.22) [l] EPANECHNIKOV, V.A. (1969). Non-parametric estimation of a multivariate probability density. Theor. Probability Appl. 14 153-158. [2] HAJEK, J. AND SmAK, Z. (1967). Theory of rank tests. Academic Press, New York. [3] NADARAYA, E.A. (1974). On the integral mean square error of some nonparametric estimates for the density function. Theor. Probability Appl. 19 133-141. [4] RosENBLATI, M. (1956). Remarks on some nonparametric estimates of a density function. Anh. Math. Statist. 27 832-837. [5] ROSENBLATI, M. (1971). Curve estimates. Ann. Math. Statist. 42 1815-1842. [6] VAN EEDEN, C. (1982). Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous. Rapport de Recherches, no. 82-7, Departement de Mathematiques et de Statistique, Universite de Montreal. 1 DEC. 1984 

Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous

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