Consistency Of A Kernel-Type Estimator Of The Intencity Of The Cyclic Poisson Process With The Linear Trend

r·

.''·'

ISSN 0854-1388

Majalah Ilmiah

l:limpU/Uln Matematika Indonesia

[MIHM/]

Published by the Indonesian Mathematical Society

with partial supports from the Departments of Mathematics at Gadjah Mada University and Bandung Institute ofTec/1110/ogy

'-"

ｾ＠

▸ Baca selengkapnya: pengertian trend non linear

Journal of the Indonesian Mathematical Society

(Majalah Ilmiah Himpunan Matematika Indonesia/MIHM!)

clo

Department of Mathematics, Institut Teknologi Bandung

Jalan Ganesa 10, Bandung 40132, Indonesia

E-mail: ｭｩｨｭｩｾｭ｡ｴｨＮｩｴ｢Ｎ｡｣Ｎｩ､＠ Website: http://mihmi.indoms-center.org

Editor-in-Chief:

Rino,·ia Simanjuntak, Dept. of.\/ath .. Institut Teknologi Bandung, Bandung 40132. Indonesia

1\lanaging Editor:

Budi Surodjo. Dept. ofJfath .. Gadjalr .\fada l/111\·ersil): Yogyakarta 55281. Indonesia

Associate Editors: Algebra and Geometrv:

Pudji Astuti. Dept. oj .• \fath., Jnstitut Teknologi Bandung, Bandung 40132, Indonesia A.D. Raza Choudry. Sc·hoo/ o_fJ!ath. Sci .. GC Unirersity. Lahore 54600. Pakistan

Zhao Dongsheng. Dept. qfJ/ath. & .\lath. Educ·ation, :Vanyang Technology l./nh:, Singapore 637616

Garret Sobczyk. Dept. de Fis. y ,\Jar .. l.lnirersidad de las An1ericas, Puebla 72820. Mexico

Sri \Vahyuni. Dept. o.l.\fath., Gadjah ;\Jada C.l11irersity, Yogyakarta 55281, Indonesia

Analysis·

Michael Co\vling. School oj".\f(lt/i .. Unil'ersiry oj',Veu· South lfQ/es. Sydney 2052. Australia

Soeparna Danna\\'ijaya. Dept. o,.l:\fath .. Gadjah li1ada l . .inil'ersiry. Yogyakarta 55281, /11do11esia

H. ｾＱＮ＠ Sri,·astava. Dept. o,.f'.\fath .. l./nil·ersity o_f'llctoria. llctoria BC VBll' 3P4, Canada

Applied ｾＱ｡ｴｨ･ｮＱ｡ｴｩ｣ｳＺ＠

Franz Kappel. Dept. oj',\fath .. l./nirel'sity o,f'Gru:. A-8010 Gra:. Austria

H.C.:\. van Tilborg. Oept. q/)fath. and Con1p. Sci. Tl.J Eindhoren, Eindhoren 5600. 1Yetherla11d

\Vidodo. Dept. o,f';\fath .. Gadjah ;\fada ｬＮｩｮｦ｜ﾷ･ｲｳｩｴ｟｜ｾ＠ }(Jgyakart" 55281. !11do11esia

Discrete tv1athematics & Combinatorics:

Jin Akiyama. R£wean:h Inst. <?(Educational D£•\: Tok"i ｬＬＧｮｦ｜ﾷ･ＱＺＭＮ［ｩｾ｜Ｚ＠ ToJ..yo 151-8677, Japun

Edy T. Baskoro. Dept. ＼ｾＯＧＮ｜Ａ＼Ｑｴｨ＠ .. lnstitut Teknologi Bandung, Bandung 40132. !11do11esia

Gyula O.H. Katona. A(/1·£·d Renyi Ins .. \lath .. Hungarian Acad. Sci. Budapest H-1053, Hu11gary

ｾﾷＱｩｲｫ｡＠ ｾｴｩｬｬ･ｲＮ＠ School ＼ｾｬｨＡＯｩｊＮ＠ Te<·h. and ,\lath. Sci .. Unh: ＼ｾｬｂ｡ｬＯ｡ｲ｡ｴＮ＠ Ballarat 3353. Australia

