AbstractIn this paper, kernel function methods are considered for estimating the intensity function of a non-homogeneous Poisson process. A least-squares cross-validation bandwidth for the kernel intensity estimator is introduced, and it is proven that this bandwidth is asymptotically optimal for kernel intensity estimation

Brooks, Maria Mori

Marron, J.Stephen

Elsevier - Publisher Connector 

Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions 

In this paper, kernel function methods are considered for estimating the intensity function of a non-homogeneous Poisson process. A least-squares cross-validation bandwidth for the kernel intensity estimator is introduced, and it is proven that this bandwidth is asymptotically optimal for kernel intensity estimation

Marron, J. Stephen

Carolina Digital Repository

Stochastic Processes and their Applications 38 (1991) 157-165 North-Holland 157 Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions Maria Mori Brooks and J. Stephen Marron” Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA Received 12 November 1989 Revised 25 July 1990 and 21 August 1990 In this paper, kernel function methods are considered for estimating the intensity function of a non- homogeneous Poisson process. A least-squares cross-validation bandwidth for the kernel intensity estimator is introduced, and it is proven that this bandwidth is asymptotically optimal for kernel intensity estimation. bandwidth selection * intensity function * cross-validation bandwidth * kernel estimation * nonstationary Poisson processes 1. Introduction Let X,, X2,. . . , XN be ordered observations on the interval [0, T] from a non- stationary Poisson process with intensity function A(x). In this paper, we consider the estimation of A(x). N, the number of observations that occur in the interval [0, T], has a Poisson distribution with E[ N] = jl A(U) du. See Cox and Isham (1980) and Diggle (1983) for further information regarding point processes. A natural kernel estimator for A(x) is: iih(X)= f Kh(X-Xi), XE[Op T], i=* (1.1) where &(x) = h-‘K(x/h). The kernel function, K (. ), is assumed here to be a probability density function, and the smoothing parameter, h, also known as the bandwidth, quantifies the smoothness of Ah(x) (see, for example, Silverman, 1986). The kernel intensity estimator differs from the typical kernel density estimator in two ways. First, i,,(x) doesn’t include a normalization factor, K’, since ji A(x) is * Research partially supported by NSF Grant DMS-8902973. 0304-4149/91/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland) 158 M.M. Brooks, J.S. Marron / Cross-validation bandwidth the expected number rather than the expected proportion of observations between s and t. Second, N, the number of observations, is a random variable in the intensity estimation setting. Theoretical properties of the kernel intensity estimator have been developed by Leadbetter and Weld (1983), Ramlau-Hansen (1983) and Diggle (1985). It is accepted that the choice of the smoothing parameter is usually far more important than the choice of the kernel function for kernel estimators. The tradeoff between smaller and larger values of h can be seen in the mean square error (MSE) of i,(x). Rosenblatt (1971) showed that a small value of h results in high variance; i,,(x) is affected by individual observations and hence is more variable. On the other hand, a large value of h results in high bias; i,,(x) is very smooth but does not include minor features of the true intensity function. Thus, it is desirable to find a data-based bandwidth that balances the effects of the bias and the variance of the estimate. We use a simple multiplicative intensity model to put a mathematical structure on this problem. The general multiplicative intensity model, introduced by Aalen (1978), is frequently used to model counting processes. See Anderson and Borgan (1985) for an overview of these models. For the related setting of kernel density estimation, Hall (1983), Burman (1985), and Stone (1984) have proven that the least-squares cross-validation bandwidth is asymptotically optimal. Hall (1983) showed that this bandwidth is optimal in probability by using asymptotic expansions of the cross-validation score function and the integrated square error (ISE) of the kernel estimator. This method requires the density function to have two continuous bounded derivatives. Burman (1985) used methods from Shibata (1980) to prove that the cross-validation