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Does anyone know where I can find the derivation of the third moment of the joint count statistic? I found this similar question answered in the past:

need derivation of join-count variance (spatial autocorrelation stat), know where it is?

but since I am interested in the skewness of the statistic I need the third moment, when the null distribution is non-free sampling (restrict to the number of locations).

Thanks in advance!

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    $\begingroup$ The answer to which you refer points out that the calculation depends on the structure of the neighborhood graph. It explains the technique needed to use that topological information to perform the combinatorial calculations required to find any moment, and it illustrates it by finding the first two moments for a particularly simple graph. It would appear, then, that (a) your question is incomplete, because it does not specify the graph and (b) that the answer you reference already provides the means to compute the skewness. It's also (c) unclear what you mean by "non-free sampling." $\endgroup$ – whuber Feb 26 '16 at 14:35
  • $\begingroup$ assuming that the neighboring graph consists of 0s and 1s where they are classifications of nearest neighbors, and by following the rational of your previous post, how can we derive the third moment J? $\endgroup$ – Akis Feb 26 '16 at 14:53
  • $\begingroup$ The "0s and 1s" are the values on the graph but they say nothing about its topology. The topology is determined by which geographic features are considered to be neighbors of which others. In some cases the features form a regular grid and the topology is determined by the grid dimensions and the size of a neighborhood ("king's case" and "rook's case" are the commonest). But in many applications the topology is determined by the adjacency relations among polygons or vertices of a travel network, in which case you need a custom calculation of the moments. $\endgroup$ – whuber Feb 26 '16 at 14:56
  • $\begingroup$ i am really interested on what you are saying is there any link that I could read about how the topology of such matrices influence the calculation of the moments? In the derivation you did in the link attached above, with BB, BW and perhaps WW, how could one proceed further to the derivation of the third moment? $\endgroup$ – Akis Feb 27 '16 at 16:37
  • $\begingroup$ The only references I know of treat just the standard grid graphs I mentioned previously. To compute the third moment exactly you would need to carry out the same kinds of calculations illustrated in my answer in the other thread. You could also estimate the third moment via simulation. That would require more computation but wouldn't take much mental effort. $\endgroup$ – whuber Feb 27 '16 at 17:21

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