Statistics & Probabilitv Thcorv:

Ma man . .\. Djauhari. Dept. of t>.-1ath .. lnstitut Teknologi Bandung. Bandung 40132. Indonesia Gopalan Nair. School of f\1ath. and Stat.. Curtin Univ. of Technology. Bentley 6102. Australia Soebanar. Dept. of f\1ath .. Gadjah f\1ada Uni\'ersity. Yogyakarta 5528 I. Indonesia

ｾｶｬ｡ｴｨ･Ｑｮ｡ｴｩ｣ｳ＠ Education:

Abdur Rah1nan As"ari. Dept. o,f'.\farh .. ,\falang State G'ni\·ersil): ,.\fa/ang 65145. Indonesia

Kaye Stacey. D£•pt. ＼ｾＨｓ｣ｩＮ＠ and ,\/cah. Ed11catio11, r...:11i\'£•rsit,\' <?f.\felhourne . .\-le/hourne 30/0. Austr

Utari Sun1armo. D£'f'l. ＼ｾＨＮ｜ｬ｡ｴｨ＠ .. b1c/onesia l_..'nirersity ＼ｾＨｅ､ＱＱ｣｡ｴｩｯＱＱ＠ . ./Ja11d1111g. buloncsia

The journal is published sen1iannuallv bv the Indonesian ｾＱ｡ｴｨ･Ｑｮ｡ｴｩ｣｡ｬ＠ Societv.

'6

ralil

J.

lndones. Math.

Soc.

(MIHMI)

Vot 15, No. l (2009), pp. 37-48.

CONSISTENCY OF A KERNEL-TYPE

ESTIMATOR OF THE INTENCITY OF

THE CYCLIC POISSON PROCESS

WITH THE LINEAR TREND

l

WAYAN MANGKU, SISWADI,

AND

RE°rNO BUDIARTI

Abstract. A consistent kernel-type nonparametric estimator of the intensity function of a cyclic Poisson process in the presence of linear trend is constructed and investigated. It is assume<!, that only a single realization of \he Poisson process is observed in a bounded window. We prove that the propoeed estimator is consistent when the size of the window indefinitely expands.

1. INTRODUCTION

Let N be a Poisson process on [O, oo) with (unknown) locally integrable

in-tensity function >.. We assume that ).. consists of two components, namely a cyclic

(periodic) component with period T

>

0 and a linear trend. In other words, for

each points e [O,oo), we can write).. as

>.(s) = >.c(s) +as, (1)

where >.c(s) is (unknown) periodic function with (known) period-rand a denotes

(unknown) slope of the linear trend. In this paper, we do not assume any parametric

form of

>..,,

except that it is periodic. That is, for e.ach point s E [O, oo) and all

k E Z, with Z denotes the set of integers, we have

(2)

Received 31.o?'-2008, Accepted 15-11-2009.

tOOO MatMmatiu Subject ｃｗＮＬ｡ｩｦｩｭｴｩｾ＠ 60055, 62G05, 62G20.

Kev won:b and Phf'l1fea: cyclic Poiseon proces11, intensity function, linear trend, nonpvametric estima-tion, consistency.

37

I I

'

ii

I

38 I.

w.

MANGKU, SJSWADI, R. BUDIARI'I

Here we consider a Poisson process on [O, oo) instead of, for example, on R because

A has to satisfy ( 1) and must be non negative. For the same reason we also restrict

our attention to the case a

;o:

O. The present paper (cf. also [2]) rums at extending

previous work for the purely cyclic case, i.e. a=

0,

(cf.

[3], [4], [6],

section

2.3

of

[7]) to the more general model (1).

Suppose now that, for some w E !1, a single realization N ( w) of the Poisson

process N defined on a probability space (!1, .:F, P) with intensity function A (cf.

