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arXiv:0706.4334v1 [math.ST] 28 Jun 2007

Power Loss for Inhomogeneous Poisson

Processes

Fazli Kh.

Department of Mathematics,

University of Kurdistan, Sanandaj

khfazli@uok.ac.ir

February 1, 2008

Abstract

In this work, based on a realization of an inhomogeneous Poisson

process whose intensity function depends on a real unknown parame-

ter, we consider a simple hypothesis against a sequence of close (con-

tiguous) alternatives. Under certain regularity conditions we obtain

the power loss of the score test with respect to the Neyman-Pearson

test. The power loss measures the performance of a second order ef-

ficient test by the help of third order asymptotic properties of the

problem under consideration.

AMS 1991 Classification: 62M05.

Key words: Inhomogeneous Poisson processes, hypotheses testing, power

loss, second order efficiency.

1 Introduction

Let X(n)be a realization of a nonhomogeneous Poisson process observed on

some increasing subsets An, n = 1,2,... of d dimensional Euclidian space

Rdwith intensity function S (ϑ,x),x ∈ An depending on one-dimensional

parameter ϑ ∈ Θ. Based on X(n)we want to test the hypotheses

H0: ϑ = ϑ0

H1: ϑ > ϑ0,

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where ϑ0is a given value in the parameter space Θ . Let us fix some α ∈ (0,1)

and define the class K(n)

K(n)

where Eϑdenotes the mathematical expectation with respect to the proba-

bility measure P(n)

most powerful test in K(n)

approach and introduce the class K′

1 − α, i.e.,

K′

It is well known that if n → ∞ for any given value ϑ of the alternative

the power of any reasonable (consistent) test tends to 1 ( see [?], [1]). In

order to compare the different tests we use the Pitman’s approach (see [9],

[7]) where instead of a fixed alternative ϑ we consider a sequence of so-

called local alternatives (close or contiguous alternatives) which converges

to ϑ0 with a certain rate and hence it is difficult to distinguish between

the null hypothesis and alternative. More precisely, let {ϕn} be a sequence

of nonnegative numbers which converges to zero with such a rate that the

likelihood ratio

Zn(u) =dP(n)

dP(n)

α

of tests at level 1 − α (size α), i.e.,

Eϑ0φn

?X(n)?= α?,

α =?φn:

ϑ. With fixed n, generally speaking, there is no uniformly

(see [8]). Therefore we turn to the asymptotic

αof sequence of tests of asymptotic level

α

α=

?

{φn} : lim

n→∞Eϑ0φn

?X(n)?= α

?

.

ϑ0+ϕnu

ϑ0

?X(n)?

has a nondegenerate limit for any u with ϑ0+ ϕnu ∈ Θ. By the Neyman-

Pearson lemma the most powerful test for H0 : ϑ = ϑ0 against the local

alternative Hu: ϑ = ϑ0+ ϕnu with u > 0, is given by

?

0,

˜φn

?X(n)?=

1, if Λn(u) > bn(u)

if Λn(u) < bn(u)

where Λn(u) = lnZn(u) and the constant bn(u) together with the contribu-

tion of the randomized part provide the size Eϑ0˜φn

of˜φnas a function of u is called the envelope power function . For any fixed

n it is the supremum of the power at the local alternative ϑu= ϑ0+ ϕnu

over all the tests at level 1 − α, i.e.

Eϑu˜φn

?X(n)?=

Notice that˜φnis not a test for the main hypotheses H0and H1because it

depends on the parameter u.

?X(n)?= α. The power

sup

φn∈K(n)

α

Eϑuφn

?X(n)?.

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Let us introduce the score statistic

∆n(ϑ0) = ϕn

?

An

˙S (ϑ0,x)

S (ϑ0,x)π(n)(dx),ϕ−2

n =

?

An

˙S (ϑ0,x)2

S (ϑ0,x)dx

where the normalizing factor ϕn= In(ϑ0)−1/2is the inverse square root of

the Fisher information

In(ϑ0) =

?

An

˙S (ϑ0,x)2

S (ϑ0,x)dx

at the point ϑ0. Here π(n)(dx) = X(n)(dx) − S (ϑ0,x)dx is the centered

Poisson process (for the definition of a stochastic integral w.r.t. a Poisson

process see the next section) and˙S (ϑ,x) denotes the derivative of S (ϑ,x)

with respect to ϑ. Based on ∆n(ϑ0) we introduce the score test

¯φn

?X(n)?=

?

1,

0,

if ∆n(ϑ0) > zα

if ∆n(ϑ0) ≤ zα

where zαis 1 − α quantile of standard Gaussian law, i.e., P{ζ > zα} = α

and ζ ∼ N(0,1).

distributions is locally asymptotically normal (LAN) at the point ϑ0, then the

test¯φn∈ K′

(or first order efficient), i.e., for any K > 0

It is well known that if the family

?

P(n)

ϑ, ϑ ∈ Θ

?

of

αis locally asymptotically uniformly most powerful (LAUMP)

sup

0≤u≤K

???Eϑu˜φn

?X(n)?− Eϑu¯φn

?X(n)???? = o(1)

as n → ∞ (see [12]). Moreover the power function of¯φnat ϑuadmits the

representation

Eϑu¯φn

?X(n)?= P{ζ > zα− u} + o(1) (1)

for any u > 0, where ζ ∼ N(0,1). For n large, hence the power of¯φn

approximates the envelope power function up to order o(1). A refinement of

(1) is given in (4). For a family

?

