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Nonparametric inference for a doubly stochastic Poisson process
Stochastic Processes North-Holland
and their Applications
331
45 (1993) 331-349
Nonparametric inference for a doubly stochastic Poisson process Klaus J. Utikal Department
of Statistics,
Uniuer.sity
of Kentucky,
Lexington,
USA
Received 14 June 1990 Revised 29 July 1991
Consider a doubly stochastic Poisson process whose intensity A, is given by A, = cu(Z,), where (Y is an unknown nonrandom function of another (covariate) process 2,. Only one continuous time observation of counting and covariate process is available. The function zJ(z)=~~ G(X) dx is estimated and the normalized estimator is shown to converge weakly to a Gaussian process as time approaches infinity. Confidence bands for 1 are given. A uniformly consistent estimator of cy is obtained. Consistent tests for independence from the covariate process are proposed. doubly stochastic Poisson process*dependence limit theorem * Gaussian processes * asymptotic
index* kernel testing
function
smoothing*martingale
central
1. Introduction
Counting process models in which the intensity of counts depends on some other random process are used in many fields. In the present work it is assumed that the counting process is doubly stochastic (or conditional) Poisson with an intensity A(t) of the form A(t) = c~(Z(t)), where (Y is a nonrandom function of some covariate process Z (sometimes called the information process). As an application of this model we consider for example a stream of photoelectrons generated by a photodetector in an optical field of an incoherent light source of randomly fluctuating intensity. We refer to Snyder (1975) for this and several other examples. We consider the problem of estimating the intensity integrated over the covariate space ti( z) = I:,, cy(x) dx. The function
(Yis only known to be nonnegative,
Lipschitz
continuous on intervals between a finite number of points at unknown locations where it may have jumps of finite but unknown size. Only one observation of counting and covariate process, taken continuously over a time interval [0, 71, is available. Herein lies the principal difference to the work of McKeague and Utikal (1990) on nonparametric intensity estimation for general nonlinear semimartingale regression models when a (preferably large) number of (ideally i.i.d.) replications are observed continuously over a finite time interval. Correspondence to: Prof. Klaus J. Utikal, Department KY 40506-0027, USA. Research supported by NSF and the Commonwealth
0304-4149/93/$06.00
0
1993-Elsevier
Science
of Statistics, of Kentucky
Publishers
University under
of Kentucky,
Epscor
B.V. All rights reserved
Grant
Lexington, RII-8610671.
332
K.J. Utikal
In Section function
2 an estimator
N is obtained
and on the dependence
/ Poisson inference
J& of .& is given.
by the method structure
of kernel
of Z needed
From
this an estimator
functions.
& of the
The assumptions
on LY
for the proofs of the results obtained
are listed next and it is shown that they hold for a substantial class of processes. Under the conditions listed in Section 2, in Theorem 3.1 we show weak convergence 1 of the appropriately normalized & to a Gaussian martingale as T + CD. As a consequence we show weak uniform consistency of cu^.Theorems 3.2, 3.3 describe the asymptotic behavior of 2, if compared to the estimator of .ti that should be used under the hypothesis that the intensity does not depend on Z. Our estimator is related to sieve estimators used by McKeague (1986) and Karr (1987) to estimate time integrated intensities in the multiplicative intensity model of Aalen (1978, 1980). The special properties of the sieve that we are using, called the histogram sieve, make the derivation of our results possible; under the assumption that the counting process is doubly stochastic, & is a martingale difference array and martingale methods are applicable. Our weak convergence arguments rely on a functional martingale central limit theorem. Applications of these results are given in Section 4. Confidence bands for the unknown parameter ti are derived. Tests for independence of the counting process from the information process are discussed and their consistency is shown. An asymptotic XI-test is written down explicitly. The main ideas of proof are given following each theorem while the technical details are worked out in the lemmas contained in the Appendix.
2. Notation
and assumptions
Let (0, 9, P) denote a complete probability space, (9,, t 2 0) a nondecreasing, right continuous family of sub-a-fields of %, where 9co contains all P-nullsets in 5. The counting process N has right-continuous sample paths and is adapted to (9,). The process
Z is predictable N(t)
=
1’0
and for simplicity
~(z(s))
scalar-valued.
N and Z are related
ds + M(t),
where (M,, 9,) is assumed to be a zero-mean local martingale. We assume is a doubly stochastic Poisson process (see BrCmaud, 1981), i.e.: (Zl)
Z(t)
is so-measurable
by
that Z
for all t 2 0.
Condition (Zl) excludes any dependence of the covariate on the counting process. It also has important technical consequences: Conditional on 9co the estimator .& defined below is a sum of independent random variables which, unconditionally is a sum of dependent random variables which can be considered a martingale difference array. This method of proof cannot be carried over to other classes of
K.J.
Utikal
/ Poisson inference
333
processes where (Zl) does not hold, such as diffusions, even though the correspond,. ing S$ can still be considered as an estimator for the integrated drift in this case. For ease of exposition
we will assume
Z to be one-dimensional.
Inference
to TV [0, 11. Extensions s~(~)=~~~cI(s)~ s will be restricted and other compact subsets are straightforward.
a(t) and dimensions Assumptions
for
to higher
on cx.
(A) (Y is a nonnegative,
nonrandom
function
number of jumps of unknown size and at unknown (on intervals between those jumps).
which,
except
locations,
maybe
is Lipschitz
for a finite continuous
We will define the estimator Gr next. We need to introduce the following notation. For a nondecreasing sequence of positive integers n = n(7), define partitions of the k=1,2 ,..., n. unit interval by 9k,7 =((k-l)/n,k/n], Assumptions
(Pl) (P2)
on the partition.
n/7+0 r/n’+0
as ~-+a, as 7+00.
Let T,,, be the total time up to 7 that the process
Z spends
in the kth subinterval,
7 T,,, =
I{Z(s)
I0
E %,,I ds,
where Z is the indicator function. The counts kth subinterval are denoted by N,,,, i.e.,
registered
at times when Z is in the
7 N,,, = As estimators
I0
Z{Z(s) E &,,> dN(s).
of ti and cy we propose
&,(z)=~‘:‘~ n k-r T,,, &(z.)=$[
(with O/0=0),
K(y)&(dx),
where [x] denotes the integer part of x and K is a nonnegative kernel function of bounded variation V(K), with bounded support, [ K(x) dx = 1 and window size b,. To assure some asymptotic uniformity of the sojurn times {T,,,, k = 1, . . , n} for T + 00 we will make assumptions on the underlying process Z. Define (2.la)
(2.lb)
K.J.
334
Assumptions (22)
on the process
There
uniformly
exists
Utikal
/
Poisson ii+wnce
2.
positive
a (strictly)
continuous
function
f such
that f7 ‘f
on [0, l] as T+ 03.
How to characterize problem
in itself.
the processes
If we assume
that satisfy
condition
Z to be stationary
(22)
is an interesting
with marginal
density
f then
under condition (22) a uniformly consistent estimator of f is provided by the histogram in which the height over each cell 9k,T is proportional to the time T,.,. That this condition, which is crucial for our model (see Lemmas 3,4,5), is indeed satisfied for a rich class of processes will briefly be illustrated next. Proposition
2.1. Let n + ~0, n/r + 0 as T + 00. Suppose
(i) for all O< s
the random
I$I,,TJ;,\,t4
ds -f(x)
uniformly
variables Z(s)
have densitiesf,,,,(x).
Suppose
= o(l),
(2.2)
in x E [0, I] as r + Co for some continuous function f;
(ii) for all 0~ s < s’< ~0 the random frC,,zC,z,(x,
that Z satisfies the conditions
y). Suppose
variables
Z(s),
that there exists a nonnegative
Z(s’)
function
have joint
densities
p(s, s’) for 0s
s < s’
such that u p( s, s’) ds’ < ~0, sup ’ 5\ IfZC\,,,Z,\I( Y I x ) -fz,c,(Y)l
s P(% 0
for all x, y E [0, 11. Under these assumptions
We note that relation
Z satisfies condition
(2.2) is trivially
satisfied
(22).
if Z is stationary.