(1)) is observed, though only within a bounded interval Wn = [O, n] C [O, oo). Our

goal in this paper is to construct a consistent (general) kernel-type estimator of

Ac at a given point s E [O, oo) using only a single realization N ( w) of the Poisson

process N observed in interval Wn = [O,

n].

There are many practical situations where we have to use only a single

re-alization for estimating intensity of a cyclic Poisson process. A review of such

applications can be seen in

[3],

and a.number of them can also be found in

[1], [5],

[7],

[9]

and .[10].

We will assume throughout that s is a Lebesgue point of A, that is we have

1

jh

'

Jim

2h IA(s

+

x) - A(s)ldx = O

hjO -h (3)

(eg. see [11], p.107-108), which automatically means thats is a Lebesgue point of

Ac as well.

Note that, since Ac is a periodic function with period T, the problem of

estimating Ac at a given points E [O. oo) can be reduced into a problem of estimating

Ac

at a given points

E

[O, r). Hence, for the rest of this paper, we will assume that

SE [0.T).

Note also that, the meaning of the asymptotic n - oo in this paper is

some-what different from the classical one. Here n does not denote our sample size, but

it denotes the length of the interval of observations. The size of our samples is a

random variable denoted by N(!O. n]).

2. Construction of the estimator and results

Let K : R - R be a real valued function, called ｫ･ｲｮ･ｾ＠ which satisfies t.he

following conditions: (Kl) K is a probability density function, (K2) K is bounded.

awl (K3) K has (closed) support

[-1.

1]. Let alw h,. be a sequence of positive real

11u1nbers converging to 0, ｴｾ｡ｴ＠ is,

h,.

10.

(!)

as n - oo.

Using the introduced notations, we may define the estimators of respectively

a and Ac at a given point s E [O, T) as follows

• 2N([O. n])

an:=

_n

•

.

(2)

39

• 1

ｾ＠

1

1n

(x

-

(s

+

kr))

. (

n

)

ｾｮＮｋＨｳＩＺ］ｬｮＨｾＩｾｫｨＬＮ＠ ₀ K h,. N(dx)-a,. ｳＫｬｮＨｾＩＮ＠ (3)

The estimator given in (3) is a generalization of the estimator discussed and

investigated in Helmers and Mangku [2] for the case that the period r is known.

A general kernel-type estimator of the intensity of a purely cyclic Poisson process

(i.e. a= O) was proposed and studied in Helmers, Mangku and Zitikis ([3], [4]).

If we are interested in estimating .>.(s) at a given points, then .>.(s) can be

estimated by

5.n,K(s) = 5.c,n,K(s) + G.,,s. (4)

To obtain the estimator

a,,

of a, it suffices to note that

EN([O, n]) =

ｾｮ

Ｒ＠

+

O(n),

as n _, oo, wJtlch directly yields the estimator given in (2). Note also that, if N

were a Poisson proses with intensity function .>.(s) = as, then

a,,

would be the

maximum likelihood estimator of q (see [8]).

Next we describe the idea behind the construction of the kernel-type estimator

5.c.n,K(s) of Ac(s). By (I) and (2) we have that, for any point sand k E N (N denotes the set of natural numbers),

Ac(s) = Ac(s + kr) = >.(s + kr) - a(s + kr). (5)

Let Bh(x) := [x-h,x+h] and

Ln

:= ｌＺｾｾＭｯｯｫＭ Ｑ ｬＨｳＫｫｲ＠ E [O,n]). By (5), we

can write

1 cc I

= L L

k

(>.c(s

+

kr)) l(s

+

kr E [O, n])

" k=1

I 00 I

=

L L

k

(>.(s + kr) - a(s + kr)) l(s + kr E [O, n])

n k=l

1 cc I

= L L;,::(>.(s+kr))l(s+krE [O,n])-as n k=l

oc

ar

'°'

[

J

- L L.,l(s+krE 0,n). n k=l

40 I. W. MANGKU, SJSWADI, R. BUDIARrl

By (1) and the assumption that s is a Lebesgue point of.>., we have

1 00 1 1

1

an

A.,(s) "'

-I:

