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If I draw i.i.d. variables from N(0,1), will the mean or the median converge faster? How much faster?

To be more specific, let $x_1, x_2, \ldots $ be a sequence of i.i.d. variables drawn from N(0,1). Define $\bar{x}_n = \frac{1}{n}\sum_{i=1}^n x_i$, and $\tilde{x}_n$ to be the median of $\{x_1, x_2, \ldots x_n\}$. Which converges to 0 faster, $\{\bar{x}_n\}$ or $\{\tilde{x}_n\}$?

For concreteness on what it means to converge faster: does $\lim_{n \to \infty} Var(\bar{X}_n)/Var(\tilde{X}_n)$ exist? If so, what is it?

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    $\begingroup$ Are you asking about the convergence in probability of a point estimate with respect to the population parameter? Or are you asking about the convergence in distribution of a random variable? $\endgroup$
    – Ryan Simmons
    Commented Feb 6, 2015 at 4:03
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    $\begingroup$ By "converge faster to 0" do you mean "which has the smaller asymptotic variance" or something else? $\endgroup$
    – Glen_b
    Commented Feb 6, 2015 at 4:37
  • $\begingroup$ @Glen_b To some extent this is motivated by a real problem : the median is more robust against outliers, so it seems like the sample median should converge more rapidly than the mean as the sample size grows. I don't really know what the best way of expressing the rate of convergence is in this situation. For concreteness, I could ask whether $\lim_{n \to \infty} Var(\bar{X}_n)/Var(\tilde{X}_n)$ exists, and if so, what it is. $\endgroup$
    – Josh Brown Kramer
    Commented Feb 6, 2015 at 5:00
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    $\begingroup$ If the data are truly sampled from a normal distribution, outliers are extremely rare - so rare that the impact on the mean leaves the sample mean as the most efficient estimate of the population mean. But you don't need a vary heavy tail to make the median competitive. That ratio you mention will indeed be about 0.63 $\endgroup$
    – Glen_b
    Commented Feb 6, 2015 at 5:08

1 Answer 1

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The mean and median are the same, in this particular case. It is known that the median is 64% efficient as the mean, so the mean is faster to converge. I can write more details but wikipedia deals with your question exactly.

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    $\begingroup$ Do you have a citation? $\endgroup$
    – Josh Brown Kramer
    Commented Feb 6, 2015 at 4:33
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    $\begingroup$ Laplace, P.S.de (1818) Deuxième supplément à la Théorie Analytique des Probabilités, Paris, Courcier -- Laplace gives the asymptotic distribution for both mean and median. See also the section on the variance of the median on Wikipedia $\endgroup$
    – Glen_b
    Commented Feb 6, 2015 at 4:42
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    $\begingroup$ @Glen_b: (+1) the ultimate reference!!! $\endgroup$
    – Xi'an
    Commented Feb 6, 2015 at 7:42
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    $\begingroup$ @Glen_b yeah that was an epic response, I laughed pretty hard. Thanks for that! $\endgroup$
    – user541686
    Commented Feb 6, 2015 at 9:58
  • $\begingroup$ @xi'an did you mean to write that the mean and median are the same quantity? $\endgroup$
    – Yair Daon
    Commented Feb 11, 2015 at 4:38

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