It holds also if
uniformly in y, which is satisfied under conditions given in Doob (1953, V.5, VI.2) when Z is Markovian with stationary transition densities. Condition (ii) of the above proposition implies that the transition density approaches the marginal at a moderately fast rate for s’-+ ~0. Multiplying (ii) by fit,,(x) and integrating (ii) over sets A, B implies (P(Z(s) s P(S,
E A, Z(s’) s’)P(Z(s)
E B) - P(Z(s)
E A)P(Z(s’)
E B)l
E A)A(B),
where A(B) is the Lebesgue measure this is the uniform mixing condition,
(2.3) of B. For families see Renyi (1958).
of uniformly
bounded
sets
K.J. Utikal
Proof
of Proposition
Substituting
2.1. For simplicity
choose
A = B = 9k,r in (2.3) and multiplying
and applying
Fubini’s
theorem
335
/ Poisson inference
x to vary over the unit by (H/T)‘,
integrating
interval. over S, s’
we obtain
or Var{j;,,> = O( L/ T), uniformly
in k as T+ CO.Therefore
(1(x E A.,),
by Jensen’s
inequality
and the orthogonality
of
k = 1, . . . , ~1,
E(]Syp, IL,(x) -.m)l
_II $)-z&W-f(x) dx+o(l), I_ !,s ,1 J~I,.
= n sup
where the last equality
holds since f(z) = I? JY,,;,f(x) dx+o(
1) for any z E $A,T.
0
Condition (ii) is used in Castellana and Leadbetter (1986) for stationary processes, where it is shown that this condition is satisfied for example for every stationary Gaussian process with zero mean and covariance function r( 7) = 1 - CIT~‘~ + o( IT)“) for 0~ (Y<2 (cy = 1 for the Ornstein-Uhlenbeck process). In order to include Gaussian processes with more ‘regular’ sample paths ( LY= 2) we recall the definition of Castellana and Leadbetter’s (1986) dependence index function
assumed satisfied.
finite for all T> y > 0. Now we can give another
Proposition
2.2. Let n + Co,
criterion
for (22) to be
0 as T + Co. Suppose that 2 is a stationary process with continuous marginal density j Let {y_, T SO} be positive constants with y7= 0(T/n2) as T+ 00. Suppose that the dependence index function &(y) for 2 satisfies p7( y7) = o( nrT) as T+ 00. Then condition (22) is satisfied. T/n2
+
K.J.
336
Proof.
Utika/
/ Poisson itzference
First we note that f& as defined
estimator
of the form (l/r)
in (2.la)
can be considered
a kernel
density
ds, where xk is the center and &(xh -. )
JoTS,( xk -Z(s))
the indicator of 9,,. It follows from Theorem 5.4 of Castellana and Leadbetter A (1986) that Var{fk,7} = o(n-yT/ r) uniformly in k. Hence by the same argument as in the proof
of the previous
proposition,
0( n’y,/r) continuity
and hence f7(x) +f( of 1: 0.
ECsup,l.fAx)
x j uniformly
=,I;=,
Var.fk,, =
in x as r + ~0 by stationarity
- $T(x)l12
of Z and
3. Main results We will assume the conditions (A), (Pl), (P2), (Zl), (22) to hold throughout. Our first result consists of the weak convergence of the appropriately normalized estimator when the interval of time during which the observation is made tends to infinity. As usual, D[O, l] denotes the spaces of real valued functions which are right-continuous with left limits, equipped with the Skorohod topology. Theorem
3.1. J;(.&&)-+m,
weakly in D[O, l] as r + ~0, where m is a Gaussian
process with mean zero and
=ln=Z Q(X)
Cov(m(z,L m(z2))=
I 0
Proof.
We decompose
2,
f(x)
dx.
into
& = &+&,
(3.1)
where
where 7 I{Z(s) E &,}(dN(s) - a(Z(s)) I0 To prove the theorem we have to show that M,,, =
Jr;,,::?,
IK(z)I
where R,(z)=&(z)-&(z)
5 0
as ~+m,
ds.
(3.2)
K.J. Utikal
/ Poisson inference
337
and ?+ J;J& weakly
+ m,
(3.3)
in D[O, l] as r + ~0. To show (3.2) we observe
that by (A),
7 I{Z(~)E&&(Z(~))
ds=(a(zk)+O(lln))T,,,
(3.4)
Jo
uniformly
in k (except
uniformly
in z. Also a(z)
uniformly
maybe
for boundedly
many),
for ZI,E &7 arbitrary.
Hence
=f;i;
in z, hence
But I;‘=, Z{Tk,T=O} 5 0 as ~+CO by (22). This shows (3.2). It follows from Lemma 1 that {M& T,_,, k = 1, . . . , n} are independent conditionally on Sz = o{Z( t), t 1 O}. Therefore by Lemma 2 it is a martingale difference array with respect to {Fk,,,,k=l ,..., n} where .YF~,~=(T{N,,~,..., N,;,,,,}vFz. Hence (3.3) follows from an application of Rebolledo’s functional martingale central limit theorem (see Helland, 1982). The details are worked out in Lemma 3. 0
As a first consequence of the above theorem we obtain the following consistency result on &. The argument is similar to that of Ramlau-Hansen (1983) given for counting integrals.
Corollary.
Suppose
that CYis Lipschitz continuous
T+ 00. Then & 5 a uniformly
on [0,11, and b, + 0, &b,
on compact subsets of (0, 1) as T+ ~0.
Proof.
a(x)dx+$l
=;j
K(y).rp(d*)+O(b,),
K(y)(,(,)-+))dx
+ a3 as
K.J.
338
by Lipschitzness
Utikal J Poisson inference
K has bounded
of (Y and because
la-&,1(z)=
I;1
support.
K(+G&)(dx)
+
(&&)(x)dK
Hence,
integrating
by
O(b)
+ O(h)
(7) T
+supi(&-d)(x)lV(K)+O(b,) T X ~~O,il/i;)+O(b,)=o,(l), * where the last inequality
follows
from the previous
theorem.
0
We will make now the additional
assumption
that (Y(Z) is constant,
i.e.:
H,: There exists some (unknown)
cr,) such that for all z E [0,11: a(z) = (Ye.
For a bounded covariate the above hypothesis specifies that the counting process is a homogeneous Poisson process (if we suppose 0~ Z(t) G 1). Under Ho a natural estimator of 22 on [0, l] is 2,(z)
= z
C;=,N~,7 I,“=,
The asymptotic
T,,,’
distribution
of the difference
between
the two estimators
is given
in the next theorem. Theorem
3.2.
If Ho holds then
J;(2T-2T)+m,, weakly in D[O, 11 as r+ ~0 where m,
is a Gaussian
process
with mean
zero and
covariance
(3.5)
K.J.
Proof.
It was shown
Utikal / Poisson inference
in the proof of Theorem
339
3.1 that
A&=&+.&+& where
and
Similarly
we decompose
under
Ho,
(3.6)
&=&+&, where “J&(z) = z
c,“=, C;=,
Hence
it remains
Mh,T Tk.7
.
to be shown
that
J;(Ji-X,)+m,,
(3.7)
weakly in D[O, I] as r + ~0. By Theorem 3.1 we already know that fi in D[O, l] as r + 00, where we can choose for m the representation integral with respect to a standard Wiener process W:
J&, + m weakly as a stochastic
m(z) =
(3.8)
Next we show that J;.&(l) where
m,(l)
c~/~,!,f(x)
Y m,(l), is a normal
(3.9) random
variable
with
mean
zero
and
Var(m,(l))
=
dx. We write J&, = A,( 1, z), where .&!J r, z) = z ‘$~CM~r(f) I 1 k.7
(with O/O = 0),
where M,,,(t)
=
’ I{Z(s) I0
E A,,l(dN(s)
- a(z(s))
ds).