.>.(x)dx- as

-Ln l=l kjDi.,,(s+kT)I s,,,,(•+kT)n(o,n] Ln

= ....!:._

f

1 EN(Bh.(s+kr)n [O,n]) _a (s+

ｾＩ＠

Ln k 2h,. Ln

k=l

ＢＧＮＮＮＮＡＺＮ｟ｾｬｎＨｂｨＮＨｳＫｫｔＩｮ｛ｏＬｮ｝Ｉ｟｡Ｈ＠

ｾＩＭ

ｌｮｾｫ＠

2h,. s+Ln (7)

In the first "' in (7) we also have used the fact that

aT 00 ar·n an

(1)

an

-L LI(s+kre[O,n])=-L (-+O(lJ)=-L +O -L

"'£·

n k=l n T n n n

From the second "'in (7) and by noting that Ln ｾ＠ ln(n/r) as n -+ oo, we

see that

, ( ＩＭＭＱＭｾｬｎＨｩｳＫｫｔＭｨＬＮＬｳＫｫｲＫｨｮ｝ｮ｛ｏＬｮ｝Ｉ｟＠ (

_n_)

(B)

-'c,n s - ln(!!) ｾ＠ k ₂h,. a s+ In(!!)

r k=1 T

can be viewed as an estimator of .Ac(s), provided the slope a of the linear trend

to be known. H a is unknown, we replace a by

a,.

(cf. (2)) and one obtains the

estimator of .Ac(s) given by

; ＨＩＭＭＱＭｾＮＡ｟ｎＨ｛ｳＫｫｲＭｨＬＬＬｳＫｫｲＫｨＬＮ｝ｮ｛ｏＬｮ｝Ｉ｟Ｎ＠ (

_n_)

(

₉₎

-'c,nS ＭｬｮＨＡＡＩｾｫ＠ ₂h,. an s+ln(!!) ·

T k=l T

Now note that the estimator 5.c.n(s) given in (9) is a special case of the estimator

>.c,n.K(s) in (3), that is in (9) we use the uniform kernel

K

= ｾＱＱＭＱＮＱＱＨｽ＠ Replacing

this uniform kernel by a general kernel K, we then obtain the estimator of Ac given

in (3).

In Helmers and Mangku [2] has been proved the following lemma.

Lema 1. Suppose tit at the intensity function A satisfies ( 1) and is locally integrable.

Then we have

(10)

and

(11)

Tl m

as

th

co

II

Ill

ft,<;

flS

"

II

Lo

.:;

'·

.

'

Consistency of • kernel-type estimator 41

as n __, oo, where IJ = ,--1

J;

_{Ac(s)ds, the global intensity of the periodic component}

Ac· Hence

a,.

is a consistent

estimator

of a. Its MSE (mean-squared-error) is given

by

MSE (a,.)= 492

+

2a

+o

ＨｾＩ＠

n2 n3 (12)

asn-oo.

Our main results are presented in the following theorem and corollary.

Theorem 1. Suppose that the intensity function A satisfies {1) and is locally

integrable. If the kernel K satisfies conditions (Kl), (K2), (K3), and h,, satisfies

assumptions ( 1) and

hn Inn-+ oo, (13)

then _{- p}

Ac.n,K(s) --> Ac(s), (14)

as n __, oo, provided s is a Lebesgue point of Ac· In other words, 5.c,. .. K(s) is a

consistent estimator of Ac(s). In addition, the MSE of 5.c.n.K(s) converges to 0, as

1l - oc.

We note that, Lemma 1 and Theorem 1 together imply the following result.

Corollary 1. Suppose that the intensity function A satisfies ( 1) rind is locally

integmble. If the kernel K satisfies conditions (Kl). (K2), (K3). and h,, satisfies

assumptions (I) and ( 13), then

>. •.

K(s) ｾ＠ A(s). (15)

a.• 11 __, oo. protrided s ;s a Lebesgue point of A. In othe1· words.

5. ...

1<(-') i11 (4) i"

a consistent esti1nntor of A(s). In addition, tlie MSE of

5.

