Approximation to the ratio of a Gaussian CDF to PDF - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T19:14:12Z http://mathoverflow.net/feeds/question/74545 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74545/approximation-to-the-ratio-of-a-gaussian-cdf-to-pdf Approximation to the ratio of a Gaussian CDF to PDF brianjd 2011-09-05T00:30:07Z 2011-09-05T20:31:39Z <p>Johnstone and Silverman (2005) claimed that for large x </p> <p>$\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$ </p> <p>where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable. </p> <p>I was able to verify the claim numerically. Q: But how would I show this analytically? This seems like it should be easy, but I can't figure it out. Also, Q: Is there a symbolic logic system (e.g., Mathematica) that can generate these sort of approximations? </p> http://mathoverflow.net/questions/74545/approximation-to-the-ratio-of-a-gaussian-cdf-to-pdf/74547#74547 Answer by Robert Israel for Approximation to the ratio of a Gaussian CDF to PDF Robert Israel 2011-09-05T01:19:21Z 2011-09-05T01:19:21Z <p>In Maple:</p> <blockquote> <p>with(Statistics): Phi:= CDF(Normal(0,1),x): phi:= PDF(Normal(0,1),x): asympt((1-Phi)/phi,x,10);</p> </blockquote> <p>$\frac{1}{x} - \frac{1}{x^3} + \frac{3}{x^5} - \frac{15}{x^7} + \frac{105}{x^9} + O\left(\frac{1}{x^{11}}\right)$</p> <p>See also <a href="http://oeis.org/A001147" rel="nofollow">http://oeis.org/A001147</a> for the sequence of coefficients</p> http://mathoverflow.net/questions/74545/approximation-to-the-ratio-of-a-gaussian-cdf-to-pdf/74550#74550 Answer by Brendan McKay for Approximation to the ratio of a Gaussian CDF to PDF Brendan McKay 2011-09-05T02:17:43Z 2011-09-05T02:17:43Z <p>If $Y(x)=(1-\Phi(x))/\phi(x)$, it is easy to check that $Y'(x)=xY(x)-1$ and from this anything you like follows by standard methods. </p> http://mathoverflow.net/questions/74545/approximation-to-the-ratio-of-a-gaussian-cdf-to-pdf/74551#74551 Answer by Deane Yang for Approximation to the ratio of a Gaussian CDF to PDF Deane Yang 2011-09-05T02:52:53Z 2011-09-05T02:52:53Z <p>If you interpret this as the existence of the limit $$\lim_{x \rightarrow \infty} \frac{x(1-\Phi(x))}{\phi(x)}$$ then it is easy to verify using l'Hopital's rule.</p> http://mathoverflow.net/questions/74545/approximation-to-the-ratio-of-a-gaussian-cdf-to-pdf/74617#74617 Answer by Yuri Bakhtin for Approximation to the ratio of a Gaussian CDF to PDF Yuri Bakhtin 2011-09-05T20:31:39Z 2011-09-05T20:31:39Z <p>Reproducing a lemma from the classic Feller book, first we can write</p> <p>$$(1-3x^{-4})\phi(x)&lt;\phi(x)&lt;(1+x^{-2})\phi(x).$$</p> <p>Integrating this from $x$ to $+\infty$, we obtain</p> <p>$$(x^{-1}-x^{-3})\phi(x)&lt;1-\Phi(x)&lt; x^{-1}\phi(x),$$ </p> <p>so you easily get an approximation rate $x^{-3}\phi(x)$, too.</p>