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In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation

$$\Phi^{-1}\left(1-\frac{1}{n}\right)\approx \sqrt{2\log n}$$

and the authors said that it follows form

$$1 - \Phi(x) \approx \frac{\phi(x)}{x}$$

But i have been unable to derive any of them. Can anyone give me some insite on the proofs or point to a source where these kind of approximations are discussed in more detail?

Mur1lo
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    For a one-line proof of $P(X>x) = 1-\Phi(x) \approx \frac{\phi(x)}{x}$ for $x > 0$, see [this answer](http://math.stackexchange.com/a/28754/15941) by Moderator cardinal over on math.SE. I expect there are proofs here on stats.SE also. – Dilip Sarwate Nov 12 '16 at 05:51
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    In fact, see [this answer](http://stats.stackexchange.com/a/7206/6633), also by Moderator cardinal, here on stats.SE. – Dilip Sarwate Nov 12 '16 at 06:00
  • Thx. The second indication was very useful – Mur1lo Nov 12 '16 at 06:07
  • Where does it have this? A page reference might help. Please also include a complete reference to the book (author, year, title, publisher) – Glen_b Nov 12 '16 at 09:26
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    Check http://math.stackexchange.com/questions/321569/approximating-the-error-function-erf-by-analytical-functions and http://math.stackexchange.com/questions/42920/efficient-and-accurate-approximation-of-error-function and http://math.stackexchange.com/questions/97/how-to-accurately-calculate-the-error-function-erfx-with-a-computer for multiple simple approximations of erfc – Tim Nov 12 '16 at 10:05
  • @user190080 I still need hints for the first approximation. The indication of Dilip Sarwate solved the first one. – Mur1lo Nov 12 '16 at 17:44
  • I assume you mean you need help with this $\displaystyle \Phi^{-1}\left(1-\frac{1}{n}\right)\approx \sqrt{2\log n}$, correct? (your comment was sort of contradicting itself :) – user190080 Nov 12 '16 at 19:05
  • You got it right :) – Mur1lo Nov 12 '16 at 19:09

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