11.H(s) co1u1c·1ycs t.o 0. a.o;

n __. oo.

3. Proofs of Theorem 1

To prove 1'hrorc1n 1, it suffices t.o verify the folJo\\·ing t.,vo le111111<.i.-.;.

Lemma 2. (Asymptotic unbiasedness) Suppos<! that. the intensity function A satisfie.•

(1) and is locally integrable. If the kernel J( satisfies conditions (h'l), (/\2), (/\3).

and h,. satisfies assumptions (I) and ( 13), then '

I. W. MAHGKU, SISWADI, R. BUDlARTI

u n - co,

provided

a

is

a

LdJeague point

of

.>..,.

Lemnia 3. (Convergence of the variance) Suppose that the intensity /unction >.

satis-fies (1) and is locally integrable. If the kernel K satisfies conditions (KI), (K2), (K3)

and h,. satisfies assumptions (1) and (13}, then

Var ( >.c,n,K(s)) --> 0, (2)

as n -->

co,

provided s is a Lebesgue point of >.c.

Proof of Lemma 2

Note that

[

E>.c,n,K(s)

］ｉｮｴｾＩ＠

ｅｫｾ＠

1n K

ＨｸＭＨｾＺｫｲＩＩ＠

EN(dx)-(s+

ｬｮｾｾＩＩｅ｡ＮＮ＠

l

(3)

First we consider the first term on the r.h.s. of (3). This term can be written as

_1_

f

_1_

r

K (x- (s

+

kr)) >.(x)dx

ｉｮＨｾＩ＠

k=I kh,.

Jo

hn

= 1

Ｎｾ＠

1

f

(x-(s+kr))

ｉｮＨｾＩｾ＠ kh,.

JR

K h. >.(x)l(x

E

IO,n])dx.

By a change of variable and using (1) and (2), we can write the r.h.s. of (4) as

ｉｮｴｾＩ＠

ｅｫｾ＠

L

K (:,.) >.(x

+

s

+

kr)l(x

+

s

+

kr E (0. nj)dx

= I (l!!)

f

ｫｾ＠

J

K (hx ) >.c(x

+

s)l(x

+

s

+

kr E (0, nj)dx

n r k=l ''n JR n

(4)

1

00

1

J.

(x)

+In(!!)

L

kh K

h

a(x

+

s + kr)l(x

+

s + kr E

IO,

n])dx. (5)

T k=l rt R rt

a

We will first show that the first term on the r.h.s. of (3), that is the r.h.s. of n

(5), is equal to c

an

>.c(s)+as+ ｉｮＨｾＩ＠ +o(l), (6)

as n

_,co,

by showing that the first term on the r.h.s. of (5) is equal to >.c(s)+o(l)

and its second term is equal to as+ an/ln(n/r)

+

o(l), as n _, oo. To check this,

(/

0

I•

!is-ｾＳＩＬ＠

(2)

..

3)

I)

f

Consistency of a kernel-type estimator

43

note that the first term on the r.h.s. of {5) is equal to

I (l!!)

f,

k! { K ( : ) (>.0(x + s) - ).,,(s)) l{x + s + kT E (0, n]}dx

n ,.. k=I n JR n

>.c(s)

ｾ＠

1

r

(

x )

+In{!.;){;;;; khn

JR

K

hn

l(x + s + kr E (0,n])dx

=

hn

ｾＨＡＮ［Ｉ＠

L

K (,:) (>.c(x + s) - >.,,(s))

ｾ＠ ｾｬｻｸ＠

+ s + kr e (0, n])dx >.c(s)

r

(

x)

ｾ＠

1

+hnln(!.;)

JR

K

hn {;;;;

kl(x+s+kr E [O,n]}dx.

Using the fact that

cc 1

(7)

L

kl(x + s + kr E [O, n]) =In(;)+ 0(1), (8)

k=l

-as n - oo uniformly in x E [-h,.,

h,,J,

the r.h.s. of (7) can be written as

= h,.