bandwidth is optimal in probability. In his proof, there are few restrictions on the kernel function, and the density is a bounded L2 function. By assuming that the kernel function is compactly supported and Holder continuous, Stone (1984) was able to show using Poissonization methods that the cross-validation bandwidth is optimal almost surely for a continuous set of bandwidths. This was the first paper to show that the optimality result applied to all bounded density functions. In this paper, we show that the least-squares cross-validation method is asymptoti- cally optimal almost surely for intensity estimation. The additional difficulty in the intensity setting is that the number of observations is now a random variable. The intensity function is required to have two continuous bounded derivatives. Moreover, the kernel function is a bounded compactly supported probability density function. We use arguments similar to the martingale methods employed by Hardle, Marron and Wand (1990) to prove the asymptotic optimality of density derivatives. This method is based on a martingale inequality given by Burkholder (1973). In Section 2, we discuss the mathematical model for the intensity function. Section 3 contains the main result regarding the asymptotic optimality of the least-squares cross-validation bandwidth. Finally, the proof of the theorem is presented in Section 4. M.M. Brooks, J.S. Marron / Cross-validation bandwidih 159 2. The simple multiplicative intensity model The simple multiplicative intensity model is a specific form of Aalen’s (1978) multiplicative intensity model. Suppose that X, , X2, . . . , XN are observations from a nonhomogeneous Poisson process with intensity h,.(x) = c(Y(x), XE [O, 7-1, (2.1) where c is a positive constant, and cr(x) is an unknown nonnegative deterministic function with Ic a(x) dx = 1. Given N, the occurrence times, X, , X2, . . . , XN, have the same distribution as the order statistics corresponding to N independent random variables with probability density function Q(X) on the interval [0, T]. The kernel estimate of h,(x) is given in (1.1). Under this model, N is a Poisson random variable that has expected value equal to c. In order to avoid boundary effects, we assume a circular design such that A,(O) = h,(T), A’,(O) = A:.( T) and A:‘(O) = A:( T). Asymptotic analysis provides a powerful tool for understanding the behavior of the kernel intensity estimator. Letting T + 00 is not appropriate since this results in all of the new observations occurring at the right endpoint. In the simple multiplica- tive intensity model, letting c + ~0 has the desirable effect of adding observations everywhere on the interval [0, T] and not changing the relative shape of the target function A,(x) in the limiting process. In other words, c-‘A,(x) is a fixed function as c+co. Since we will consider convergence properties for sequences of random variables, we require the values of c to come from the sequence of positive real numbers {c,}T=, such that c,/s+ r for some constant T> 0 as S+CO. Thus, for the sequence of intensity functions A,,(x) = C,CY(X) indexed by s, we construct a corresponding sequence of kernel estimators i;;(x). For these kernel estimators, the bandwidth h is dependent on s. It follows from Ramlau-Hansen (1983) that ii(x) is uniformly consistent and asymptotically normal as c,, + CO, h + 0 and hc, + co. 3. Asymptotic optimality of the cross-validation bandwidth We are interested in finding a data based bandwidth that approximately minimizes the integrated square error (ISE) of ih where I T ISE,(h) = [l,,(x)-A(x)12dx 0 ZZ i 7- I T x,,(x)’ dx -2 i,,(x)A(x) dx+ 0 0 I T A(x)’ dx. (3.1) 0 For kernel density estimates, Rudemo (1982) and Bowman (1984) suggested using the method of least-squares cross-validation for selecting the bandwidth. In the intensity estimation setting, the cross-validation score function is defined as: CV*(h) = I 7- xh(x)’ dx -2 ; ihr(Xi) (3.2) 0 i=l 160 MM. Brooks, J.S. Marron / Cross-validation bandwidth where ahi is the leave-one-out estimator, ihi( i Kh(X-XJ). (3.3) j=l j#i Since CL, A,i(Xi) is a method of moments estimator of jl ih(x dx, and ,,’ Am dx is independent of h, CV,(h) is a reasonable unbiased estimate of the terms in ISE, (h) that depend on h. Therefore, the bandwidth that minimizes CV, (h) should be close to the bandwidth that minimizes ISE,(h). Let & be any bandwidth that minimizes ISE,(h) and &, any bandwidth that minimizes CV,(h) (these minima always