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IndoMS Journal on Statistics
Vol. 2, No. 2 (2014), Page 1-10
CONSISTENT ESTIMATION OF THE MEAN FUNCTION OF A
COMPOUND CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR
TREND
Bonno Andri Wibowo, I Wayan Mangku, Siswadi
Department of Mathematics, Bogor Agricultural University
Jalan Meranti, Kampus IPB Dramaga, Bogor, 16680, Indonesia
Email: [email protected], [email protected], [email protected]
Abstract
This manuscript is concerned with investigation of a consistent estimator for the
mean function of a compound cyclic Poisson process in the presence of linear trend.
The cyclic component of intensity function of this process is not assumed to have
any parametric form, but its period is assumed to be known. The slope of the linear
trend is assumed to be positive, but its value is unknown. Morever, we consider the
case when there is only a single realization of the Poisson process is observed in a
bounded interval. Bias as well as variance of the estimator has been proved
converges to zero, when the size of interval indefinitely expands.
Keywords: coumpound cyclic Poisson, consistency, mean function,
linear trend.
Abstrak
Pada karya ilmiah ini dikaji penduga konsisten bagi fungsi nilai harapan pada proses
Poisson periodik majemuk dengan tren linier. Komponen periodik dari fungsi
intensitas pada proses ini tidak diasumsikan memiliki bentuk parametrik apa pun,
namun periodenya diasumsikan diketahui. Kemiringan dari tren linier diasumsikan
positif namun nilainya tidak diketahui. Kemudian, kita fokus pada kasus dimana
hanya ada realisasi tunggal dari proses Poisson yang diamati pada interval terbatas.
Telah dibuktikan bahwa bias dan ragam penduga konvergen ke nol, ketika panjang
interval pengamatan menuju takhingga.
Kata kunci: Fungsi nilai harapan, kekonsistenan, Poisson periodik majemuk, tren
linier.
1. Introduction
Let
be a Poisson process with (unknown) locally integrable intensity
function which is assumed to consist of two components, namely, a periodic or cyclic
component with period
and a linear trend component. In other words, for any point > 0,
the intensity function can be written as
2010 Mathematics Subject Classification: 60G55, 62G05, 62G20.
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Bonno Andri Wibowo, I Wayan Mangku, Siswadi
(1)
where
is a periodic function with period and denotes the slope of the linear trend which
is assumed
(2)
> 0.
We do not assume any (parametric) form of
except that it is periodic, that is, the equality
(3)
holds for all
Let
0 and
, where denotes the set of natural numbers.
be a process with
(4)
where
is a sequence of independent and identically distributed random variabels with
mean
and variance
, which is also independent of the process
. The
process
is said to be a compound cyclic Poisson process with linear trend. The
model presented in (4) is an extension of the model presented in Ruhiyat et al. (2013). We refer
to [1], [3], [6] and [7] for some applications of the compound Poisson process.
Suppose that, for some
, a single realization
of the process
defined on probability space
with intensity function is observed, though only within
a bounded interval
Futhermore, suppose that for each data point in the observed
realization
, say i-th data point, i = 1, 2, ...,
its corresponding random
variable
is also observed. The mean function (expected value) of
denoted by
is
given by:
with
.
Let
where for any real number x,
than or equal to x, and let also
with
of the Poisson process
Then, for any given
which implies
Let
, we have
denotes the largest integer that less
Then, for any given real number , we can write
that is the global intensity of the cyclic component
. We assume that
(5)
Con
3
The rest of this paper is organized as follows. The estimator and main results are presented
in Section 2, and the proof of this main result is presented in Section 3.
2. The Estimator and Main Results
The estimator of the mean function
using the available data set at hand is given by:
(6)
where
and
with the understanding that
when
. Thus,
when
Our main results are presented in the following theorems. These theorems are about weak
consistency of the estimator and the rate of convergence of respectively bias and variance of the
estimator of
. We refer to [2], [4], [5] and [8] for some related work in estimation of , ,
for different purposes.
Theorem 1 (Weak consistency). Suppose that intensity function
integrable. If, in additon,
satisfies condition (4), then
as
. Hence
is a weakly consistent estimator of
.
satisfies (1) and is locally
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Bonno Andri Wibowo, I Wayan Mangku, Siswadi
Theorem 2 (Rate convergence of the bias). Suppose that intensity function satisfies (1) and is
locally integrable. If, in additon,
satisfies condition (4), then
as
(7)
.
Theorem 3 (Rate convergence of the variance). Suppose that intensity function
and is locally integrable. If, in additon,
satisfies condition (4), then
as
.
satisfies (1)
(8)
3. Proof of Theorms
In this section, we present the proof of the theorems and some lemmas which are needed
in the proof of our theorems.
Lemma 1. Suppose that the intensity function
and
(10)
as
Proof. We refer to [2].