ｉｾＨＡＮ［Ｉ＠

L

K (:,.) (>.c(x + s) - >.c(s))

(1n{;)

+ 0(1)) dx

ＫｨＬｾｾｾｾｾＩ＠

LK

(,;..)

ＨＱＱＱｻｾＩＫＰＨＱＩＩ､ｸ＠

=

L

K

(:,.)hi,.

(>..,(x+s)->.c(s))dx

+>.c(s) { K(x.)dx

+

0 (

-1 1

1 ) , (9)

la

i,, n n

as n - oo. Since s is a Lehesquc of >.0 (cf. (3)) and the kernel K satisfies

conditions (K2) and (K3), it easily seen that the first term on the r.h.s. of (9) is

o(I), as n - oo. By the assumption:

JR

K(x)dx = 1 {cf. {Kl)), the second term

on the r.h.s. of (9) is equal to >.0(s). A simple argument using assumption (13)

shows that the third term on the r.h.s. of {9) is o(l), as n - oo. Hence, the first

term on the r.h.s. of (5) is equal to >.c(s)

+

o(l), as n - oo.

Next we show that the second term on the r.h.s. of (5) is equal to

I

I

I

written as

l. W. MAJ(mro, SISWADI, R.

BuDIAKil

ｉｮｾ［ＧＩ＠

f;,

ｫｾ＠

L

K ( , : ) xl(x

+

s+ kT

E

[O,n])dx

ＫｉｮｾＩ＠

f;,

ｫｾ＠

L

K (,:) I(x

+

s+ kr

E

[O,n])dx

Ｋｉｮｾ［Ｉ＠

f;,

ｫｾ＠

L

K (,:) kl(x

ｾ＠

s

+

kT

E

[o,n])dx

= h,,

ｾＨ［ＧＩ＠

L

K (,:) x

f;,

ｾｉＨｸ＠

+

s

+

kT E [O, n])dx

+ h,,:(;')

L

K (,:.)

f;,

ｾｉＨｸＫｳ＠

+kr E [O,n]}dx

+

h..

:C;')

k

K (,:)

f;;1cx +

s

+

kr

E

[O,n])dx. Using (8), and the fact that

oc

LI(x +s+ kT E [O,n])

=;

+

0(1),

k=I

as n - oo uniformly in x E [-h,,,

h,.J,

the quantity in {10) can be written as

h,.

ｾＨ［ＧＩ＠

(1nc;i +

ｾＨＱＩＩ＠

L

K (,:) xdx

+

h..;:C;>l

(1nc;i+oc1i)

L

K(:,.)dx

(10)

+ ,, ..

ｾＺＨｾＩ＠

(;+oc1i)

L/((,:.)dx.

(11)

Since K is bounded and ｊｾ＠ ₁ xdx = 0, the first term of ( 11) is equal t.o zero. A

simple calculation shows that the second term of (11) is equal to ns

+

o( I) and the

third term of (11) is equal to an/ln(n/r)

+

o(l) as n - oo. Hence, we have that

the second term on the r.h.s. of (5) is equal to as+ an/ln(n/r)

+

o(l) as n - oo.

Combining this with the previous result, we obtain (G).

Finally we consider the second term on the r.h.s. of (3). I3y (10) of Lemma

1, this term can be computed as follows

-

(s

+ ....!:...)

(a+

20

+

0

(2.))

=-as -

ｾ＠

+

o(l),

{12)

Inn n n2 _Jn(;')

as n - oo. Combining

(6)

and

(12)

we obtain

(1).

This completes the proof of

Lemma

2.

Tl

\

- - -

-Consistency of a kernel-type estimator

Proof of Lemma 3

The variance of Xc,n,K ( s) can be computed as follows

Var (>.c,n,K(s)) =Var

ＨｉｮｴｾＩｾ＠

ｫｾｮ＠

ion K ( x-

ＨｾＫ＠

kT)) N(dx)

)-+Var

(a,.