We note that ~%~(t, z) for fixed LY,r is a martingale with respect to s,,, 0~ t Q T by (Zl). Then (3.9) follows easily from (22) and a central limit theorem for counting integrals as it is stated for instance in Karr (1986). Next we observe that (3.9) implies
K.J. Urikal / Poisson in,ference
340
6 .a, + ml weakly in D[O, 11, where m,(z) = zm,(l). the representation
‘,r(~(1 m
m,(z) = z where
W is the same process
J;(&, weakly
s;f(x,
dx
We choose for the process
m,
(3.10)
d W(x),
as in (3.8). Now we show (3.11)
J-6)‘+ (m, m,)‘,
in (D[O, 11)’ as T + ~0. This implies vG(R,-X,)+m-m,,
weakly
in (D[O, l])l as T + CO,where (~-~r)(z)=~(j-z$=qdW(x)-zj-O’],~dxdW(x)). 0
(3.12)
0
Since Cov{(m -m,)(z,), (m -m,)(z2)}=Cov{m,(z,), m,(z*)} for all OGz,
; 4(&(x,+,)-&x,))+b,(&(x,+,)-&(x4) I-1 3
ii, Q,(m(x,+,)-
foranyO=x,,
Details
as ~-+a. side of (3.13) it can be seen that this is a sum over a to which a martingale central limit theorem can be out in Lemma 5. 0
are carried
The next theorem considered
shows that under deviation
in the previous
(3.13)
m(xi))+b,(m,(x,+,)-m,(xi)),
theorem
from Ho the difference
of estimators
will tend to infinity.
Theorem 3.3. Let n -3 ~0, n/ r2 + 0 as r + CO. Suppose that Z satisfies the conditions qf Proposition 2.1 and .fic,,(x) uniformly
ds-f(x)
in x E: [0, l] as 7 + Co for some continuous function J;@(z)-.Tqz)l:
(3.14)
=0(1/J;),
co
f: Then
as r+co,
for cdl z E [0, l] for which Z
1 a(x) dx # z
a(x)_/“(x) dx.
(3.15)
K.J. Utikal
Condition Proposition Proof.
(3.14) is similar
in.ferrnce
/ Poisson
341
to (2.2) and the same comments
as those
following
2.1 apply.
As in (3.6) we have &?,=~*+&+R,,
where *S&:(z) = z
’ a(x)f(x) 50
R,(z) = 2;(z)
dx,
- 9l*,
where
IT
I{Z(s)
Th=1
E
&,}a(Z(s))
ds.
c,
BY (3.4), %!2:(2)=2
uniformly
7
a(zA)T,,,+O(l/n)
in z. Also &c4”(z)=z
uniformly
i h-l
; a(zl) k=I
f(x)
dx+O(lln)
in z, hence
c ,,_> sup a(z)- ;,‘1,c 2c ,
I+-
n {,,.f(d
dxl+O(lln)
.L, ,,(x) ds -f(x) where the last equality
follows
from the same arguments
+0(1/J;),
as those given in the proof
of Proposition 2.1. It follows that 6 sup, lR1-(z)l = O,(l). Also, as indicated in the proof of Theorem 3.2, it is easy to see that &J@, + mz weakly in D[O, l] as T+ 00, where m?(z) = z j: Ja(x)f(x) d W(x) (compare (3.9)). Hence it follows from Theorem
3.1 that ~~~~(z)-~(z)~=~)~(z)-~*(z)~+Op(l)~~cc.
0
4. Applications Conjidence
bands
As a first application of Theorem 3.1 we construct asymptotic the unknown parameter ti, We introduce the function H(z) =Var(m(z))
=
confidence
bands
for
K.J. Urikal
342
/ Poisson irzference
Then (4.1) weakly
W,, is the Brownian
in D[O, l] as T+CO, where
in Lemma
4 that a uniformly
consistent
estimator
bridge
process.
for the unknown
It is shown
function
H is
given by
fi for H in (4.1) we obtain the following band for Sp:
Substituting confidence
.a4 * cc,pp(
1+~).
asymptotic
ZE[O, 11,
where P[sup,,. ,. ,,-I W”( t)( > c,,] = N. A table for the distribution can be found in Hall and Wellner (1980). Testing for the independence
from
lOO( 1 - a)-percent
of sup,,. ,. rlrl W”( t)l
a covariate
As a second application we derive tests for the null hypothesis Ho: a(z) is constant for all z E [0, 11. Under H,,, 2, and & are both estimators for &. On the other hand, if H,, is not true, 2, remains a valid estimator for .PI while 8, is a suitable estimator for a different parameter Op”. Therefore we will base any test statistic on the difference between the two. Its asymptotic distribution is given in Theorem 3.2. From this theorem the asymptotic distribution of any continuous functional of of the continuity theorem (G&z) - B(z)), 0 G z G 1) can be found by application (Billingsley,
1968). As examples
5, = fi
,,sup, lk7(z)
we suggest
- 2T(z)\
(s&(z) - &( z))I dz
&=T Hence
(Kolmogorov-Smirnow),
(Cramer-von
f (I,-~,(z,~,)-~~(z,)+~~(z,-,))’ ,=I
(chi-square).
we reject H,, if 5, > ci:‘, i = 1, 2, 3, where
= (Y, ,,1
P
(
c (m,,(z,) - m,,(z,_,))‘> ,=I
CX’
Mises),
)
= a.
K.J.
Utikal / Poisson inference
343
For computing c,, one needs to substitute the unknown parameters c~,f(x) by some uniformly consistent estimators in (3.5), for example by & = L&( 1) and f by its smoothened histogram method see McKeague
as in assumption (22). For a rigourous and Utikal (1990, Proposition 4.2).
justification
of this
A difference between 5,) & and & is that the latter only involves finite-dimensional distributions, hence ci:’ can be computed explicitly by transformation of a multivariate normal distribution. ci,’ ’ , cbf’ can be obtained by simulating the process m, given in (3.5), for which its representation (3.12) could be helpful. The consistency of the tests using .$, or & follows from Theorem 3.3. We now indicate briefly how to conduct the asymptotic ,$-test. (1) Divide the region ((0, l), say) into n subintervals of equal length such that T,,, > 0 for k = 1,. . . , n without any T,,, being very close to zero. (2) Compute the (column) vector X, = (x,.,), k = 1,. . . , n, where N L.7 N, XL,, =---) TA.7 T, where N7=Cz=, NL,,, T,=C~=,TL,,. (3) Compute a (generalized) inverse of the matrix V ,,,T =
V, = (u,,,,), i,j = 1,. . . , n, where
if i=j=k,
(l/T,,,)-(l/T,),
if i#j.
1 -(l/T,),
(4) Reject H,, if 5 = ( T7/ N,)X: VFX, > c,,, where c, is the (1 - Lu)-percent tile of the x’-distribution with q degrees of freedom, where q is the rank of the matrix V=(v,;), i,j=l,..., n, where uii =
j:klr,,ln
l/f(x)
1 -(ll~z~:,f(x,
dx - (l/n’Ixf(x)
dx)
dx),
if i =j = k, if i#j.
It follows from Theorem 3.2 that the above test will have approximately size 1 - (Y if r is large. Usually we would expect V to be of full rank (not in the important special case f(x) = I,,,,,,(x)). In the above V, was obtained by estimating f(x) in (3.5) by nTJr if xE $r,T. The use of a more sophisticated (and less simple) density estimator
may increase
the performance
of the test.
Appendix Lemma
1. ne
random
variables
{ Nk,T, k = 1, . . . , n} are independent
conditionally
on So. Proof.
Conditioned on So, N,,,(t) = 5: Z{Z( s) E 9,,} d N( s) are Poisson processes with respective rates hk,J t) = I{Z( t) E 9a,,}a (Z( t)), for k = 1, . . . , n. For any subfamily NkI,,, . . . , N,,,,, we can derive the backward Kolmogorov equations for
K.J.
