Take the 2-minute tour ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

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?

share|improve this question
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
+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
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

1 Answer 1

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?

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.