Testing renewal point processes versus point processes with

Testing renewal point processes versus point processes
with dependent intervals between consecutive events
Franz Streit
Section de Mathematiques de l'Universite de Geneve,
2-4 rue du Lievre,
CH-1211 Geneve 24, Switzerland
[email protected]
This paper deals with statistical tests helpful to distinguish various type of point processes dened on the real line. A small survey of notions and procedures useful when developing
tests of renewal point processes against point processes with Markov dependent lengths of consecutive interevent intervals is rst given. This complements the results in (Streit, 1997), a
paper discussing tests of renewal point processes versus a) point processes with independent
locations or versus b) time series type point process models as alternatives.
Then a locally most powerful test to discriminate a homogeneous Poisson point process with
known global rate from a particular Wold type point process f generated by the conditional
density function for an interval length Y following an interval of length X specied by
(1) fY jX (yjx : ; ) = (1 , ),1 exp[,(1 , ),1 (x + y] I0(2(1 , ),1 (xy)1=2)
for 0 < 1; > 0 and for x; y 0 g is constructed. Here I0 designates the modied Bessel
function of order 0 . Classes of density functions given by expression (1) or similar formulae
were analysed by ( Kibble,W.F. 1940/41 and Lancaster,H.O. 1963 ) and were used in point
process theory in ( Daley, D. and Vere-Jones, D., 1988).
Based on (1) we want to test the hypothesis H0 : = 0 versus H1 : > 0; near 0, given
the observation of of the lengths ui[i = 1; : : : ; n0] of n0 consecutive interevent intervals and
conditional on u1 as observed. The likelihood and the loglikelihood function are obtained and
the score statistic (Le, 1992) of the locally optimal test T (U (1); : : : ; U (n) is computed yielding
(2) T (U (1); : : : ; U (n0 ) =
X(1 , U (k , 1))(1 , U (k)):
n0
k=2
Some of its properties are shown. Using results from (Billingsley, 1968, chapter 4) it is established that for n0 ! 1 and under H0 the score statistic follows asymptotically a normal
distribution. This allows to nd a locally most powerful large sample test.
The problem analyzed is representative for many similar problems, which can be investigated
by analogous techniques.
REFERENCES
Billingsley,P. (1968) Convergence of probability measures, J.Wiley, New York.
Chung,K.L. (1974) A course in probability theory, Academic Press, New York.
Cox,D.R. & Isham,V.(1980) Point processes, Chapman & Hall, London.
Daley,D. & Vere-Jones, D. (1988) An introduction to the theory of point processes, SpringerVerlag, New York.
Kibble,W.F. (1940/41) A two-variate gamma type distribution. Sankhya 5, 137{150.
Lancaster, H.O. (1963) Correlations and canonical forms of bivariate distributions. Ann. Math.
Stat. 34, 532{538.
Le,C.T. (1992) Fondamentals of biostatistical inference, M.Dekker, New York.
Streit,F. (1997) Statistical tests for distinguishing renewal point processes from alternative
stochastic models. Rendiconti del Circulo Matematico di Palermo, Serie II, Suppl. 50, 375{
385.
Szego,G. (1939) Orthogonal polynomials. American Mathematical Society, New York.
RESUM
E
Dans ce travail on continue les recherches commencees par l'article (Streit,1997) concernant des
tests statistiques utiles a distinguer entre des classes de processus ponctuels du type renouvellement et d'autres types de processus ponctuels. On traite ici le cas ou les modeles stochastiques
alternatives sont des processus ponctuels pour lesquels les longueurs d'intervalles entre evenements consecutifs forment une cha^ne de Markov.
On developpe un test localememt le plus puissant pour comparer un processus homogene de
Poisson a taux global connua un processus du type Wold engendre par la fonction de densite
(1). On indique la fonction de vraisemblance relevante, la statistique (2) du score ecace et
quelques-unes de ses proprietes. Cela permet d'obtenir un test asymptotique localement le plus
puissant pour tester H0 : = 0 versus H1 : > 0 et proche a 0.