In the help file for the Kest function in spatstat there is a warning section stating:
"The estimator of K(r) is approximately unbiased for each fixed r. Bias increases with r and depends on the window geometry. For a rectangular window it is prudent to restrict the r values to a maximum of 1/4 of the smaller side length of the rectangle. Bias may become appreciable for point patterns consisting of fewer than 15 points."
I would like to know in what sense the estimator of K(r) becomes biased with increasing r and for point patterns with fewer than 15 points?
Any advice on this matter would be greatly appreciated!
I have read the book "Spatial point patterns" (Baddeley et al., 2015) but I can't seem to find the answer there (or in any other literature). I may of course have missed that section of the book, if so please let me know.
I don't know the historical facts about where n=15 comes from, but this is probably related to the fact that the estimate of K(r) is only ratio-unbiased. Typically what we can estimate directly is X(r) = lambda^2*K(r) where lambda is the the true intensity of the process. Then we use the estimate of this quantity, X_est(r) say, together with an estimate of lambda^2, lambda^2_est say, and then estimate K(r) as K_est(r) = X_est(r) / lambda^2_est. Thus the numerator and denominator are unbiased estimates of the right things, but the ratio isn't. The problem is worst when lambda^2 is poorly estimated, i.e., when we have few data points.
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