<p>This seems to be addressed in the following paper by Joseph P. Romano, Section 3  [1] (specifically, Example 1):</p>
<blockquote>
<p><strong>Example 1</strong> (Finite versus not finite mean). Let <span class="math-container">$X$</span> be <span class="math-container">$X_1, \dots, X_n$</span>, <span class="math-container">$n$</span> i.i.d. observations on the real line. As remarked by Bahadur &amp; Savage (1956), &quot;it would be interesting to know whether, in comparable non-parametric situations, tests of the existence of <span class="math-container">$\mu$</span> are equally unsuccessful&quot;; here, <span class="math-container">$\mu$</span> refers to the mean of an observation. So, let <span class="math-container">$P_0$</span> be the family of distributions on the real line with a finite mean, and let <span class="math-container">$P_1$</span> be the distributions without a finite mean. Condition B readily holds. [...] Hence, the conclusion of theorem 1 holds, and so it is impossible to construct a test with power greater than the size of the test.</p>
</blockquote>
<p>[1] Romano, Joseph P. <em>“On Non-Parametric Testing, the Uniform Behaviour of the t-Test, and Related Problems.”</em> Scandinavian Journal of Statistics, vol. 31, no. 4, 2004, pp. 567–584. JSTOR, <a href="http://www.jstor.org/stable/4616851" rel="nofollow noreferrer">www.jstor.org/stable/4616851</a>. Accessed 15 Sept. 2020.</p>

CC BY-SA 4.0
2020-09-15T15:55:50.977Z
2020-09-15T15:55:50.977Z
Clement C.
<p>The mean of a positive random variable <span class="math-container">$X$</span> is either finite or infinite; define <span class="math-container">$J(X)$</span> to be <span class="math-container">$0$</span> in the former case and <span class="math-container">$1$</span> in the latter case. Claim: there does not exist a function <span class="math-container">$J_n$</span> from the iid sample of size <span class="math-container">$n$</span> to <span class="math-container">$\{0,1\}$</span> such that
<span class="math-container">$J_n\to J(X)$</span> in probability.</p>
<p>This claim (which is undoubtedly true) appears to be folklore knowledge among statisticians, but I have not been able to find a formal proof in the literature. The closest result I could find is the one by
<a href="https://www.sciencedirect.com/science/article/abs/pii/S0167715296000843" rel="noreferrer">Hawkins</a>, which proves the impossibility of a sequential test with finite stopping time.</p>
<p>Can anyone give a pointer to a book/paper where the folklore claim is formally proven?</p>

CC BY-SA 4.0
2020-09-15T14:40:55.323Z
I'm guilty of slightly abusing notation: $J(X)$ is a function of the *distribution* and hence is a deterministic scalar taking values in $\{0,1\}$.
Aryeh
CC BY-SA 4.0
2020-09-15T14:41:22.807Z
$J_n$ is a (possibly randomized) function of the sample, and hence is a random variable.
Aryeh
<reference-request><machine-learning><lg.learning><st.statistics>
Testing for finite expectation
Aryeh
# Testing for finite expectation

(Asked by: username_0 on 2020-09-15)

The mean of a positive random variable $X$ is either finite or infinite; define $J(X)$ to be $0$ in the former case and $1$ in the latter case. Claim: there does not exist a function $J\_n$ from the iid sample of size $n$ to $\{0,1\}$ such that
$J\_n\to J(X)$ in probability.

This claim (which is undoubtedly true) appears to be folklore knowledge among statisticians, but I have not been able to find a formal proof in the literature. The closest result I could find is the one by
[Hawkins](https://www.sciencedirect.com/science/article/abs/pii/S0167715296000843), which proves the impossibility of a sequential test with finite stopping time.

Can anyone give a pointer to a book/paper where the folklore claim is formally proven?

**Tags:** reference-request, machine-learning, lg.learning, st.statistics

> **Comments:**
> **username_0**: I'm guilty of slightly abusing notation: $J(X)$ is a function of the *distribution* and hence is a deterministic scalar taking values in $\{0,1\}$.  
> **username_0**: $J_n$ is a (possibly randomized) function of the sample, and hence is a random variable.  

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## 1 answers

### Answer by username_1 on 2020-09-15. Score: 4 | Accepted answer

This seems to be addressed in the following paper by Joseph P. Romano, Section 3 [1] (specifically, Example 1):

> **Example 1** (Finite versus not finite mean). Let $X$ be $X\_1, \dots, X\_n$, $n$ i.i.d. observations on the real line. As remarked by Bahadur & Savage (1956), "it would be interesting to know whether, in comparable non-parametric situations, tests of the existence of $\mu$ are equally unsuccessful"; here, $\mu$ refers to the mean of an observation. So, let $P\_0$ be the family of distributions on the real line with a finite mean, and let $P\_1$ be the distributions without a finite mean. Condition B readily holds. [...] Hence, the conclusion of theorem 1 holds, and so it is impossible to construct a test with power greater than the size of the test.

[1] Romano, Joseph P. *“On Non-Parametric Testing, the Uniform Behaviour of the t-Test, and Related Problems.”* Scandinavian Journal of Statistics, vol. 31, no. 4, 2004, pp. 567–584. JSTOR, [www.jstor.org/stable/4616851](http://www.jstor.org/stable/4616851). Accessed 15 Sept. 2020.


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