344
Utikal / Poisson inference
P;::,.-;.-,,;:,‘,,(f)%‘P( A$, ,,( t) = m, , . . ) IV,,,,,(t) = m, 1.FJ, for all t 2 0. The solution can be shown to satisfy p;I,‘ ‘.‘..,‘L;?(t)=II
k,
P;C,(t).
Lemma 2. {(W.,lTk,,, K,,,,),
observing
that
II,,&,(t)
= 0
q
k= 1,. .
n 1 IS a martingale
dlfereerence arrqy.
Proof.
It follows from the theory of counting processes that 56 I{Z(s) E .J4a,,}~(Z(s)) ds is the s,-predictable compensator of Jh Z{Z(s) E $k,,} dN(s), hence (since
T,,, E SC]),
’ I{Z(s)
E
E A,,>
TIl.7
(see Bremaud,
dN(s)
= E
1981). But the right-hand
’ I{Z(s)
I
o-sup :- 1 Q(Z) Hence
ElM,,,/T,,,I
Lemma
3. &
0
E A,,>
T A.7
’ IW(s)
E -%,,I TX,7
a(Z(s))
ds
side is less than or equal to
ds 4 ,,syp, a(z)
property
follows
from Lemma
1.
q
J&, + m weakly in D[O, l] as T + ~0.
Proof.
Since (..G,(z), z E [0, 11) is not necessarily it by an L’-integrable martingale 2:. Define
for some 6 > 0. Since
T,,, E .F,, the martingale
Moreover
it is clearly
Lh-integrable
cZ{T,,,~i3
forsome
L’-integrable,
property
for k = 1,2,.
k: lsk
we will approximate
is conserved
by Lemma
2.
. . Also
* 1Mk.I p 1 --1-+O k=l
Tk,,
as r+ Co, since I{T,,,sS
for some k: l~k
as r+co
(by assumption
(22)).
(Note that I+,,, s 0 implies Y,,Z+,,,.J%0 for any random variables Y, defined on the martingale same space as Z,,,.) To prove that A.,&: + m we apply Rebolledo’s functional CLT (see Helland, 1982, Theorem 3.2a). Set MA,,Z{ T,,,> S} = Mf,,. We have to verify the conditions (A.la)
K.J.
345
Utikal / Poisson inference
for each z as r + ~0, (A.lb) for all e > 0 as T+ CO.First verify (A.la):
(by Lemma
2 and Theorem
5; Z{Z(.r) E &,&(Z(s)) Z-E,7 =
uniformly Hence
in k (except
where the convergence we observe that
maybe
@(zh)+o(lln)
boundedly
in probability
many)
follows
9.2.1 of Chung, ds Z{T,,,>
S>
1974)
I 01 9
Z{Th,T>61 (by (Zl)), as V-+ CO,where zk E ,ak,., arbitrary.
by assumption
(22). To verify (A.lb)
/
3k_‘,,n,7]) “2
‘2
But by Johnson
and Kotz (1969, Chapter
4),
gol=h
(A.2)
E[M4,,,(S0]=h+3h2~4(h+1)2,
(A.3)
E[M:,,I and
where A=
Z{Z(s) E &}a(Z(s))
ds s
sup
o- z- I
city,,.
K.J. lltikal
346
1 Poisson inference
Therefore (A.4) in k by condition
uniformly Lemma
4.
Proof.
By uniform
uniformly
(22). This proves the lemma.
continuity
of a/f
(except
in z with xk E .9k,T arbitrary. -A__
maybe
But
NkT U(Xk)
7
nTk,,
T,,,
f(xk)
_?I__
1
s z1+12+13+147
where I,=
Nk,r
II
nTk.7 f(xk 1 12=
I
--n;.,
1
=hk) 4
f(Xk)
3
k.7
n--
II
ftXk)
,-cy(xk) T LT
1 a(%),
Tf(Xk)
Tk.7 n7fo-’
.
But by (22),
uniformly
in k as T + ~0. Also, as in Lemma +-
cx(Xk)
z+0,
k,T
uniformly
in k. This proves
the lemma.
0
3,
0
in finitely
many
points),
K.J. Utikal / Poisson inference
Lemma5.
347
F~rO=x,,<...
J;
I?(A,(x,+~)-~/i(xi))+b,(JU,(x,+,)-JIZ,(xi)) i=,
Proof.
Similarly as in the proof of Lemma 3, M,,, has to be replaced by a truncated and square integrable martingale difference array. Suppose this has been done. Also suppose x,+, -x,_, > l/n for all i. We will show equivalently J;
; 4&(x,+,)-&x,))+c,J;&(l) ,=,
where K
co=-
C b,(x,+,-xi). r=,
Now
To apply a martingale the conditions
CLT (see Helland,
1982, Theorem
2Sa)
we have to verify
(A.5a) and secondly
(ASb)
K.J. Utikal / Poisson inference
348
To show (i), using (3.8) and (3.10), note that Var{ .
’
m l;f(x)
dx d w(x)
where z,=ao;
[nx,+,l TZ{TQ > a> 1
a$ i=,
n
k=[nx,]+, [“X,+1]
I,=&:,
Tk,r
>
61
by assumption
;
,=,
a,’
n
CC;=,
[?+,I c k-[nx,]+l
M;,,
(
---,
Td2
“““‘~;f(:,
dx’
7Z{ Tk,T> 61 I;=,
(22). This shows (ASa).
[
af
p
c
z,=2ct”co
where
7Tk,J{
c i=,
nTk,,
k=rnx,l+~
,=I
;rE
K ~acu,
Th.7 Next show the Lindeberg
condition
(A.5b).
K.J.
by (A.3), (A.4) and (22).
Utikal
/ Poisson in,ference
349
Similarly
Finally
This proves
the lemma.
0
Acknowledgement
The problem of estimating (Ywas brought to my attention by I.W. McKeague whom I also want to thank for many instructive conversations. My thanks also go to the referees whose comments and suggestions have led to an improved version of this article.
References 0.0. 0.0.
Aalen, Nonparametric inference for a family of counting processes, Ann. Statist. 6 (1978) 701-726. Aalen, A model for nonparametric regression analysis of counting processes, Lecture Notes in Statist. No. 2 (Springer, New York, 1980). P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968). P. BrCmaud, Point Processes and Queues (Springer, New York, 1981). J.V. Castellana and M.R. Leadbetter, On smoothed probability density estimation for stationary processes, Stochastic Process. Appl. 21 (1986) 179-193. K.L. Chung, A Course in Probability Theory (Academic Press, New York, 1974). J.L. Doob, Stochastic Processes (Wiley, New York, 1953). W.J. Hall and J.A. Wellner, Confidence bands for a survival curve with censored data, Biometrika 67 (1980) 133-143. I.S. Helland, Central limit theorems for martingales with discrete or continuous time, Stand. J. Statist. 9 (1982) 79-94. N.L. Johnson and S. Katz, Discrete Distributions (Houghton Mifflin, Boston, MA, 1969). A.F. Karr, Point Processes and their Statistical Inference (Dekker, New York, 1986). A.F. Karr, Maximum likelihood estimation in the multiplicative intensity model via sieves, Ann. Statist. 15 (1987)473-490. I.W. McKeague, Estimation for a semimartingale regression model using the method of sieves, Ann. Statist. 14 (1986) 579-589. I.W. McKeague and K.J. Utikal, Identifying nonlinear covariate effects in semimartingale regression models, Probab. Theory Rel. Fields 87 (1990) I-25. H. Ramlau-Hansen, Smoothing counting process intensities by means of kernel functions, Ann. Statist. 11 (1983) 453-466. A. RCnyi, On mixing sequences of sets, Acta Math. Acad. Sci. Hung. 9 (1958) 215-228. D. Snyder, Random Point Processes (Wiley, New York, 1975).