(s

+

ｬｮｺｾＩＩＩ＠

+2Cov

ＨｉｮｴｾＩｾ＠

ｫｾ＠

Ion

K ( x-

ＨｾＫ＠

kT)) N(dx),a,. (s

+

ｬｮｺｾＩＩＩＮ＠

45

(13)

We will prove Lemma 3 by showing that each term on the r.h.s. of (13) is o(l) as

ｮｾｯｯＮ＠

First we check that the first term on the r.h.s. of (13) is o(l) as n __, oo.

To do this, we argue as follows. By ( 1), for sufficiently large n, we have that the

intervals

[s

+

kT - h,,, s

+

kT

+

hn]

and

[s

+

jT - hn, s

+

jT

+

h,.J

are not overlap

for all k ｾ＠ j. This implies, for all k ｾ＠ j,

K (x - (s

+

kT)) N(dx) and K (x - (s

+

jT))

N(dx)

hn hn

are independent. Hence, the variance in the first term on the r.h.s. of (13) can be computed as follows

1

ｾ＠

1

1"

2 _(x-(s+kT))

(h,.

ｬｮＨｾＩＩＲ＠

6

k2 o K hn Var(N(dx))

= l"

2-f:

\[K2(x-(s+kT))EN(dx)

(h,.ln(:;:)) k=I k o h,.

=

1

n

2-f 121" K2

(x - (s

+

kT)) >.(x)dx. (14)

(h,. In(:;:)) >=I k 0 h,.

Dy a change of variable and using (1) and (2). the r.h.s. of (14) can be written as

(h,.

ｉＺＨｾ＠

))2

t. :

2

L

K 2

( : . . ) >.(x

+

s

+

kr)l(x

+

s

+

kT E

IO,

nJ)dx

=

(hn

ＱＺＨｾＩＩＲ＠

t. :

2

JR

K' (:.) >.<(x

+

s)l(x

+

s

+

kr E

IO,

nj)dx

+

(hn

ｉＺＨｾＩＩＧ＠

t. :.

L

K2 ( : . ) a(x

+

s

+

kT)l(x

+

s

+

kT E

IO,

nJ)dx. (15)

I

46 I. W. MANGKU, SISWADI, R. BUDIAKI'I

The first term on the r.h.s. of (15) is equal to

(h,.

ｾＨ［ＧＩＩＲ＠

L

K2 ( , : . ) (>.c(x

+

s) - >.c(s)

+

>.c(s))

t, :.

l(x

+

s

+

kT E [0, n])dx = (h,.

ｾＨ［Ｇ＠

)}'

L

K2 ( , : . ) (>.c(x

+

s) - >.c(s))

t, :.

l(x

+

s

+

kT E [O, n])dx

>.c(s)

J.

2 ( x ) .;;.. 1

+

(h,. ln(;'))' R K h,.

ｾ＠

k2l(x

+

s

+

kT E [O,n])dx. (16)

Note that

00 1

:E

k₂ l(x

+

s

+

kT E (0, n]) = 0(1),

ｫｾｬ＠

(17)

as n __, oo,.uniformly in x E [-hnohn]- Since the kernel K is bounded and has

support in [-1, l], by (3) and (17) we see that the first term on the r.h.s. of (16)

is of order o((lnn)-2

(hn)-

_{1 )) = o(l), as n __, oo (cf. (13)). A similar argument}

shows that the second term on the r.h.s. of {16) is of order O((hn lnn)-2) = o(l),

as ｮｾ＠ oo.

Next we consider the second term on the r.h.s. of (15). This term can be

written &S

(hn:(';))2

L

K2

(:n)

(x

+

s)

ｾ＠

:2I(x

+

s

+

kT E [O,n])dx

ar

J.

Ｒ

ＨｸＩｾｬ＠

+

(hn In('; ))2 R K hn

ｾ＠

j/(x

+

s

+

kr E [O, n])dx.

By (17), the first term of {18) reduces to

O(l)(hnl:(';))2

LK

2

(:n)

(x+s)dx

= 0(1)

ＨｬｮＨＧ［ｾＩ

Ｒ

ｨＬＮ＠

L

K2(x)(xh,,

+

s)dx

= 0

C1n(';\)2hn)

= o(l),

(18)

(19)

as n __, oo. By a similar argument and using (8), we see that the second term of

(18) is of order O((hnlnn)-1 ) = o(l), as n __, oo (cf. (13)). Hence we have proved

that the first term on the r.h.s. of (13) is o(l), as n __, oo.