Lemma 2. Suppose that the intensity function
and
as
, where
and
Proof. We refer to [4].
satisfies (1) and is locally integrable. Then
(9)
is
for all
.
satisfies (1) and is locally integrable. Then
(11)
Con
5
Lemma 3. Suppose that the intensity function
satisfies (1) and is locally integrable. Then
(12)
and
(13)
as
.
Proof . We refer to [5].
Lemma 4. Suppose that the intensity function
=
and
as
, where
Proof . The value of
=
satisfies (1) and is locally integrable. Then
(14)
(15)
=
.
can be computed as follows:
A simple calculation shows that
=
as
as
(16)
(17)
Now note that, by Lemma 1 we have
(18)
Substituting (17) and (18) into the r.h.s of (16) we obtain (14).
The value of
=
A simple calculation shows that
can be computed as follows:
.
(19)
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Bonno Andri Wibowo, I Wayan Mangku, Siswadi
=
(20)
as
Substituting (18) and (20) into the r.h.s of (19) we obtain (15). This completes the
proof of Lemma 4.
Lemma 5. Suppose that the intensity function
as
, where
Proof . To compute
=
satisfies (1) and is locally integrable. Then
.
(21)
we argue as follows. Let
Ten we have
Notes that
and
Now we see that
=
=
are independent.
(22)
By substituting (11), (12) and (13) into the r.h.s of (22), then we obtain (21). This completes the
proof of Lemma 5.
Lemma 6. Suppose that the intensity function satisfies (1) and is locally integrable. If
condition (2) and (5) are satisfied, then with probability 1,
(23)
as
.
Proof.
=
=
which is
as
proof of Lemma 6.
. Then, by the Borel-Cantelli Lemma, we have (23). This completes the
Con
7
Proof of Theorem 1. By (6), to prve Theorem 1, it suffices to check
(24)
(25)
(26)
and
(27)
as
. By Lemma 1, we have (24), by Lemma 2, we have (25), and by Lemma 3, we have
(26), By Lemma 6 and the weak law of large numbers, we have (27). This completes the proof
of Theorem 1.
Proof of Theorem 2. The expected value of
can be computed as follows:
=
=
=
=
(28)
Substituting (9) of Lemma 1, (11) of Lemma 2 and (12) of Lemma 3 into the r.h.s. of (28), and
after some algebras, we obtain that
=
A simple calculation shows that
(29)
(30)
as
Substituting (30) into the r.h.s. of (29) and after some simplification we obtain
as
This completes the proof of Theorem 2.
=
,
Proof of Theorem 3. To prove Theorem 3, first we compute
as follows:
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Bonno Andri Wibowo, I Wayan Mangku, Siswadi
=
=
Since
mean
(31)
is a sequence of independent and identically distributed random variabels with
and variance , a simple calculation shows that
(32)
Now note that, by Lemma 2 we have
and by Lemma 3 we have
=
(33)
(34)
as
Substituting (18), (32), (33), (34), Lemma 4, Lemma 5 into the r.h.s. of (31), after
some simplification we have
=
The first term on the r.h.s. of (35) is equal to
as
=
as
while its second term can be simplied as
Substituting (36) and (37) into the r.h.s. of (35), then we have
(35)
(36)
(37)
Con
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=
(38)
as
By the r.h.s. of (38) and the r.h.s. of (7) we obtain (8). This completes the proof of
Theorem 3.
References
[1] Byrne J., 1969, Properties of compound Poisson processes with applications in statistical
physics, Physica 41, 575-587.
[2] Helmers R, Mangku IW., 2009, Estimating the intensity of a cyclic Poisson process in the
presence of linear trend, Annals Institute of Statistical Mathematics, 61, 599-628.
[3] Kegler SR., 2007, Applying the compound Poisson process model to reporting of injuryrelated mortality rates, Epidemiologic Perspectives & Innovations, 4, 1-9.
[4] Mangku IW., 2005, A note on estimation of the global intensity of a cyclic Poisson process
in the presence of linear trend, Journal of Mathematics and Its Application, 4(2), 1-12.
[5] Mangku IW., 2010, Consistent estimation of the distribution function and the density of
waiting time of a cyclic Poisson process with linear trend, Far East Journal of Theoretical
Statistics, 33(1), 81-91
[6]
, 2008, The probability function of the compound Poisson process and an
application to aftershock sequence in Turkey, Environtmetrics, 19, 79-85.
[7] Puig P, Barquinero JF., 2011, An application of compound Poisson modeling to biological
dosimetry, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467(2127), 897-910.
[8] Ruhiyat, Mangku IW, Purnaba IGP., 2013, Consistent estimation of the mean function of a
compound cyclic Poisson process, Far East Journal of Mathematical Sciences, 77(2),
183-194.
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Bonno Andri Wibowo, I Wayan Mangku, Siswadi