Dear researcher,
Doing CSR test (based on Ripley K-function) for an observed
homogeneous point process in a Window W with intensity \lambda(=n/a) ,
we accept the null hypothesis of CSR if sup|\sqrt(\hat{K(t)}/\pi
-t|)<=c for t<=t_0 (c is unknown) which yields
max(0,(t-c)^2<=\hat{K}(t)<=\pi(c+t)^2 (**)
The problem is understanding the details of what Ripley has done
successfully based on simulation in ( [ref 1]) and (ref [2], p. 46) to
obtain the unknown factor c.
One of facts we have is that \hat{K}(t) converges to a homogeneous
PPP of rate 2\pi*t *n^2 /a ( [ref 1]).
(ref [2], p. 46) has mentioned with no details(!) that "simulation
based on equation (**) shows c=1.45*\sqrt(a./n)" . My question is how?
Ripley in ( [ref 1]), speaks also about “first exit time
distribution” of a homogeneous Poisson point process (PPP) from
moving boundary max(0,(t-c)^2<=\hat{K}(t)<=\pi(c+t)^2 . I don’t know
whether or not this fact has been used in simulation to obtain the
factor c. If yes how?
Sorry for disturb with this long (unrelated) question.
Yours,
Hamid
[1] Ripley, B.D. (1979) Tests of ‘randomness’ for spatial point
patterns. J. Roy. Statist. Soc. B
41, 368–374.
[2] Ripley, B.D. (1991) Statistical Inference for Spatial Processes.
Cambridge University Press, Cambridge.
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