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If $X \sim N\left(\mu, \frac{\sigma^2}{n}\right)$, what can we say about the behaviour of $\text{Var}\{\Phi(X)\}$ with respect to $n$?

Is it true that this variance behaves as $O(n^{-1})$ for $n$ large enough?

I would show this using the Maclaurin series on $\Phi(x)$, assuming $x \ll 1$. Would this be correct? If so, could I restrict to the first terms of the series?

Thanks.

user79097
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  • You could simply use the Delta method. – Xi'an Jan 31 '18 at 17:16
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    Sorry but if $\Phi(X)$ is the CDF, its distribution should be uniform(0,1) independently for $n$ and its variance should be $1/12$. Am I missing something? – gioxc88 Jan 31 '18 at 17:23
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    I think this would be the case if $X$ was standard normal, which is not the case here. – user79097 Jan 31 '18 at 18:03
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    An explicit formula for the variance in terms of $\mu$ and $\sigma/\sqrt{n}$ is given at https://stats.stackexchange.com/questions/212421/variance-of-a-cumulative-distribution-function-of-normal-distribution/237374#237374. Compute that through first order in $n^{-1}$. – whuber Jan 31 '18 at 19:20

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