Extreme Tails of the Normal Distribution

Estimating extreme quantiles of the normal distribution.

Approximating Tails of the Normal Distribution

Feller vol 1. Chapter VII Lemma 2 page 175 gives \[ (x^{-1}-x^{-3})\phi(x) < 1-\Phi(x) < x^{-1}\phi(x). \]

See also Abramowitz and Stegun asymptotic series 26.2.12 \[ \Phi(x) = \frac{\phi(x)}{x}\left( 1-\frac{1}{x^2} + \frac{1\cdot 3}{x^4} +\cdots +\frac{(-1)^n1\cdot 3\cdot \dots \cdot (2n-1)}{x^{2n}} \right). \]

Using \(\log\) for log to base 10 we get \(\log(\phi(x)) = -\log(\sqrt{2\pi}) - \displaystyle\frac{x^2}{2\ln(10)}\). The value \(\ln(10)=2.302585\) and, miraculously, \(\log(\sqrt{2\pi})=0.399090\). Therefore \[ \log(1-\Phi(x)) \approx -0.399 - \frac{x^2}{2\cdot 2.302585} - \log(x) \approx -0.4 - \frac{x^2}{4.6} - \log(x). \]

The table below compares various estimates for \(\log(1-\Phi(x))\).

\(-\log(1-\Phi(x))\) compared to approximations.

\(x\)	AS Table 26.2	\(\phi(x)/x\)	\(0.4-x^2/4.6-\log(x)\)	\((x^{-1}-x^{-3})\phi(x)\)
2	1.64302	1.56871	1.57060	1.69365
3	2.86970	2.83054	2.83364	2.88169
4	4.49933	4.47551	4.48032	4.50353
5	6.54265	6.52674	6.53375	6.54447
6	9.00586	8.99454	9.00424	9.00678
7	11.89285	11.88440	11.89727	11.89336
8	15.20614	15.19960	15.21613	15.20644
9	18.94746	18.94226	18.96294	18.94765
10	23.11805	23.11381	23.13913	23.11818
15	50.43522	50.43331	50.48913	50.43524
20	88.56010	88.55902	88.65755	88.56010
25	137.51475	137.51406	137.66751	137.51475
50	544.96634	544.96616	545.57723	544.96634
100	2,173.87154	2,173.87150	2,176.31304	2,173.87154
200	8,688.58977	8,688.58976	8,698.35320	8,688.58977
500	54,289.90830	54,289.90830	54,350.92506	54,289.90830

Hence the famous Goldman Sachs CFO quote about 25 deviations from the mean “several days in a row” (Kevin Dowd et al “How Unlucky is 25-sigma?”) is ridiculous.