exist since ISE,(h) and CV,(h) are continuous and bounded functions). Assume that: (a) The kernel function, K ( -), is a bounded compactly supported probability density function. (b) The true intensity function, A ( * ), has two continuous bounded derivatives. (c) The bandwidths under consideration come from a set H, where for each s and some constants p, 6 > 0, #(H,) = {the number of elements in H,} s cf , and for h E H,, cJltS s h s CL”. (d) For some constant T>O, c,/s+ r as s+cc. Assumption (b) is a common technical assumption on A(x) which allows Taylor expansion methods to be used. With this assumption, the true intensity function can have any amount of underlying smoothness. Assumption (c) can be weakened so that H, is a continuous interval by using a continuity argument found in Hardle and Marron (1985). This set of possible bandwidths nearly covers the range of consistent bandwidths. Under these assumptions, the ISE obtained with the cross-validation bandwidth converges almost surely to the minimum ISE. In this sense, the least-squares cross-validation bandwidth is asymptotically optimal for kernel intensity estimation under the simple multiplicative intensity model. This result is stated in Theorem 1. Theorem 1. If assumptions (a), (b), (c) and (d) hold, then, under the simple muftiplica- tive intensity model, I=, Ckv, IsE,o+ l a.s. ass+m. (3.4) The mean integrated square error (MISE), [S 7 MISE,(h) = E (~,,(x)-~(x))~ dx , (3.5) 0 1 is another error criterion that is used to evaluate bandwidth selection procedures. Let h, be the bandwidth that minimizes MISE,(h). By Lemma 1 in Section 4, the ISE and MISE are essentially the same for large s. As a result of Theorem 1, icy is M. M. Brooks, J.S. Marron / Cross-validation bandwidth 161 also asymptotically optimal with respect to MISE in the sense that MISE&) --, 1 MISE,(h,) a’s’ as s+co’ Details of the proof of Theorem 1 are presented in Section 4. (3.6) 4. Proof of Theorem 1 In this section, we outline the proof of Theorem 1. The mean integrated square error (MISE) of ih can be decomposed into a variance term and a squared bias term. Using Taylor expansion methods similar to those in Silverman (1986, pp. 39-40), it is straightforward to show that the MISE of ih is MISE,(h)=h-ir(j X.)+h4c:[:lu’K]21:,~f~(x)12+o(h-1c,+hlci) (4.1) as h + 0, c, +KJ and hc, + ~0. Thus, the asymptotic mean integrated square error (AMISE) is AMISE,(h)=h’c.(j K2)+h4ci[:Iu’K]iI:[o”(x)l’. (4.2) The two lemmas below are used to prove statement (3.4). Lemma 1. sup BE,(h)-AMISE,(h) ~0 AMISE, (h) a.s. ass+co. htH, Lemma 2. sup CV,(b)-ISE,(h)+[CV,(b)-ISE,(b)l -,. AMISE, (h) + AMISE, (b) a.s. ass+m. h,bt H, Lemma 1 says that the ISE and the AMISE of ih(X) are asymptotically equivalent, and the two lemmas together imply that sup cv,(h)-ISE,(h)-[CV,(b)-ISE,(b)l -,. a.s. as s+co. h,bt H, ISE,(h)+ISE,(b) (4.3) Since ISE(&) 5 ISE(c_) and CV(&,) s CV(&), Theorem 1 follows for the simple multiplicative intensity model. Now, we must prove Lemma 1 and Lemma 2. The details of the proof of Lemma 2 are given below; Lemma 1 is proven using similar martingale methods. Proof of Lemma 2. Let g( 1,2,. . . , IV) + (1,2, . . . , IV) be a random permutation of the numbers 1, 2,. . . , N. Define Y, = XgciJ. Essentially, the Y’s are the ‘unordered’ Xi’s. Since the Xi’s are observations from a nonhomogeneous Poisson process with intensity h(x), then given AJ, the Y(‘s are i.i.d. random variables with density 162 M.M. Brooks, J.S. Marron / Cross-validation bandwidth (Y(x)~~~,~,(x). As a result, kernel density methods developed by Hardle, Marron and Wand (1990) can be used to study a(x). Define: &h(X) = c,‘x,(x), &hi(X) = C,‘i,i(X), CV,(b) = I 7- a*,(X)’ dX-2cy’ ; &hi(X,) = c12CV,(h). 0 i=l Hence, ISE,( h) = cJ2 ISE,( b), and MISE, (b) = cl2 MISE, (b). It is not difficult to show that CV,(h)-ISE,(b)-[CV,(b)-ISE,(b)] sup h,bc H, AMISE,(h)+AMISE,(b) = sup CV,(b)-ISE,(h)-[CV,(b)-ISE,(b)] h,be H, AMISE,(b)+AMISE,(b) s2 sup c~‘CE~ a*,,(Xi)-jl &ha-G hsH, AMISE,(h) where G=[~(N-~)/c,-~]c~‘C,N_~ ~(Xi)-[N(N-l)/cf]S,T(y2. Thus, it suffices to prove the following: sup CL’C,“=, a*hr(Xi)-.IoT(I*ha-G +. as ass-,co AMISE,(h) . . hEH, (4.4) Define: I I 7 &j(h)=K,(Y,-Y,)- &(y-y)a(y)dy-(Y(I/;)+ a’(y) dy, 0 x(h)=E(UGI Yi) = &(x-.J’b’(J’)dY- I II &(Y -Z)a(y)a(Z) dy dz-(Y(Y) + l T a’(y) dy, 0 WV(h)= u,- v, =&(yi-y)- I &,(y-Y;)~(y)dy- Kh(y-yk(y)dy I + 1s &(Y -z)a(y)a(z) dy da, Ri(b)=[(N-1)/c,-l] 2 &(Y-y)a(y)dy [5 - Kh(Y-Z)cu(y)a(z) dy dz