and their Applications
331
45 (1993) 331-349
Nonparametric inference for a doubly stochastic Poisson process Klaus J. Utikal Department
of Statistics,
Uniuer.sity
of Kentucky,
Lexington,
USA
Received 14 June 1990 Revised 29 July 1991
Consider a doubly stochastic Poisson process whose intensity A, is given by A, = cu(Z,), where (Y is an unknown nonrandom function of another (covariate) process 2,. Only one continuous time observation of counting and covariate process is available. The function zJ(z)=~~ G(X) dx is estimated and the normalized estimator is shown to converge weakly to a Gaussian process as time approaches infinity. Confidence bands for 1 are given. A uniformly consistent estimator of cy is obtained. Consistent tests for independence from the covariate process are proposed. doubly stochastic Poisson process*dependence limit theorem * Gaussian processes * asymptotic
index* kernel testing
function
smoothing*martingale
central
1. Introduction
Counting process models in which the intensity of counts depends on some other random process are used in many fields. In the present work it is assumed that the counting process is doubly stochastic (or conditional) Poisson with an intensity A(t) of the form A(t) = c~(Z(t)), where (Y is a nonrandom function of some covariate process Z (sometimes called the information process). As an application of this model we consider for example a stream of photoelectrons generated by a photodetector in an optical field of an incoherent light source of randomly fluctuating intensity. We refer to Snyder (1975) for this and several other examples. We consider the problem of estimating the intensity integrated over the covariate space ti( z) = I:,, cy(x) dx. The function
(Yis only known to be nonnegative,
Lipschitz
continuous on intervals between a finite number of points at unknown locations where it may have jumps of finite but unknown size. Only one observation of counting and covariate process, taken continuously over a time interval [0, 71, is available. Herein lies the principal difference to the work of McKeague and Utikal (1990) on nonparametric intensity estimation for general nonlinear semimartingale regression models when a (preferably large) number of (ideally i.i.d.) replications are observed continuously over a finite time interval. Correspondence to: Prof. Klaus J. Utikal, Department KY 40506-0027, USA. Research supported by NSF and the Commonwealth
0304-4149/93/$06.00
0
1993-Elsevier
Science
of Statistics, of Kentucky
Publishers
University under
of Kentucky,
Epscor
B.V. All rights reserved
Grant
Lexington, RII-8610671.
332
K.J. Utikal
In Section function
2 an estimator
N is obtained
and on the dependence
/ Poisson inference
J& of .& is given.
by the method structure
of kernel
of Z needed
From
this an estimator
functions.
& of the
The assumptions
on LY
for the proofs of the results obtained
are listed next and it is shown that they hold for a substantial class of processes. Under the conditions listed in Section 2, in Theorem 3.1 we show weak convergence 1 of the appropriately normalized & to a Gaussian martingale as T + CD. As a consequence we show weak uniform consistency of cu^.Theorems 3.2, 3.3 describe the asymptotic behavior of 2, if compared to the estimator of .ti that should be used under the hypothesis that the intensity does not depend on Z. Our estimator is related to sieve estimators used by McKeague (1986) and Karr (1987) to estimate time integrated intensities in the multiplicative intensity model of Aalen (1978, 1980). The special properties of the sieve that we are using, called the histogram sieve, make the derivation of our results possible; under the assumption that the counting process is doubly stochastic, & is a martingale difference array and martingale methods are applicable. Our weak convergence arguments rely on a functional martingale central limit theorem. Applications of these results are given in Section 4. Confidence bands for the unknown parameter ti are derived. Tests for independence of the counting process from the information process are discussed and their consistency is shown. An asymptotic XI-test is written down explicitly. The main ideas of proof are given following each theorem while the technical details are worked out in the lemmas contained in the Appendix.
2. Notation
and assumptions
Let (0, 9, P) denote a complete probability space, (9,, t 2 0) a nondecreasing, right continuous family of sub-a-fields of %, where 9co contains all P-nullsets in 5. The counting process N has right-continuous sample paths and is adapted to (9,). The process
Z is predictable N(t)
=
1’0
and for simplicity
~(z(s))
scalar-valued.
N and Z are related
ds + M(t),
where (M,, 9,) is assumed to be a zero-mean local martingale. We assume is a doubly stochastic Poisson process (see BrCmaud, 1981), i.e.: (Zl)
Z(t)
is so-measurable
by
that Z
for all t 2 0.
Condition (Zl) excludes any dependence of the covariate on the counting process. It also has important technical consequences: Conditional on 9co the estimator .& defined below is a sum of independent random variables which, unconditionally is a sum of dependent random variables which can be considered a martingale difference array. This method of proof cannot be carried over to other classes of
K.J.
Utikal
/ Poisson inference
333
processes where (Zl) does not hold, such as diffusions, even though the correspond,. ing S$ can still be considered as an estimator for the integrated drift in this case. For ease of exposition
we will assume
Z to be one-dimensional.
Inference
to TV [0, 11. Extensions s~(~)=~~~cI(s)~ s will be restricted and other compact subsets are straightforward.
a(t) and dimensions Assumptions
for
to higher
on cx.
(A) (Y is a nonnegative,
nonrandom
function
number of jumps of unknown size and at unknown (on intervals between those jumps).
which,
except
locations,
maybe
is Lipschitz
for a finite continuous
We will define the estimator Gr next. We need to introduce the following notation. For a nondecreasing sequence of positive integers n = n(7), define partitions of the k=1,2 ,..., n. unit interval by 9k,7 =((k-l)/n,k/n], Assumptions
(Pl) (P2)
on the partition.
n/7+0 r/n’+0
as ~-+a, as 7+00.
Let T,,, be the total time up to 7 that the process
Z spends
in the kth subinterval,
7 T,,, =
I{Z(s)
I0
E %,,I ds,
where Z is the indicator function. The counts kth subinterval are denoted by N,,,, i.e.,
registered
at times when Z is in the
7 N,,, = As estimators
I0
Z{Z(s) E &,,> dN(s).
of ti and cy we propose
&,(z)=~‘:‘~ n k-r T,,, &(z.)=$[
(with O/0=0),
K(y)&(dx),
where [x] denotes the integer part of x and K is a nonnegative kernel function of bounded variation V(K), with bounded support, [ K(x) dx = 1 and window size b,. To assure some asymptotic uniformity of the sojurn times {T,,,, k = 1, . . , n} for T + 00 we will make assumptions on the underlying process Z. Define (2.la)
(2.lb)
K.J.
334
Assumptions (22)
on the process
There
uniformly
exists
Utikal
/
Poisson ii+wnce
2.
positive
a (strictly)
continuous
function
f such
that f7 ‘f
on [0, l] as T+ 03.
How to characterize problem
in itself.
the processes
If we assume
that satisfy
condition
Z to be stationary
(22)
is an interesting
with marginal
density
f then
under condition (22) a uniformly consistent estimator of f is provided by the histogram in which the height over each cell 9k,T is proportional to the time T,.,. That this condition, which is crucial for our model (see Lemmas 3,4,5), is indeed satisfied for a rich class of processes will briefly be illustrated next. Proposition
2.1. Let n + ~0, n/r + 0 as T + 00. Suppose
(i) for all O< s
the random
I$I,,TJ;,\,t4
ds -f(x)
uniformly
variables Z(s)
have densitiesf,,,,(x).
Suppose
= o(l),
(2.2)
in x E [0, I] as r + Co for some continuous function f;
(ii) for all 0~ s < s’< ~0 the random frC,,zC,z,(x,
that Z satisfies the conditions
y). Suppose
variables
Z(s),
that there exists a nonnegative
Z(s’)
function
have joint
densities
p(s, s’) for 0s
s < s’
such that u p( s, s’) ds’ < ~0, sup ’ 5\ IfZC\,,,Z,\I( Y I x ) -fz,c,(Y)l
s P(% 0
for all x, y E [0, 11. Under these assumptions
We note that relation
Z satisfies condition
(2.2) is trivially
satisfied
(22).
if Z is stationary.
