Can I write $\Phi(ax)$ as $b\Phi(x)$ for some constant $b$, please? FYI, $\Phi(\cdot)$ is the cdf for standard normal random variable. Is there such relationship?

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    $\begingroup$ Note that the first one will always be between 0 and 1 and the second one will always be between 0 and b. $\endgroup$ – power Jun 5 '14 at 7:30

Your question doesn't specify but I assume you mean for some given constant $a$ and that the relationship should hold for all $x$ - i.e. that we should have a single $b$ that works for any $x$. (If you want $b$ to be a function of $x$, that's quite a different question)

This is quite clearly impossible, since $\Phi(ax)$ is bounded between 0 and 1. Since any value of $b$ other than $1$ will either make $b\Phi(x)>1$ for some $x$, or will make it unable to attain all values (because it will have a maximum smaller than the maximum value of $\Phi(ax)$).

Hence $b$ can only be $1$.

This then implies that $a$ can only be $1$.

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That is, the only way for this to be true across all $x$ is for both $a$ and $b$ to be 1.

  • $\begingroup$ Sorry, I certainly mis-represented my question here. I am still working on this question stats.stackexchange.com/questions/101258/… $\endgroup$ – LaTeXFan Jun 5 '14 at 9:26
  • $\begingroup$ And I am trying to re-arrange the given integral so that I can find an unbiased estimator for $\Phi(\mu)$. Any ideas? $\endgroup$ – LaTeXFan Jun 5 '14 at 9:28

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