M.M. Brooks, J.S. Marron / Cross-validation bandwidth 163 Note that E( V, 1 N) = 0 and E( W, 1 yI, N) = E( Wl, ( q, N) = 0 for i,j = 1, 2,. . . , N. For sums over i=l, 2,..., N, Yi (the unordered observation) can be replaced by Xi (the ordered observation) in the summand. Thus, Hence, statement (4.4) holds when (4.5), (4.6) and (4.7) hold. ccl”=, V,(h) +. h”t”r: AMISE,( h) a.s. as s+co, sup c,*CIN_I Cjfi Wjth) +O as hEH, AMISE,(h) . . ass+co, C.;’ I;“=, K(h) ~ o f:,$), AMISE, (h) a.s. as s+co. (4.5) (4.6) (4.7) Therefore, in order to prove Lemma 2, it is sufficient to prove statements (4.5), (4.6) and (4.7). Conditional on N, {CF=, Vi},“=, and {CF=, C,:: Wti}kN=I are martingales with respect to the o-fields generated by {Y,, Y2,. . . , Yk}. Burkholder’s (1973, p. 40) inequality implies that for some constant A: E [(kyfNj, g2m~N]~AE[(,~*E[V:JNl)mIN] , / +A f HI K12”7 Nl, i=l (4.8) (4.9) Since N is a Poisson random variable with mean c,, E[ N”] s A,&’ where A,,, is constant for each m. Hence, c cy2m E[A,Nmh4”‘+A,N] (by (4.8)) s Am(c,mh4m + c;‘~+‘) 164 M.M. Brooks, J.S. Marron / Cross-validation bandwidth and 1 2m s c;4”E[A,N2mh~m +A,Nm+2h-2m] (by (4.9)) s A,(c;2mh~m + ~;~~+~h-~“‘). As a result, for h E H,, m sufficiently large and some y > 0, E cTlCfv=, K 1 2m ~A,(c,mh4m+c;2m+‘) AMISE,(h) (c;‘h-‘+ h4)2m s A,[h” + csh2”‘] s A,p,c,““. (4.10) Using Chebychev’s theorem, sup I’[ I&, v;l >c;“‘AMISE,(h)] sA,,,c;(Y’2)m. htH, (4.11) Recall that #(H,) s ct, and choose m such that rn > 2(p +2)/y. By assumption (d), one can show that for the sequence {c,}, Thus, the Borel-Cantelli Lemma implies that c3’C,N,, K +. ;:;, AMISE,(h) a.s. as s+a. (4.12) This proves (4.5). Following a similar procedure, statement (4.6) can be verified. Finally, consider (4.7). Since N is Poisson(c,s), it is known that [(N - c,)/&] is asymptotically normal. Using this fact and a Taylor expansion of Ri, one can show that for any O< E <$, Hence, for each m, there exists a constant c, such that whenever c, > c,, I 2m ~ &~T’h4m-2Fm ~ A,hc,-2e.,m < A -?,,, (c,‘h-‘+ h4)2m . ms (4.13) for h E H, and some y > 0. (4.7) follows from (4.13) as seen above. Therefore, we have proven Lemma 2. 0 M.M. Brooks, J.S. Marron / Cross-validation bandwidth 165 References 0. Aalen, Nonparametric inference for a family of counting processes, Ann. Statist. 6 (1978) 701-726. P.K. Andersen and 0. Borgan, Counting process models for life history data: A review, Stand. J. Statist. 12 (1985)97-l%. A.W. Bowman, An alternative of cross-validation for the smoothing of density estimates, Biometrika 71 (1984) 353-360. D.L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973) 19-42. P. Burman, A data dependent approach to density estimation, Z. Wahrsch. Verw. Gebiete 69 (1985) 609-628. D.R. Cox, and V. Isham, Point Processes (Chapman and Hall, London, 1980). L. Devroye and L. GyBrfi, Nonparametric Density Estimation: The L, View (Wiley, New York, 1984). P.J. Diggle, Statistical Analysis of Spatial Point Patterns (Academic Press, London, 1983). P. Diggle, A kernel method for smoothing point process data, Appl. Statist. 34 (1985) 138-147. W. HLrdle and J.S. Marron, Optimal bandwidth selection in nonparametric regression estimation, Ann. Statist. 13 (1985) 1465-1481. W. Hlrdle, J.S. Marron and M.P. Wand, Bandwidth choice for density derivatives, J. Roy. Statist. Sot. Ser. B 52 (1990) 223-232. P. Hall, Large sample optimality of least squares cross-validation in density estimation, Ann. Statist. 11 (1983) 1156-1174. M.R. Leadbetter and D. Wold, On estimation of point process intensities, in: P.K. Sen, ed., Contributions to Statistics: Essays in Honour of Norman L. Johnson (North-Holland, Amsterdam, 1983) pp. 299- 312. J.S. Marron and W. Hirdle, Random approximations to some measures of accuracy in nonparametric curve estimation, J. Multivariate Anal. 20 (1986) 91-113. H. Ramlau-Hansen, Smoothing counting process intensities by means of kernel functions, Ann. Statist. 11 (1983) 453-466. M. Rosenblatt, Curve estimates, Ann. Math. Statist. 42 (1971) 1815-1842. M. Rudemo, Empirical choice of histograms and kernel density estimators, Stand. J. Statist. 9 (1982) 65-78. R. Shibata, Asymptotically efficient selection of the order of the model for estimating parameters of a linear process, Ann. Statist. 8 (1980) 147-164. B.W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, New York, 1986). C. Stone, An asymptotically optimal window selection rule for kernel density estimates, Ann. Statist. 12 (1984) 1285-1297. 

