The Rao-Blackwell theorem, Lemma 2.3 and the fact that ~ is an unbiased estimator yield that ... The following corollary reformulates Theorem 2.1 for point.
Ann. Inst. Statist. Math. Vol. 57, No. 1, 71-81 (2005) Q2005 The Institute of Statistical Mathematics
OPTIMISATION OF LINEAR UNBIASED INTENSITY ESTIMATORS FOR POINT PROCESSES TOM~,.~ MRKVI~KA 1. AND ILYA MOLCHANOV2
1Mathematical Department, Pedagogical Faculty, University of South Bohemia, Jeron~mova 10, 17001 ~eskd Bud~jovice, Czech Republic 2Department of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, CH_ 3012 Bern, Switzerland (Received December 1, 2003; revised April 27, 2004) Abstract. A general n o n - s t a t i o n a r y point process whose intensity function is given up to an unknown numerical factor A is considered. As an a l t e r n a t i v e to t h e conventional e s t i m a t o r of A b a s e d on counting t h e points, we consider general linear unbiased e s t i m a t o r s of A given by sums of weights associated with i n d i v i d u a l points. A necessary a n d sufficient condition for a linear, u n b i a s e d e s t i m a t o r for t h e intensity A to have t h e m i n i m u m variance is d e t e r m i n e d . It is shown t h a t t h e r e are "nearly" no o t h e r processes t h a n Poisson and Cox for which t h e unweighted e s t i m a t o r of A, which counts the points only, is o p t i m a l . T h e p r o p e r t i e s of t h e o p t i m a l e s t i m a t o r are i l l u s t r a t e d by simulations for the Mat~rn cluster a n d t h e Mat~rn h a r d - c o r e processes.
Key words and phrases: Intensity estimation, Poisson process, linear e s t i m a t o r s , Mat~rn cluster process; Mat~rn hard-core process.
1.
Introduction
A typical task of spatial statistic is to study properties of estimators of parameters of point processes. One of the most important parameters of a point process is its intensity. In this work we consider a point process 9 with a known non-stationary structure. It means that the intensity measure of ~ has the form A(B) = ~ AV(x)dx, where A is unknown and V(x) is a known function that determines the non-stationary structure of ~. 9 is said to be a process with unknown scaling. If V(x) is constant, then 9 is stationary. To prevent the over-parametrisation of the model, assume that fw V(x)dx IW], where W C R d is the observation window and IWI is the d-dimensional Lebesgue measure of W. Processes with unknown scaling are important itself but even more because they are related to stationary processes of compact sets with a known distribution of primary grain, see Stoyan et al. (1995), Molchanov (1997). In fact the reference points of compact sets which hit the observation window form a process with unknown scaling where the function V may be computed from the distribution of the primary grain. *This research was partially supported by Grant Agency of Czech Republic, project No. 201/03/D062. 71
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TOM.AS MRKVICKA AND ILYA MOLCHANOV
There are various estimators of the intensity A. One often used is
(1.1)
r
IWi
where O(W) denotes the number of points of 9 in W. The estimator ~ is strongly consistent as W T ]~d if the process 9 is ergodic. Other estimators based on the interpoint distance method are discussed in Byth (1982) and Diddle (1983). An estimation method in which a fraction of the points is independently marked (and thus counted) followed by consideration of the ranks of the nearest marked point is described by Ss163 (1992). In this work we consider a family of unbiased estimators (called first order or linear, unbiased estimators) (1.2)
Af:
~
f(x),
xEapNW
where f : W --~ ]R is a weight function. Condition
/wf(
X)V(x)dx = 1
ensures that ~/ is unbiased. The estimator (1.1) corresponds to f ( x ) = Iwl-'. An important task is to compare the existing estimators and find an optimal estimator. For stationary Poisson and mixed Poisson processes (Stoyan et al. (1995)) the exact likelihood is available and the maximum likelihood estimator and the minimum variance unbiased estimator is ~. However for other more complicated processes the problem of determining the best estimator is still open. In Section 2 we find a necessary and sufficient condition on f for estimator (1.2) to have the minimal variance among all linear, unbiased estimators. Then the question, when the common constant estimator ~ given by (1.1) is optimal, is discussed. It is shown that ~ is a minimum variance unbiased estimator for non-stationary Poisson and mixed Poisson point processes and that there are "nearly" no other processes than Poisson and Cox for which A is optimal. In Section 3 we present simulation studies which compare the constant estimator with the optimal one.
2. Necessary and sufficient condition for optimal estimator Assume that the process 9 is a second-order point process in I~d. Let tt(2)(d(x, y)) (respectively a(2)(d(x, y))) denote the second-order moment (respectively factorial moment) measure of O. Let .Tr be the family of real-valued measurable functions on W such that Jr Furthermore, let
and
= / dw •
f ( x ) f ( Y ) P ( 2 ) ( d ( x , Y ) ) < oo.
OPTIMISATION OF INTENSITY ESTIMATORS
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The variance of ~S is Var(~/) : J
