Why Finite Samples Cannot Establish an Infinite Mean Without a Model
Summary
The document asks whether observations of a hitting time can establish that its population mean is infinite, even when the exact distribution is unknown and a Lévy fit is poor. The response argues that no distribution-free test can distinguish an infinite-mean law from a distribution that agrees with it over the observed range but is truncated beyond the largest sample value. With finite data, the unobserved tail can contain radically different amounts of probability and expectation.
The practical lesson is that an inference about an infinite mean requires assumptions restricting the possible distributions. The response recommends considering a broader, better-fitting family connected theoretically to hitting times for the processes under study. It does not propose a test, estimator, or finite-sample procedure, and it does not establish that the particular hitting time has infinite expectation. Conclusions would depend on the chosen model and its tail assumptions.
Key ideas
- Finite samples cannot distinguish an infinite-mean distribution from a sufficiently distant truncation without additional assumptions.
- Testing whether a mean is infinite requires restricting the family of possible distributions.
- A poor fit to one proposed distribution does not identify the true tail behavior.
- A useful model should fit the data and connect theoretically to the process that generates the hitting time.
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Full text
# How to test that a distribution has infinite mean?
# How to test that a distribution has infinite mean?
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?
## Answer by jwg (score 2)
https://quant.stackexchange.com/a/14594
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.