English

Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions

Stochastic Processes and their Applications 38 (1991) 157-165 
North-Holland 
157 
Asymptotic optimality of the least-squares 
cross-validation bandwidth for kernel 
estimates of intensity functions 
Maria Mori Brooks and J. Stephen Marron” 
Department of Statistics, University of North Carolina, Chapel Hill, NC 27599-3260, USA 
Received 12 November 1989 
Revised 25 July 1990 and 21 August 1990 
In this paper, kernel function methods are considered for estimating the intensity function of a non- 
homogeneous Poisson process. A least-squares cross-validation bandwidth for the kernel intensity 
estimator is introduced, and it is proven that this bandwidth is asymptotically optimal for kernel intensity 
estimation. 
bandwidth selection * intensity function * cross-validation bandwidth * kernel estimation * nonstationary 
Poisson processes 
1. Introduction 
Let X,, X2,. . . , XN be ordered observations on the interval [0, T] from a non- 
stationary Poisson process with intensity function A(x). In this paper, we consider 
the estimation of A(x). N, the number of observations that occur in the interval 
[0, T], has a Poisson distribution with E[ N] = jl A(U) du. See Cox and Isham 
(1980) and Diggle (1983) for further information regarding point processes. 
A natural kernel estimator for A(x) is: 
iih(X)= f Kh(X-Xi), XE[Op T], 
i=* 
(1.1) 
where &(x) = h-‘K(x/h). The kernel function, K (. ), is assumed here to be a 
probability density function, and the smoothing parameter, h, also known as the 
bandwidth, quantifies the smoothness of Ah(x) (see, for example, Silverman, 1986). 
The kernel intensity estimator differs from the typical kernel density estimator in 
two ways. First, i,,(x) doesn’t include a normalization factor, K’, since ji A(x) is 
* Research partially supported by NSF Grant DMS-8902973. 
0304-4149/91/$03.50 0 1991-Elsevier Science Publishers B.V. (North-Holland) 
158 M.M. Brooks, J.S. Marron / Cross-validation bandwidth 
the expected number rather than the expected proportion of observations between 
s and t. Second, N, the number of observations, is a random variable in the intensity 
estimation setting. Theoretical properties of the kernel intensity estimator have been 
developed by Leadbetter and Weld (1983), Ramlau-Hansen (1983) and Diggle 
(1985). 
It is accepted that the choice of the smoothing parameter is usually far more 
important than the choice of the kernel function for kernel estimators. The tradeoff 
between smaller and larger values of h can be seen in the mean square error (MSE) 
of i,(x). Rosenblatt (1971) showed that a small value of h results in high variance; 
i,,(x) is affected by individual observations and hence is more variable. On the 
other hand, a large value of h results in high bias; i,,(x) is very smooth but does 
not include minor features of the true intensity function. Thus, it is desirable to find 
a data-based bandwidth that balances the effects of the bias and the variance of the 
estimate. 
We use a simple multiplicative intensity model to put a mathematical structure 
on this problem. The general multiplicative intensity model, introduced by Aalen 
(1978), is frequently used to model counting processes. See Anderson and Borgan 
(1985) for an overview of these models. 
For the related setting of kernel density estimation, Hall (1983), Burman (1985), 
and Stone (1984) have proven that the least-squares cross-validation bandwidth is 
asymptotically optimal. Hall (1983) showed that this bandwidth is optimal in 
probability by using asymptotic expansions of the cross-validation score function 
and the integrated square error (ISE) of the kernel estimator. This method requires 
the density function to have two continuous bounded derivatives. Burman (1985) 
used methods from Shibata (1980) to prove that the cross-validation bandwidth is 
optimal in probability. In his proof, there are few restrictions on the kernel function, 
and the density is a bounded L2 function. By assuming that the kernel function is 
compactly supported and Holder continuous, Stone (1984) was able to show using 
Poissonization methods that the cross-validation bandwidth is optimal almost surely 
for a continuous set of bandwidths. This was the first paper to show that the 
optimality result applied to all bounded density functions. 
In this paper, we show that the least-squares cross-validation method is asymptoti- 
cally optimal almost surely for intensity estimation. The additional difficulty in the 
