So I'm reading all these theorems about U-statistics and thinking this could all work except for one really important detail: they all require that the random variable in question has a finite second moment. Aside from Normals (which my data is definitely not), there are no stable distributions with second moments.
It's not clear from a first reading how big a deal this is. There might be a way to get some useful results even with that condition relaxed. If not, this line of inquiry is going to be a dead end.
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