Next we consider the second term on the r.h.s. of (13). By (11) of Lemma 1,

this term can be computed as follows

ＨＫｉｮｾＧ［ＩＩＧ＠

Var(ii,,) = (•2

+

Ｈｬｮｾ［ＩＩ

Ｒ＠

+

ＡｾＩＩ＠

Ｈｾ＠

+O

Ｈｾ Ｓ ＩＩ＠

= o(l), (20)

ﾷﾷｾ＠

•

Consistency of a lciemel-type estimator

47

as n-+ oo.

Finally, we consider the third. term on the r.h.s. of (13). Since' the fust and

second terms on the r.h.s. of (13) are both of order o(l) as n --> oo, by

Cauchy-Schwarz, it easily seen that the third term on the r.h.s. of (13) is o(l) as n --> oo.

Therefore, all terms on the r.h.s. of (13) are indeed of order o(l) as n--> oo, which

i)nply (2). This completes the proof of Lemma :t.

Acknowledgement. This research was funded by program hibah kompetisi A2,

Departemen Matematika, FMIPA-IPB, tahun anggaron 2005.

REFERENCES

!. J. D. DALEY AND D. VERE-JONES, An Introduction to the Theory of Point Processes,

Springer, Ne·w York, 1988.

2. R. HELMERS AND I W. MANGKU, "Estimating the intensity of a cyclic Poisson process in the presence of linear trend", submitted to An. Inst. of Stat. Math.

3. R. HELMERS, I W. MANGKU, AND R. ZITIKIS, "Consistent estimation of the intensity function of a cyclic Poisson process", J. Multivariate Anol. 84 (2003), 19-39.

4. R. HELMERS, I W. MANG KU, AND R. ZITJKIS, "Statistical properties of a kernel-type estimator of the intensity function of a cyclic Poisson process", J. Multi11ariate Anal.

92 (2005), 1-23.

5. A. F. KARR, Point Processes and their Statistical Inference Second Edition, Marcel Dekker, New York, 1991.

6. Y. A. KUTOYANTS, 11

0n nonparametric estimation of intensity function of inhomoge-neous Poisson Processes", ProbL Contr. Infonn. Theory 13:4 (1984}, 253-258. 7. Y. A. KUTOYANTS, Statistical Inference for Spatial Poisson Processes, Lecture Notes

in Statistics1 Volume 134, Springer-Verlag, New-York, 1998.

8. I W. MANGKU, I. \\'IDIYASTUTI, AND I G. P. PURNABA, '1_{Estimating the intensity}

in the form of a power function of an inhon1ogeneous Poisson process'', Submitted for publication.

9. R. D. REISS, A Course on Point Processes, Springer, Nev.· York, 1993.

10. D. R. SNYDER AND rvi. I. MILLER, Random Point Processes in Time and Space,

Springer-Verlag, New-York, 1991.

II. R. L. WHEEDEN AND A. ZYCMUND, Measure and Integrol: An Introduction to Real Analysis, Maree] Dekker, Inc., New York, 1977.

I WAYAN MANCKU: Department of Mathematics, Faculty of Mathematics and Natural Sciences, Bogar Agricultural University, JI. Meranti, Kampus IPB Darmaga, Bogar 16680, Indonesia.

I

I

I

r

I

I

I.

w.

MANGKU, 11lswADI, R. BUD!Altrl

S1sWADI: Department of Mathematics, Faculty of Mathematics and Natural Sciences, Bo-· gor Agricultural University, JI. Meranti, Kampus IPB Dannaga. Bogor 16680, Indonesia.

RETNO BUDIARTI: Department of Mathematics, Faculty of Mathematics and Natural Sciences, Bogor Agricultural University, JI. Meranti, Kampus IPB Darmaga, Bogor 16680,

Indonesia..