intensity setting is that the number of observations is now a random variable. The 
intensity function is required to have two continuous bounded derivatives. Moreover, 
the kernel function is a bounded compactly supported probability density function. 
We use arguments similar to the martingale methods employed by Hardle, Marron 
and Wand (1990) to prove the asymptotic optimality of density derivatives. This 
method is based on a martingale inequality given by Burkholder (1973). 
In Section 2, we discuss the mathematical model for the intensity function. 
Section 3 contains the main result regarding the asymptotic optimality of the 
least-squares cross-validation bandwidth. Finally, the proof of the theorem is 
presented in Section 4. 
M.M. Brooks, J.S. Marron / Cross-validation bandwidih 159 
2. The simple multiplicative intensity model 
The simple multiplicative intensity model is a specific form of Aalen’s (1978) 
multiplicative intensity model. Suppose that X, , X2, . . . , XN are observations from 
a nonhomogeneous Poisson process with intensity 
h,.(x) = c(Y(x), XE [O, 7-1, (2.1) 
where c is a positive constant, and cr(x) is an unknown nonnegative deterministic 
function with Ic a(x) dx = 1. Given N, the occurrence times, X, , X2, . . . , XN, have 
the same distribution as the order statistics corresponding to N independent random 
variables with probability density function Q(X) on the interval [0, T]. The kernel 
estimate of h,(x) is given in (1.1). Under this model, N is a Poisson random variable 
that has expected value equal to c. In order to avoid boundary effects, we assume 
a circular design such that A,(O) = h,(T), A’,(O) = A:.( T) and A:‘(O) = A:( T). 
Asymptotic analysis provides a powerful tool for understanding the behavior of 
the kernel intensity estimator. Letting T + 00 is not appropriate since this results in 
all of the new observations occurring at the right endpoint. In the simple multiplica- 
tive intensity model, letting c + ~0 has the desirable effect of adding observations 
everywhere on the interval [0, T] and not changing the relative shape of the target 
function A,(x) in the limiting process. In other words, c-‘A,(x) is a fixed function 
as c+co. 
Since we will consider convergence properties for sequences of random variables, 
we require the values of c to come from the sequence of positive real numbers 
{c,}T=, such that c,/s+ r for some constant T> 0 as S+CO. Thus, for the sequence 
of intensity functions A,,(x) = C,CY(X) indexed by s, we construct a corresponding 
sequence of kernel estimators i;;(x). For these kernel estimators, the bandwidth h 
is dependent on s. It follows from Ramlau-Hansen (1983) that ii(x) is uniformly 
consistent and asymptotically normal as c,, + CO, h + 0 and hc, + co. 
3. Asymptotic optimality of the cross-validation bandwidth 
We are interested in finding a data based bandwidth that approximately minimizes 
the integrated square error (ISE) of ih where 
I 
T 
ISE,(h) = [l,,(x)-A(x)12dx 
0 
ZZ 
i 
7- 
I 
T 
x,,(x)’ dx -2 i,,(x)A(x) dx+ 
0 0 I 
T 
A(x)’ dx. (3.1) 
0 
For kernel density estimates, Rudemo (1982) and Bowman (1984) suggested using 
the method of least-squares cross-validation for selecting the bandwidth. In the 
intensity estimation setting, the cross-validation score function is defined as: 
CV*(h) = 
I 
7- 
xh(x)’ dx -2 ; ihr(Xi) (3.2) 
0 i=l 
160 MM. Brooks, J.S. Marron / Cross-validation bandwidth 
where ahi is the leave-one-out estimator, 
ihi( i Kh(X-XJ). (3.3) 
j=l 
j#i 
Since CL, A,i(Xi) is a method of moments estimator of jl ih(x dx, and 
,,’ Am dx is independent of h, CV,(h) is a reasonable unbiased estimate of the 
terms in ISE, (h) that depend on h. Therefore, the bandwidth that minimizes CV, (h) 
should be close to the bandwidth that minimizes ISE,(h). 
Let & be any bandwidth that minimizes ISE,(h) and &, any bandwidth that 
minimizes CV,(h) (these minima always exist since ISE,(h) and CV,(h) are 
continuous and bounded functions). Assume that: 
(a) The kernel function, K ( -), is a bounded compactly supported probability 
density function. 
(b) The true intensity function, A ( * ), has two continuous bounded derivatives. 
(c) The bandwidths under consideration come from a set H, where for each s 
and some constants p, 6 > 0, 
#(H,) = {the number of elements in H,} s cf , 
and for h E H,, cJltS s h s CL”. 
(d) For some constant T>O, c,/s+ r as s+cc. 
Assumption (b) is a common technical assumption on A(x) which allows Taylor 
