# Are there correct papers describing tests for finiteness of ($k$th) moments?

Note: A previous question asking about the finiteness of second moments was asked in 2010 -- this question is not a duplicate of that one because (1) this question is about tests for the finiteness of arbitrary moments, including the first moment (2) all of the papers I have found regarding this question were published after 2010, again, when the other question was asked.

Question: Does anyone know what the state of current literature is for hypothesis tests regarding the finiteness of moments of empirical distributions?

I have found at least three papers with Google, (1)(2)(3), all published after 2010, which claim to describe tests to address this question. However, the accepted answer to the previous question claims that any such test is impossible. So are all three papers wrong?

(Considering how often incorrect papers are published, I don't consider that a remote possibility.)

Has anyone read one of these papers (1)(2)(3), or a similar paper, such that they are able to say whether any of the methods suggested therein are actually correct?

Motivation/Background: The Strong Law of Large Numbers requires finite first moments, and the Central Limit Theorem requires finite second moments. These results are crucial to the validity of most statistical tests used in practice, as far as I am aware. However, I do not know of any widely used methods to test the validity of these assumptions in empirical datasets. This is especially concerning to me since the typical real-valued continuous probability distribution does not have finite first moments, and of the (atypical) real-valued continuous probability distributions which do have finite first moments, most in turn do not have finite second moments.

Thus the state of the art of such hypothesis-testing procedures seems to be absolutely essential to the correct practice of statistics, unless for example a concerted effort is made to change all statistical tests used in practice to ones with the default assumption that no moments are finite.

• The conclusion of reference (2) is "This short note advises caution in using the test for the existence of finite moments." (3) limits its test to distributions for which the "tail index" exists. Although (1) claims to be universal, it limits its evaluation to power laws. All three make it clear that there's no important distinction between testing the second moment and testing general $k^\text{th}$ moments. Thus, I believe (a) this question does not address anything new and (b) none of the references actually contradict the conclusion I drew seven years ago. – whuber Jun 29 '17 at 13:53
• @whuber Thank you for your timely and thorough expert appraisal -- I appreciate it. – Chill2Macht Jun 30 '17 at 12:00
• @whuber Just to clarify, is the problem only for nonparametric tests, or is it also impossible for parametric tests? This paper seems to make a formal argument why it is impossible for nonparametric tests, so that combined with your previous answer does make it seem quite plausible that it is entirely impossible for nonparametric tests, I don't know about parametric ones however. sciencedirect.com/science/article/pii/S0167715296000843 – Chill2Macht Jun 30 '17 at 12:50
• I think you've hit on a good resolution of the apparent contradictions. For parametric models it's certainly possible to test for finiteness of a moment, especially when the threshold between finiteness and non-finiteness is in the interior of the distribution space. In just about any practical situation, that will amount to one component of the parameter determining the tail behavior of the distribution. E.g., in the Pareto family with power parameter $\alpha\in[0,\infty)$, any test of $H_0:\alpha\le 1$ vs $H_A:\alpha\gt 1$ is a test of finiteness of the first moment. – whuber Jun 30 '17 at 13:24