It holds also if
uniformly in y, which is satisfied under conditions given in Doob (1953, V.5, VI.2) when Z is Markovian with stationary transition densities. Condition (ii) of the above proposition implies that the transition density approaches the marginal at a moderately fast rate for s’-+ ~0. Multiplying (ii) by fit,,(x) and integrating (ii) over sets A, B implies (P(Z(s) s P(S,
E A, Z(s’) s’)P(Z(s)
E B) - P(Z(s)
E A)P(Z(s’)
E B)l
E A)A(B),
where A(B) is the Lebesgue measure this is the uniform mixing condition,
(2.3) of B. For families see Renyi (1958).
of uniformly
bounded
sets
K.J. Utikal
Proof
of Proposition
Substituting
2.1. For simplicity
choose
A = B = 9k,r in (2.3) and multiplying
and applying
Fubini’s
theorem
335
/ Poisson inference
x to vary over the unit by (H/T)‘,
integrating
interval. over S, s’
we obtain
or Var{j;,,> = O( L/ T), uniformly
in k as T+ CO.Therefore
(1(x E A.,),
by Jensen’s
inequality
and the orthogonality
of
k = 1, . . . , ~1,
E(]Syp, IL,(x) -.m)l
_II $)-z&W-f(x) dx+o(l), I_ !,s ,1 J~I,.
= n sup
where the last equality
holds since f(z) = I? JY,,;,f(x) dx+o(
1) for any z E $A,T.
0
Condition (ii) is used in Castellana and Leadbetter (1986) for stationary processes, where it is shown that this condition is satisfied for example for every stationary Gaussian process with zero mean and covariance function r( 7) = 1 - CIT~‘~ + o( IT)“) for 0~ (Y<2 (cy = 1 for the Ornstein-Uhlenbeck process). In order to include Gaussian processes with more ‘regular’ sample paths ( LY= 2) we recall the definition of Castellana and Leadbetter’s (1986) dependence index function
assumed satisfied.
finite for all T> y > 0. Now we can give another
Proposition
2.2. Let n + Co,
criterion
for (22) to be
0 as T + Co. Suppose that 2 is a stationary process with continuous marginal density j Let {y_, T SO} be positive constants with y7= 0(T/n2) as T+ 00. Suppose that the dependence index function &(y) for 2 satisfies p7( y7) = o( nrT) as T+ 00. Then condition (22) is satisfied. T/n2
+
K.J.
336
Proof.
Utika/
/ Poisson itzference
First we note that f& as defined
estimator
of the form (l/r)
in (2.la)
can be considered
a kernel
density
ds, where xk is the center and &(xh -. )
JoTS,( xk -Z(s))
the indicator of 9,,. It follows from Theorem 5.4 of Castellana and Leadbetter A (1986) that Var{fk,7} = o(n-yT/ r) uniformly in k. Hence by the same argument as in the proof
of the previous
proposition,
0( n’y,/r) continuity
and hence f7(x) +f( of 1: 0.
ECsup,l.fAx)
x j uniformly
=,I;=,
Var.fk,, =
in x as r + ~0 by stationarity
- $T(x)l12
of Z and
3. Main results We will assume the conditions (A), (Pl), (P2), (Zl), (22) to hold throughout. Our first result consists of the weak convergence of the appropriately normalized estimator when the interval of time during which the observation is made tends to infinity. As usual, D[O, l] denotes the spaces of real valued functions which are right-continuous with left limits, equipped with the Skorohod topology. Theorem
3.1. J;(.&&)-+m,
weakly in D[O, l] as r + ~0, where m is a Gaussian
process with mean zero and
=ln=Z Q(X)
Cov(m(z,L m(z2))=
I 0
Proof.
We decompose
2,
f(x)
dx.
into
& = &+&,
(3.1)
where
where 7 I{Z(s) E &,}(dN(s) - a(Z(s)) I0 To prove the theorem we have to show that M,,, =
Jr;,,::?,
IK(z)I
where R,(z)=&(z)-&(z)
5 0
as ~+m,
ds.
(3.2)
K.J. Utikal
/ Poisson inference
337
and ?+ J;J& weakly
+ m,
(3.3)
in D[O, l] as r + ~0. To show (3.2) we observe
that by (A),
7 I{Z(~)E&&(Z(~))
ds=(a(zk)+O(lln))T,,,
(3.4)
Jo
uniformly
in k (except
uniformly
in z. Also a(z)
uniformly
maybe
for boundedly
many),
for ZI,E &7 arbitrary.
Hence
=f;i;
in z, hence
But I;‘=, Z{Tk,T=O} 5 0 as ~+CO by (22). This shows (3.2). It follows from Lemma 1 that {M& T,_,, k = 1, . . . , n} are independent conditionally on Sz = o{Z( t), t 1 O}. Therefore by Lemma 2 it is a martingale difference array with respect to {Fk,,,,k=l ,..., n} where .YF~,~=(T{N,,~,..., N,;,,,,}vFz. Hence (3.3) follows from an application of Rebolledo’s functional martingale central limit theorem (see Helland, 1982). The details are worked out in Lemma 3. 0
As a first consequence of the above theorem we obtain the following consistency result on &. The argument is similar to that of Ramlau-Hansen (1983) given for counting integrals.
Corollary.
Suppose
that CYis Lipschitz continuous
T+ 00. Then & 5 a uniformly
on [0,11, and b, + 0, &b,
on compact subsets of (0, 1) as T+ ~0.
Proof.
a(x)dx+$l
=;j
K(y).rp(d*)+O(b,),
K(y)(,(,)-+))dx
+ a3 as
K.J.
338
by Lipschitzness
Utikal J Poisson inference
K has bounded
of (Y and because
la-&,1(z)=
I;1
support.
K(+G&)(dx)
+
(&&)(x)dK
Hence,
integrating
by
O(b)
+ O(h)
(7) T
+supi(&-d)(x)lV(K)+O(b,) T X ~~O,il/i;)+O(b,)=o,(l), * where the last inequality
follows
from the previous
theorem.
0
We will make now the additional
assumption
that (Y(Z) is constant,
i.e.:
H,: There exists some (unknown)
cr,) such that for all z E [0,11: a(z) = (Ye.
For a bounded covariate the above hypothesis specifies that the counting process is a homogeneous Poisson process (if we suppose 0~ Z(t) G 1). Under Ho a natural estimator of 22 on [0, l] is 2,(z)
= z
C;=,N~,7 I,“=,
The asymptotic
T,,,’
distribution
of the difference
between
the two estimators
is given
in the next theorem. Theorem
3.2.
If Ho holds then
J;(2T-2T)+m,, weakly in D[O, 11 as r+ ~0 where m,
is a Gaussian
process
with mean
zero and
covariance
(3.5)
K.J.
Proof.
It was shown
Utikal / Poisson inference
in the proof of Theorem
339
3.1 that
A&=&+.&+& where
and
Similarly
we decompose
under
Ho,
(3.6)
&=&+&, where “J&(z) = z
c,“=, C;=,
Hence
it remains
Mh,T Tk.7
.
to be shown
that
J;(Ji-X,)+m,,
(3.7)
weakly in D[O, I] as r + ~0. By Theorem 3.1 we already know that fi in D[O, l] as r + 00, where we can choose for m the representation integral with respect to a standard Wiener process W:
J&, + m weakly as a stochastic
m(z) =
(3.8)
Next we show that J;.&(l) where
m,(l)
c~/~,!,f(x)
Y m,(l), is a normal
(3.9) random
variable
with
mean
zero
and
Var(m,(l))
=
dx. We write J&, = A,( 1, z), where .&!J r, z) = z ‘$~CM~r(f) I 1 k.7
(with O/O = 0),
where M,,,(t)
=
’ I{Z(s) I0
E A,,l(dN(s)
- a(z(s))
ds).