expansion methods to be used. With this assumption, the true intensity function 
can have any amount of underlying smoothness. Assumption (c) can be weakened 
so that H, is a continuous interval by using a continuity argument found in Hardle 
and Marron (1985). This set of possible bandwidths nearly covers the range of 
consistent bandwidths. 
Under these assumptions, the ISE obtained with the cross-validation bandwidth 
converges almost surely to the minimum ISE. In this sense, the least-squares 
cross-validation bandwidth is asymptotically optimal for kernel intensity estimation 
under the simple multiplicative intensity model. This result is stated in Theorem 1. 
Theorem 1. If assumptions (a), (b), (c) and (d) hold, then, under the simple muftiplica- 
tive intensity model, 
I=, Ckv, 
IsE,o+ l a.s. ass+m. (3.4) 
The mean integrated square error (MISE), 
[S 
7 
MISE,(h) = E (~,,(x)-~(x))~ dx , (3.5) 
0 1 
is another error criterion that is used to evaluate bandwidth selection procedures. 
Let h, be the bandwidth that minimizes MISE,(h). By Lemma 1 in Section 4, the 
ISE and MISE are essentially the same for large s. As a result of Theorem 1, icy is 
M. M. Brooks, J.S. Marron / Cross-validation bandwidth 161 
also asymptotically optimal with respect to MISE in the sense that 
MISE&) --, 1 
MISE,(h,) a’s’ as s+co’ 
Details of the proof of Theorem 1 are presented in Section 4. 
(3.6) 
4. Proof of Theorem 1 
In this section, we outline the proof of Theorem 1. 
The mean integrated square error (MISE) of ih can be decomposed into a variance 
term and a squared bias term. Using Taylor expansion methods similar to those in 
Silverman (1986, pp. 39-40), it is straightforward to show that the MISE of ih is 
MISE,(h)=h-ir(j X.)+h4c:[:lu’K]21:,~f~(x)12+o(h-1c,+hlci) (4.1) 
as h + 0, c, +KJ and hc, + ~0. Thus, the asymptotic mean integrated square error 
(AMISE) is 
AMISE,(h)=h’c.(j K2)+h4ci[:Iu’K]iI:[o”(x)l’. (4.2) 
The two lemmas below are used to prove statement (3.4). 
Lemma 1. 
sup 
BE,(h)-AMISE,(h) ~0 
AMISE, (h) 
a.s. ass+co. 
htH, 
Lemma 2. 
sup 
CV,(b)-ISE,(h)+[CV,(b)-ISE,(b)l -,. 
AMISE, (h) + AMISE, (b) 
a.s. ass+m. 
h,bt H, 
Lemma 1 says that the ISE and the AMISE of ih(X) are asymptotically equivalent, 
and the two lemmas together imply that 
sup 
cv,(h)-ISE,(h)-[CV,(b)-ISE,(b)l -,. 
a.s. as s+co. 
h,bt H, ISE,(h)+ISE,(b) 
(4.3) 
Since ISE(&) 5 ISE(c_) and CV(&,) s CV(&), Theorem 1 follows for the simple 
multiplicative intensity model. 
Now, we must prove Lemma 1 and Lemma 2. The details of the proof of Lemma 
2 are given below; Lemma 1 is proven using similar martingale methods. 
Proof of Lemma 2. Let g( 1,2,. . . , IV) + (1,2, . . . , IV) be a random permutation of 
the numbers 1, 2,. . . , N. Define Y, = XgciJ. Essentially, the Y’s are the ‘unordered’ 
Xi’s. Since the Xi’s are observations from a nonhomogeneous Poisson process with 
intensity h(x), then given AJ, the Y(‘s are i.i.d. random variables with density 
162 M.M. Brooks, J.S. Marron / Cross-validation bandwidth 
(Y(x)~~~,~,(x). As a result, kernel density methods developed by Hardle, Marron and 
Wand (1990) can be used to study a(x). Define: 
&h(X) = c,‘x,(x), 
&hi(X) = C,‘i,i(X), 
CV,(b) = 
I 
7- 
a*,(X)’ dX-2cy’ ; &hi(X,) = c12CV,(h). 
0 i=l 
Hence, ISE,( h) = cJ2 ISE,( b), and MISE, (b) = cl2 MISE, (b). It is not difficult to 
show that 
CV,(h)-ISE,(b)-[CV,(b)-ISE,(b)] 
sup 
h,bc H, AMISE,(h)+AMISE,(b) 
= sup 
CV,(b)-ISE,(h)-[CV,(b)-ISE,(b)] 
h,be H, AMISE,(b)+AMISE,(b) 
s2 sup 
c~‘CE~ a*,,(Xi)-jl &ha-G 
hsH, AMISE,(h) 
where G=[~(N-~)/c,-~]c~‘C,N_~ ~(Xi)-[N(N-l)/cf]S,T(y2. 
Thus, it suffices to prove the following: 
sup 
CL’C,“=, a*hr(Xi)-.IoT(I*ha-G +. as ass-,co 
AMISE,(h) 
. . 
hEH, 
(4.4) 
Define: 
I I 
7 
&j(h)=K,(Y,-Y,)- &(y-y)a(y)dy-(Y(I/;)+ a’(y) dy, 
0 
x(h)=E(UGI Yi) 
= &(x-.J’b’(J’)dY- 
I II 
&(Y -Z)a(y)a(Z) dy dz-(Y(Y) 
+ 
l 
T 
a’(y) dy, 
0 
WV(h)= u,- v, 
=&(yi-y)- 
I 
&,(y-Y;)~(y)dy- Kh(y-yk(y)dy 
I 
+ 
1s 
&(Y -z)a(y)a(z) dy da, 
Ri(b)=[(N-1)/c,-l] 2 &(Y-y)a(y)dy 
[5 
- 
Kh(Y-Z)cu(y)a(z) dy dz 
M.M. Brooks, J.S. Marron / Cross-validation bandwidth 163 
Note that E( V, 1 N) = 0 and E( W, 1 yI, N) = E( Wl, ( q, N) = 0 for i,j = 1, 2,. . . , N. 
For sums over i=l, 2,..., N, Yi (the unordered observation) can be replaced 
by Xi (the ordered observation) in the summand. Thus, 
Hence, statement (4.4) holds when (4.5), (4.6) and (4.7) hold. 
ccl”=, V,(h) +. 
h”t”r: AMISE,( h) 
a.s. as s+co, 
sup 
c,*CIN_I Cjfi Wjth) +O as 
hEH, AMISE,(h) 
. . ass+co, 
C.;’ I;“=, K(h) ~ o 
f:,$), AMISE, (h) 
