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I observe a sample from a distribution that I expect to be the hitting time

$$\tau = \inf\{t>0| X(t)>a\}$$

where $X(t)$ is a Lévy process with $X(0)=0$ and $a$ is some constant. $X$ is not a Brownian motion and the experimental fit to the Lévy distribution is poor.

However, I do not need to know the exact formula for the law of $\tau$. For my needs I only need to know that the expectation of $\tau$ is infinite (as in the case of $\tau$ for a Brownian motion). Is it possible to formulate and test this as a statistical hypothesis?

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Perhaps you could take increasing subsets $A_n$ of your sample, test on $A_n$ whether the mean is greater than $a_n$, where $a_n \uparrow \infty$. If the likelihood of these hypotheses is "stable" as $n$ increases, maybe that would support the mean being infinite. –  quasi Aug 26 '13 at 20:55
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+1 but I would suggest moving that to one of mathematical websites: MSE or MO. Perhaps, cross-validated also could be useful –  Ilya Sep 25 '13 at 19:05
    
@quasi, surely the results would vary based on what sequence $a_n$ you chose? –  jwg Sep 1 at 10:49
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For the Brownian motion case, I think an application of Dynkin's formula should do the job (en.wikipedia.org/wiki/Dynkin's_formula). If the gernator of your Levy process is of a handy form then maybe Dynkin helps in your case too. Maybe check out the Brownian case first (see also the section in Oksendal's book about this amazon.com/…). –  Richard Sep 1 at 15:50

2 Answers 2

I don't think it is possible to do this without having a specific model or family of distributions which you assume that you are observing.

For any finite sample, the greatest probability estimate for the population mean is the sample mean. If you had a sample, you would never be able to distinguish statistically between a distribution with infinite mean, and the truncation of the same distribution at some point higher than the maximum of your sample (or a discrete disribution, with all the probability distributed between the points you observed). For you to establish that the former is more likely than the latter, you would have to restrict yourself to some family of possible distributions for which this is the case.

If your sample doesn't fit a Levy distribution well, is there some larger family of distributions which it might fit better, and for which you can establish a theoretical link to hitting times for some general set of processes?

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You cannot test a sample for infinite mean, but you can test if the sample likely belongs to a distribution with infinite mean, e.g. by Kolmogorov-Smirnov test statistic:

$$T^*= \sup_x |F_n(x)-F(x)|$$

Where $F_n(x)$ being the empirical distribution function.

You could set $F(x)$ to the Cauchy distribution, which has an undefined mean due to its heavy-tailedness.

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The Cauchy-distribution is distribution on the real line. The OP asks about a hitting time (non-negative). Furthermore trying one distribution is a bit crude, isn't it? –  Richard Sep 1 at 15:38
    
@Richard Thanks; the Cauchy distribution is symmetric so one could double $2F(x)$ to get a positive distribution with infinite mean. I agree it may be a bad guess, least it may work if nothing else. –  emcor Sep 1 at 20:24

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