We note that ~%~(t, z) for fixed LY,r is a martingale with respect to s,,, 0~ t Q T by (Zl). Then (3.9) follows easily from (22) and a central limit theorem for counting integrals as it is stated for instance in Karr (1986). Next we observe that (3.9) implies
K.J. Urikal / Poisson in,ference
340
6 .a, + ml weakly in D[O, 11, where m,(z) = zm,(l). the representation
‘,r(~(1 m
m,(z) = z where
W is the same process
J;(&, weakly
s;f(x,
dx
We choose for the process
m,
(3.10)
d W(x),
as in (3.8). Now we show (3.11)
J-6)‘+ (m, m,)‘,
in (D[O, 11)’ as T + ~0. This implies vG(R,-X,)+m-m,,
weakly
in (D[O, l])l as T + CO,where (~-~r)(z)=~(j-z$=qdW(x)-zj-O’],~dxdW(x)). 0
(3.12)
0
Since Cov{(m -m,)(z,), (m -m,)(z2)}=Cov{m,(z,), m,(z*)} for all OGz,
; 4(&(x,+,)-&x,))+b,(&(x,+,)-&(x4) I-1 3
ii, Q,(m(x,+,)-
foranyO=x,,
Details
as ~-+a. side of (3.13) it can be seen that this is a sum over a to which a martingale central limit theorem can be out in Lemma 5. 0
are carried
The next theorem considered
shows that under deviation
in the previous
(3.13)
m(xi))+b,(m,(x,+,)-m,(xi)),
theorem
from Ho the difference
of estimators
will tend to infinity.
Theorem 3.3. Let n -3 ~0, n/ r2 + 0 as r + CO. Suppose that Z satisfies the conditions qf Proposition 2.1 and .fic,,(x) uniformly
ds-f(x)
in x E: [0, l] as 7 + Co for some continuous function J;@(z)-.Tqz)l:
(3.14)
=0(1/J;),
co
f: Then
as r+co,
for cdl z E [0, l] for which Z
1 a(x) dx # z
a(x)_/“(x) dx.
(3.15)
K.J. Utikal
Condition Proposition Proof.
(3.14) is similar
in.ferrnce
/ Poisson
341
to (2.2) and the same comments
as those
following
2.1 apply.
As in (3.6) we have &?,=~*+&+R,,
where *S&:(z) = z
’ a(x)f(x) 50
R,(z) = 2;(z)
dx,
- 9l*,
where
IT
I{Z(s)
Th=1
E
&,}a(Z(s))
ds.
c,
BY (3.4), %!2:(2)=2
uniformly
7
a(zA)T,,,+O(l/n)
in z. Also &c4”(z)=z
uniformly
i h-l
; a(zl) k=I
f(x)
dx+O(lln)
in z, hence
c ,,_> sup a(z)- ;,‘1,c 2c ,
I+-
n {,,.f(d
dxl+O(lln)
.L, ,,(x) ds -f(x) where the last equality
follows
from the same arguments
+0(1/J;),
as those given in the proof
of Proposition 2.1. It follows that 6 sup, lR1-(z)l = O,(l). Also, as indicated in the proof of Theorem 3.2, it is easy to see that &J@, + mz weakly in D[O, l] as T+ 00, where m?(z) = z j: Ja(x)f(x) d W(x) (compare (3.9)). Hence it follows from Theorem
3.1 that ~~~~(z)-~(z)~=~)~(z)-~*(z)~+Op(l)~~cc.
0
4. Applications Conjidence
bands
As a first application of Theorem 3.1 we construct asymptotic the unknown parameter ti, We introduce the function H(z) =Var(m(z))
=
confidence
bands
for
K.J. Urikal
342
/ Poisson irzference
Then (4.1) weakly
W,, is the Brownian
in D[O, l] as T+CO, where
in Lemma
4 that a uniformly
consistent
estimator
bridge
process.
for the unknown
It is shown
function
H is
given by
fi for H in (4.1) we obtain the following band for Sp:
Substituting confidence
.a4 * cc,pp(
1+~).
asymptotic
ZE[O, 11,
where P[sup,,. ,. ,,-I W”( t)( > c,,] = N. A table for the distribution can be found in Hall and Wellner (1980). Testing for the independence
from
lOO( 1 - a)-percent
of sup,,. ,. rlrl W”( t)l
a covariate
As a second application we derive tests for the null hypothesis Ho: a(z) is constant for all z E [0, 11. Under H,,, 2, and & are both estimators for &. On the other hand, if H,, is not true, 2, remains a valid estimator for .PI while 8, is a suitable estimator for a different parameter Op”. Therefore we will base any test statistic on the difference between the two. Its asymptotic distribution is given in Theorem 3.2. From this theorem the asymptotic distribution of any continuous functional of of the continuity theorem (G&z) - B(z)), 0 G z G 1) can be found by application (Billingsley,
1968). As examples
5, = fi
,,sup, lk7(z)
we suggest
- 2T(z)\
(s&(z) - &( z))I dz
&=T Hence
(Kolmogorov-Smirnow),
(Cramer-von
f (I,-~,(z,~,)-~~(z,)+~~(z,-,))’ ,=I
(chi-square).
we reject H,, if 5, > ci:‘, i = 1, 2, 3, where
= (Y, ,,1
P
(
c (m,,(z,) - m,,(z,_,))‘> ,=I
CX’
Mises),
)
= a.
K.J.
Utikal / Poisson inference
343
For computing c,, one needs to substitute the unknown parameters c~,f(x) by some uniformly consistent estimators in (3.5), for example by & = L&( 1) and f by its smoothened histogram method see McKeague
as in assumption (22). For a rigourous and Utikal (1990, Proposition 4.2).
justification
of this
A difference between 5,) & and & is that the latter only involves finite-dimensional distributions, hence ci:’ can be computed explicitly by transformation of a multivariate normal distribution. ci,’ ’ , cbf’ can be obtained by simulating the process m, given in (3.5), for which its representation (3.12) could be helpful. The consistency of the tests using .$, or & follows from Theorem 3.3. We now indicate briefly how to conduct the asymptotic ,$-test. (1) Divide the region ((0, l), say) into n subintervals of equal length such that T,,, > 0 for k = 1,. . . , n without any T,,, being very close to zero. (2) Compute the (column) vector X, = (x,.,), k = 1,. . . , n, where N L.7 N, XL,, =---) TA.7 T, where N7=Cz=, NL,,, T,=C~=,TL,,. (3) Compute a (generalized) inverse of the matrix V ,,,T =
V, = (u,,,,), i,j = 1,. . . , n, where
if i=j=k,
(l/T,,,)-(l/T,),
if i#j.
1 -(l/T,),
(4) Reject H,, if 5 = ( T7/ N,)X: VFX, > c,,, where c, is the (1 - Lu)-percent tile of the x’-distribution with q degrees of freedom, where q is the rank of the matrix V=(v,;), i,j=l,..., n, where uii =
j:klr,,ln
l/f(x)
1 -(ll~z~:,f(x,
dx - (l/n’Ixf(x)
dx)
dx),
if i =j = k, if i#j.
It follows from Theorem 3.2 that the above test will have approximately size 1 - (Y if r is large. Usually we would expect V to be of full rank (not in the important special case f(x) = I,,,,,,(x)). In the above V, was obtained by estimating f(x) in (3.5) by nTJr if xE $r,T. The use of a more sophisticated (and less simple) density estimator
may increase
the performance
of the test.
Appendix Lemma
1. ne
random
variables
{ Nk,T, k = 1, . . . , n} are independent
conditionally
on So. Proof.
Conditioned on So, N,,,(t) = 5: Z{Z( s) E 9,,} d N( s) are Poisson processes with respective rates hk,J t) = I{Z( t) E 9a,,}a (Z( t)), for k = 1, . . . , n. For any subfamily NkI,,, . . . , N,,,,, we can derive the backward Kolmogorov equations for
K.J.