a.s. as s+co. 
(4.5) 
(4.6) 
(4.7) 
Therefore, in order to prove Lemma 2, it is sufficient to prove statements (4.5), (4.6) 
and (4.7). 
Conditional on N, {CF=, Vi},“=, and {CF=, C,:: Wti}kN=I are martingales with respect 
to the o-fields generated by {Y,, Y2,. . . , Yk}. Burkholder’s (1973, p. 40) inequality 
implies that for some constant A: 
E [(kyfNj, g2m~N]~AE[(,~*E[V:JNl)mIN] , / 
+A f HI K12”7 Nl, 
i=l 
(4.8) 
(4.9) 
Since N is a Poisson random variable with mean c,, E[ N”] s A,&’ where A,,, is 
constant for each m. Hence, 
c cy2m E[A,Nmh4”‘+A,N] (by (4.8)) 
s Am(c,mh4m + c;‘~+‘) 
164 M.M. Brooks, J.S. Marron / Cross-validation bandwidth 
and 
1 
2m 
s c;4”E[A,N2mh~m +A,Nm+2h-2m] (by (4.9)) 
s A,(c;2mh~m + ~;~~+~h-~“‘). 
As a result, for h E H,, m sufficiently large and some y > 0, 
E 
cTlCfv=, K 1 2m ~A,(c,mh4m+c;2m+‘) AMISE,(h) (c;‘h-‘+ h4)2m s A,[h” + csh2”‘] s A,p,c,““. (4.10) 
Using Chebychev’s theorem, 
sup I’[ I&, v;l >c;“‘AMISE,(h)] sA,,,c;(Y’2)m. 
htH, 
(4.11) 
Recall that #(H,) s ct, and choose m such that rn > 2(p +2)/y. By assumption (d), 
one can show that for the sequence {c,}, 
Thus, the Borel-Cantelli Lemma implies that 
c3’C,N,, K +. 
;:;, AMISE,(h) 
a.s. as s+a. (4.12) 
This proves (4.5). Following a similar procedure, statement (4.6) can be verified. 
Finally, consider (4.7). Since N is Poisson(c,s), it is known that [(N - c,)/&] is 
asymptotically normal. Using this fact and a Taylor expansion of Ri, one can show 
that for any O< E <$, 
Hence, for each m, there exists a constant c, such that whenever c, > c,, 
I 
2m ~ &~T’h4m-2Fm ~ A,hc,-2e.,m < A -?,,, 
(c,‘h-‘+ h4)2m . ms 
(4.13) 
for h E H, and some y > 0. (4.7) follows from (4.13) as seen above. Therefore, we 
have proven Lemma 2. 0 
M.M. Brooks, J.S. Marron / Cross-validation bandwidth 165 
References 
0. Aalen, Nonparametric inference for a family of counting processes, Ann. Statist. 6 (1978) 701-726. 
P.K. Andersen and 0. Borgan, Counting process models for life history data: A review, Stand. J. Statist. 
12 (1985)97-l%. 
A.W. Bowman, An alternative of cross-validation for the smoothing of density estimates, Biometrika 71 
(1984) 353-360. 
D.L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973) 19-42. 
P. Burman, A data dependent approach to density estimation, Z. Wahrsch. Verw. Gebiete 69 (1985) 
609-628. 
D.R. Cox, and V. Isham, Point Processes (Chapman and Hall, London, 1980). 
L. Devroye and L. GyBrfi, Nonparametric Density Estimation: The L, View (Wiley, New York, 1984). 
P.J. Diggle, Statistical Analysis of Spatial Point Patterns (Academic Press, London, 1983). 
P. Diggle, A kernel method for smoothing point process data, Appl. Statist. 34 (1985) 138-147. 
W. HLrdle and J.S. Marron, Optimal bandwidth selection in nonparametric regression estimation, Ann. 
Statist. 13 (1985) 1465-1481. 
W. Hlrdle, J.S. Marron and M.P. Wand, Bandwidth choice for density derivatives, J. Roy. Statist. Sot. 
Ser. B 52 (1990) 223-232. 
P. Hall, Large sample optimality of least squares cross-validation in density estimation, Ann. Statist. 11 
(1983) 1156-1174. 
M.R. Leadbetter and D. Wold, On estimation of point process intensities, in: P.K. Sen, ed., Contributions 
to Statistics: Essays in Honour of Norman L. Johnson (North-Holland, Amsterdam, 1983) pp. 299- 
312. 
J.S. Marron and W. Hirdle, Random approximations to some measures of accuracy in nonparametric 
curve estimation, J. Multivariate Anal. 20 (1986) 91-113. 
H. Ramlau-Hansen, Smoothing counting process intensities by means of kernel functions, Ann. Statist. 
11 (1983) 453-466. 
M. Rosenblatt, Curve estimates, Ann. Math. Statist. 42 (1971) 1815-1842. 
M. Rudemo, Empirical choice of histograms and kernel density estimators, Stand. J. Statist. 9 (1982) 65-78. 
R. Shibata, Asymptotically efficient selection of the order of the model for estimating parameters of a 
linear process, Ann. Statist. 8 (1980) 147-164. 
B.W. Silverman, Density estimation for statistics and data analysis (Chapman and Hall, New York, 1986). 
C. Stone, An asymptotically optimal window selection rule for kernel density estimates, Ann. Statist. 12 
(1984) 1285-1297. 


file:///data/core-remote/dit/data/elsevier/pdf/c08/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMzA0NDE0OTkxOTAwNzZv.pdf

Asymptotic optimality of the least-squares cross-validation bandwidth for kernel estimates of intensity functions

Abstract

Similar works

Full text

Available Versions

Elsevier - Publisher Connector

Carolina Digital Repository

Elsevier - Publisher Connector