Poisson process with intensity functions {S (ϑ,·), ϑ ∈ Θ} the conditions of

LAN for multidimensional parameter ϑ are obtained by Yu. A. Kutoyants,

[6]. The first order efficiency of¯φn follows from the LAN representation

which implies in turn the asymptotic normalities of ∆n(ϑ0) and Λn(u) under

both H0and Hu. Therefore the refinement of the central limit theorem, by

taking into account one term after the Gaussian term, improves the situation.

P(n)

ϑ, ϑ ∈ Θ

?

of distributions related to a

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This can be done by the help of Edgeworth type expansion of the distribution

function of the stochastic integral ∆n(ϑ0) and Λn(u) under H0and Hu. Under

certain regularity conditions related to second order asymptotic properties

of the family

?

i.e., a test φ∗

P(n)

ϑ, ϑ ∈ Θ

?

, we can construct a second order efficient test,

nsuch that for any K > 0

sup

0≤u≤K

???Eϑu˜φn

nis given by Eϑ0φ∗

?X(n)?− Eϑuφ∗

n

?X(n)???? = O(ε2

?X(n)?

n),(2)

for some sequence εn → 0 (see [4]). Furthermore the probability of the

first type error of φ∗

follows that the power function of φ∗

approximates the envelope power function up to order O(ε2

works as good as˜φnup to this order. Second order efficiency of φ∗

to the fact that the two first terms in the Edgeworth expansions of the

distributions functions of ∆n(ϑ0) and Λn(u) under the local alternative are

equal up to the order O(ε2

order efficient test, and especially φ∗

of φ∗

n

= α + O(ε2

n). From (2) it

n(a second order efficient test, generally)

n) and hence it

nis related

n). Hence to measure the performance of a second

n, it is natural to consider the power loss

nwith respect to the most powerful test˜φn, which is defined by

r(u) = lim

n→∞ε−2

n

?

Eϑu˜φn

?X(n)?− Eϑuφ∗

n

?X(n)??

, (3)

for u > 0. This requires to take into account higher order terms in the

Edgeworth expansions of the distribution functions. See [1], chapter 3, for

a general theorem and the power loss results for the tests based on L−,R−

and U− statistics in the i.i.d. case. The main object of this work is to

obtain the power loss of the score test φ∗

nonhomogeneous Poisson process with intensity function S (ϑ,x),x ∈ An

and to give the explicit representation of r(u).

nbased on a realization X(n)of a

2Preliminaries

Let us remind several facts from a Poisson process. A Poisson process X(n)

is a random point measure which on the set B ⊂ Ancan be written as

X(n)(B) =

?

where {xi} are the events (random points) of the Poisson process and χ{D}is

the indicator function of the event D. The Poisson process with (parametric)

intensity function S(ϑ,x), x ∈ An (with respect to Lebesgue measure) is

entirely defined by the following two conditions:

xi∈An

χ{xi∈B},

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• for any collection of disjoint sets B1,...,Bm⊆ Anthe random variables

X(n)(B1),...,X(n)(Bm) are independent,

• the random variable X(n)(B) for any B ⊆ Anhas Poisson distribution

with parameter Λ(n)

ϑ(B) =?

BS(ϑ,x) dx.

This and other definitions of the spatial Poisson processes as well as their

properties and examples can be found in many books devoted to point pro-

cesses (see, e.g., Daley and Vere-Jones [2], Krickeberg [5], Reiss [10], Ripley

[11], Snyder and Miller [13]). The spatial Poisson processes are widely used

in many fields. In particular this is a good mathematical model for the prob-

lems of image restoration when the optical signal is weak and statistics of

photons is well described by an inhomogeneous Poisson process [13].

By the definition, X(n)is a random element of the set M(n)

integer valued measures defined on the set An. Let B(M(n)

σ−field with respect to which all the mappings:

ΠB: M(n)

are measurable. Here Anis the σ−field of Borel subsets of An. Let P(n)

note the probability law induced by the random element (realization) X(n)of

a Poisson process with intensity function S(ϑ,x), x ∈ Anon the measurable

space (M(n)

Λ(n)

P(n)

0, containing all

0) be the smallest

0

→ {0,1,2,...,∞},ΠB(X(n)) = X(n)(B),

B ∈ An,

ϑ

de-

0,B(M(n)

0)). We remind that if the intensity measures Λ(n)

ϑ2are equivalent then the corresponding probability measures P(n)

ϑ2are equivalent and the likelihood ratio is given by

ϑ1and

ϑ1and

dP(n)

dP(n)

ϑ2

ϑ1

?X(n)?=

??

= exp

An

lnS(ϑ2,x)

S(ϑ1,x)X(n)(dx) −

?

An

[S(ϑ2,x) − S(ϑ1,x)] dx

?

.

For a proof see [6] page 28. Here the stochastic integral

?

An

f(x) X(n)(dx) =

?

xi∈An

f(xi)

where {xi} are the events (random points) of the Poisson process.

Since the main tool used in this work is based on the Edgeworth expan-

sion, here we present the conditions under which the distribution function

Fn(y) = P(n)

ϑ{In(f) < y},

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