344
Utikal / Poisson inference
P;::,.-;.-,,;:,‘,,(f)%‘P( A$, ,,( t) = m, , . . ) IV,,,,,(t) = m, 1.FJ, for all t 2 0. The solution can be shown to satisfy p;I,‘ ‘.‘..,‘L;?(t)=II
k,
P;C,(t).
Lemma 2. {(W.,lTk,,, K,,,,),
observing
that
II,,&,(t)
= 0
q
k= 1,. .
n 1 IS a martingale
dlfereerence arrqy.
Proof.
It follows from the theory of counting processes that 56 I{Z(s) E .J4a,,}~(Z(s)) ds is the s,-predictable compensator of Jh Z{Z(s) E $k,,} dN(s), hence (since
T,,, E SC]),
’ I{Z(s)
E
E A,,>
TIl.7
(see Bremaud,
dN(s)
= E
1981). But the right-hand
’ I{Z(s)
I
o-sup :- 1 Q(Z) Hence
ElM,,,/T,,,I
Lemma
3. &
0
E A,,>
T A.7
’ IW(s)
E -%,,I TX,7
a(Z(s))
ds
side is less than or equal to
ds 4 ,,syp, a(z)
property
follows
from Lemma
1.
q
J&, + m weakly in D[O, l] as T + ~0.
Proof.
Since (..G,(z), z E [0, 11) is not necessarily it by an L’-integrable martingale 2:. Define
for some 6 > 0. Since
T,,, E .F,, the martingale
Moreover
it is clearly
Lh-integrable
cZ{T,,,~i3
forsome
L’-integrable,
property
for k = 1,2,.
k: lsk
we will approximate
is conserved
by Lemma
2.
. . Also
* 1Mk.I p 1 --1-+O k=l
Tk,,
as r+ Co, since I{T,,,sS
for some k: l~k
as r+co
(by assumption
(22)).
(Note that I+,,, s 0 implies Y,,Z+,,,.J%0 for any random variables Y, defined on the martingale same space as Z,,,.) To prove that A.,&: + m we apply Rebolledo’s functional CLT (see Helland, 1982, Theorem 3.2a). Set MA,,Z{ T,,,> S} = Mf,,. We have to verify the conditions (A.la)
K.J.
345
Utikal / Poisson inference
for each z as r + ~0, (A.lb) for all e > 0 as T+ CO.First verify (A.la):
(by Lemma
2 and Theorem
5; Z{Z(.r) E &,&(Z(s)) Z-E,7 =
uniformly Hence
in k (except
where the convergence we observe that
maybe
@(zh)+o(lln)
boundedly
in probability
many)
follows
9.2.1 of Chung, ds Z{T,,,>
S>
1974)
I 01 9
Z{Th,T>61 (by (Zl)), as V-+ CO,where zk E ,ak,., arbitrary.
by assumption
(22). To verify (A.lb)
/
3k_‘,,n,7]) “2
‘2
But by Johnson
and Kotz (1969, Chapter
4),
gol=h
(A.2)
E[M4,,,(S0]=h+3h2~4(h+1)2,
(A.3)
E[M:,,I and
where A=
Z{Z(s) E &}a(Z(s))
ds s
sup
o- z- I
city,,.
K.J. lltikal
346
1 Poisson inference
Therefore (A.4) in k by condition
uniformly Lemma
4.
Proof.
By uniform
uniformly
(22). This proves the lemma.
continuity
of a/f
(except
in z with xk E .9k,T arbitrary. -A__
maybe
But
NkT U(Xk)
7
nTk,,
T,,,
f(xk)
_?I__
1
s z1+12+13+147
where I,=
Nk,r
II
nTk.7 f(xk 1 12=
I
--n;.,
1
=hk) 4
f(Xk)
3
k.7
n--
II
ftXk)
,-cy(xk) T LT
1 a(%),
Tf(Xk)
Tk.7 n7fo-’
.
But by (22),
uniformly
in k as T + ~0. Also, as in Lemma +-
cx(Xk)
z+0,
k,T
uniformly
in k. This proves
the lemma.
0
3,
0
in finitely
many
points),
K.J. Utikal / Poisson inference
Lemma5.
347
F~rO=x,,<...
J;
I?(A,(x,+~)-~/i(xi))+b,(JU,(x,+,)-JIZ,(xi)) i=,
Proof.
Similarly as in the proof of Lemma 3, M,,, has to be replaced by a truncated and square integrable martingale difference array. Suppose this has been done. Also suppose x,+, -x,_, > l/n for all i. We will show equivalently J;
; 4&(x,+,)-&x,))+c,J;&(l) ,=,
where K
co=-
C b,(x,+,-xi). r=,
Now
To apply a martingale the conditions
CLT (see Helland,
1982, Theorem
2Sa)
we have to verify
(A.5a) and secondly
(ASb)
K.J. Utikal / Poisson inference
348
To show (i), using (3.8) and (3.10), note that Var{ .
’
m l;f(x)
dx d w(x)
where z,=ao;
[nx,+,l TZ{TQ > a> 1
a$ i=,
n
k=[nx,]+, [“X,+1]
I,=&:,
Tk,r
>
61
by assumption
;
,=,
a,’
n
CC;=,
[?+,I c k-[nx,]+l
M;,,
(
---,
Td2
“““‘~;f(:,
dx’
7Z{ Tk,T> 61 I;=,
(22). This shows (ASa).
[
af
p
c
z,=2ct”co
where
7Tk,J{
c i=,
nTk,,
k=rnx,l+~
,=I
;rE
K ~acu,
Th.7 Next show the Lindeberg
condition
(A.5b).
K.J.
by (A.3), (A.4) and (22).
Utikal
/ Poisson in,ference
349
Similarly
Finally
This proves
the lemma.
0
Acknowledgement
The problem of estimating (Ywas brought to my attention by I.W. McKeague whom I also want to thank for many instructive conversations. My thanks also go to the referees whose comments and suggestions have led to an improved version of this article.
References 0.0. 0.0.
Aalen, Nonparametric inference for a family of counting processes, Ann. Statist. 6 (1978) 701-726. Aalen, A model for nonparametric regression analysis of counting processes, Lecture Notes in Statist. No. 2 (Springer, New York, 1980). P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968). P. BrCmaud, Point Processes and Queues (Springer, New York, 1981). J.V. Castellana and M.R. Leadbetter, On smoothed probability density estimation for stationary processes, Stochastic Process. Appl. 21 (1986) 179-193. K.L. Chung, A Course in Probability Theory (Academic Press, New York, 1974). J.L. Doob, Stochastic Processes (Wiley, New York, 1953). W.J. Hall and J.A. Wellner, Confidence bands for a survival curve with censored data, Biometrika 67 (1980) 133-143. I.S. Helland, Central limit theorems for martingales with discrete or continuous time, Stand. J. Statist. 9 (1982) 79-94. N.L. Johnson and S. Katz, Discrete Distributions (Houghton Mifflin, Boston, MA, 1969). A.F. Karr, Point Processes and their Statistical Inference (Dekker, New York, 1986). A.F. Karr, Maximum likelihood estimation in the multiplicative intensity model via sieves, Ann. Statist. 15 (1987)473-490. I.W. McKeague, Estimation for a semimartingale regression model using the method of sieves, Ann. Statist. 14 (1986) 579-589. I.W. McKeague and K.J. Utikal, Identifying nonlinear covariate effects in semimartingale regression models, Probab. Theory Rel. Fields 87 (1990) I-25. H. Ramlau-Hansen, Smoothing counting process intensities by means of kernel functions, Ann. Statist. 11 (1983) 453-466. A. RCnyi, On mixing sequences of sets, Acta Math. Acad. Sci. Hung. 9 (1958) 215-228. D. Snyder, Random Point Processes (Wiley, New York, 1975).