naginterfaces.library.stat.inv_​cdf_​normal

naginterfaces.library.stat.inv_cdf_normal(p, tail='L')

inv_cdf_normal returns the deviate associated with the given probability of the standard Normal distribution.

For full information please refer to the NAG Library document for g01fa

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/g01/g01faf.html

Parameters

pfloat

$p$, the probability from the standard Normal distribution as defined by $tail$.

tailstr, length 1, optional

Indicates which tail the supplied probability represents.

$tail=\text{'L'}$

The lower probability, i.e., $P(X\le x_p)$.

$tail=\text{'U'}$

The upper probability, i.e., $P(X\ge x_p)$.

$tail=\text{'S'}$

The two tail (significance level) probability, i.e., $P(X\ge \left|x_p\right|)+P(X\le -\left|x_p\right|)$.

$tail=\text{'C'}$

The two tail (confidence interval) probability, i.e., $P(X\le \left|x_p\right|)-P(X\le -\left|x_p\right|)$.

Returns

xfloat

The deviate associated with the given probability of the standard Normal distribution.

Raises

NagValueError

(errno $1$)

On entry, $tail=⟨value⟩$.

Constraint: $tail=\text{'L'}$, $\text{'U'}$, $\text{'S'}$ or $\text{'C'}$.

(errno $2$)

On entry, $p=⟨value⟩$.

Constraint: $p<1.0$.

(errno $2$)

On entry, $p=⟨value⟩$.

Constraint: $p>0.0$.

Notes

The deviate, $x_p$ associated with the lower tail probability, $p$, for the standard Normal distribution is defined as the solution to

$$P(X\le x_p)=p={\int }_{-\infty }^{x_p}Z\left(X\right)dX\text{,}$$

where

$$Z\left(X\right)=\frac{1}{\sqrt{2\pi }}{e}^{-X^2/2}\text{,}-\infty <X<\infty \text{.}$$

The method used is an extension of that of Wichura (1988). $p$ is first replaced by $q=p-0.5$.

1. If $\left|q\right|\le 0.3$, $x_p$ is computed by a rational Chebyshev approximation

   $$x_p=s\frac{A\left(s^2\right)}{B\left(s^2\right)}\text{,}$$

   where $s=\sqrt{2\pi }q$ and $A$, $B$ are polynomials of degree $7$.

2. If $0.3<\left|q\right|\le 0.42$, $x_p$ is computed by a rational Chebyshev approximation

   $$x_p=sign\left(q\right)\left(\frac{C\left(t\right)}{D\left(t\right)}\right)\text{,}$$

   where $t=\left|q\right|-0.3$ and $C$, $D$ are polynomials of degree $5$.

3. If $\left|q\right|>0.42$, $x_p$ is computed as

   $$x_p=sign\left(q\right)[\left(\frac{E\left(u\right)}{F\left(u\right)}\right)+u]\text{,}$$

   where $u=\sqrt{-2\times \log \left(min(p,1-p)\right)}$ and $E$, $F$ are polynomials of degree $6$.

For the upper tail probability $-x_p$ is returned, while for the two tail probabilities the value $x_{p^{\ast }}$ is returned, where $p^{\ast }$ is the required tail probability computed from the input value of $p$.

References

NIST Digital Library of Mathematical Functions

Hastings, N A J and Peacock, J B, 1975, Statistical Distributions, Butterworth

Wichura, 1988, Algorithm AS 241: the percentage points of the Normal distribution, Appl. Statist. (37), 477–484