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Let $X_1, \ldots, X_n$ be a set of $n$ i.i.d. samples of a non-negative random variable $X$ with $\mathrm{E}(X)=\mu$ and $\mathrm{Var}(X)=\sigma^2$. By the central limit theorem, the sample mean $\hat{\mu}_n=n^{-1}\sum_{i=1}^n$ converges in distribution to a normal distribution, i.e. $$ \hat{\mu}_n\stackrel{d}{\rightarrow}N\left(\mu,\frac{\sigma^2}{n}\right). $$

Given that $X$ is non-negative I would expect a distribution $L$ supported on the positive real line with right-skew to be a better approximation than a Normal approximation provided that $L$ approaches the normal distribution in the limit $n\rightarrow\infty$. The general idea was presented in, for example, The Lognormal Central Limit Theorem for Positive Random Variables but I don't find their argument to be particularly rigorous--no proofs are given.

Is it possible to show that a gamma or lognormal distribution provides a better approximation to the sampling distribution of $\hat{\mu}_n$ if $X$ is non-negative?

Till Hoffmann
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    That is not quite what the central limit theorem says. It is something closer to $\sqrt{n}\left(\hat{\mu}_n - \mu\right)\ \xrightarrow{d}\ N(0,;\sigma^2)$ – Henry Sep 17 '15 at 12:55
  • Apologies, I forgot to include the factor of $n$. That's fixed. – Till Hoffmann Sep 17 '15 at 13:02
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    How can $\hat{ \mu}_n$ converges to $N \left( \mu , \frac{\sigma^2}{n} \right)$? The RHS has to be free of $n$. – Hetebrij Sep 17 '15 at 13:06
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    A very funny article, thanks for sharing. But mathematically it is crap. No, these distributions do not provide better approximation unless one uses some meaningless MAD benchmarks to estimate accuracy. – zhoraster Sep 17 '15 at 15:18
  • Indeed, the article seems more like "I found something fun" than a proper paper. Nevertheless, I agree with their intuitive arguments. – Till Hoffmann Sep 17 '15 at 15:25
  • First of all, you should clearly distinguish between non-negative and positive real numbers, because it is essential to know if the additive identity, i.e. zero, is included or not. In the case of non-negative real numbers I would argue for the $\chi^2$ distribution, whereas in the case of positive real numbers I would do so for the log-normal distribution. – Björn Friedrich Oct 24 '15 at 19:41
  • @TillHoffmann If you want this discussion to be productive, you might want to explain what $\hat\mu_n$ converging to something depending on $n$ means (Hetebrij's comment) and to recognize that even the abstract of the linked paper is crap, confusing the restriction that the random variables are nonnegative (quite irrelevant) with considering the regime of not so many increments (which might indeed imply that, for some specific distributions, non normal approximations work better). – Did Oct 25 '15 at 09:56
  • @BjörnFriedrich Another misleading comment, I am afraid. That the increments are almost surely nonnegative (or positive) is completely irrelevant to the validity of CLT and one frankly does not even care whether, in your rather strange parlance, "the additive identity, i.e. zero, is included or not". Note that "In the case of non-negative real numbers I would argue for the χ2 distribution, whereas in the case of positive real numbers I would do so for the log-normal distribution" is pure fantasy. – Did Oct 25 '15 at 09:57

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