naginterfaces.library.stat.inv_​cdf_​normal_​vector

naginterfaces.library.stat.inv_cdf_normal_vector(tail, p, xmu, xstd)

inv_cdf_normal_vector returns a number of deviates associated with given probabilities of the Normal distribution.

For full information please refer to the NAG Library document for g01ta

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01taf.html

Parameters

tailstr, length 1, array-like, shape $\left(\text{ltail}\right)$

Indicates which tail the supplied probabilities represent. Letting $Z$ denote a variate from a standard Normal distribution, and $z_i=\frac{x_{p_i}-{\mu }_i}{{\sigma }_i}$, then for $j=\left(mod(\text{i}-1,\text{ltail})\right)+1-1$, for $\text{i}=1,2,\dots ,max(\text{ltail},\text{lp},\text{lxmu},\text{lxstd})$:

$tail\left[j\right]=\text{'L'}$

The lower tail probability, i.e., $p_i=P(Z\le z_i)$.

$tail\left[j\right]=\text{'U'}$

The upper tail probability, i.e., $p_i=P(Z\ge z_i)$.

$tail\left[j\right]=\text{'C'}$

The two tail (confidence interval) probability, i.e., $p_i=P(Z\le \left|z_i\right|)-P(Z\le -\left|z_i\right|)$.

$tail\left[j\right]=\text{'S'}$

The two tail (significance level) probability, i.e., $p_i=P(Z\ge \left|z_i\right|)+P(Z\le -\left|z_i\right|)$.

pfloat, array-like, shape $\left(\text{lp}\right)$

$p_i$, the probabilities for the Normal distribution as defined by $tail$ with $p_i=p\left[j\right]$, $j=mod(i-1,\text{lp})$.

xmufloat, array-like, shape $\left(\text{lxmu}\right)$

${\mu }_i$, the means.

xstdfloat, array-like, shape $\left(\text{lxstd}\right)$

${\sigma }_i$, the standard deviations.

Returns

xfloat, ndarray, shape $(max(\text{ltail},\text{lxmu}))$

$x_{p_i}$, the deviates for the Normal distribution.

ivalidint, ndarray, shape $(max(\text{ltail},\text{lxmu}))$

$ivalid[i-1]$ indicates any errors with the input arguments, with

$ivalid[i-1]=0$

No error.

$ivalid[i-1]=1$

On entry, invalid value supplied in $tail$ when calculating $x_{p_i}$.

$ivalid[i-1]=2$

On entry, $p_i\le 0.0$, or, $p_i\ge 1.0$.

$ivalid[i-1]=3$

On entry, ${\sigma }_i\le 0.0$.

Raises

NagValueError

(errno $2$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{ltail}>0$.

(errno $3$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lp}>0$.

(errno $4$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lxmu}>0$.

(errno $5$)

On entry, $\text{array size}=⟨value⟩$.

Constraint: $\text{lxstd}>0$.

Warns

NagAlgorithmicWarning

(errno $1$)

On entry, at least one value of $tail$, $xstd$ or $p$ was invalid.

Check $ivalid$ for more information.

Notes

The deviate, $x_{p_i}$ associated with the lower tail probability, $p_i$, for the Normal distribution is defined as the solution to

$$P(X_i\le x_{p_i})=p_i={\int }_{-\infty }^{x_{p_i}}{Z}_i\left(X_i\right)d{X}_i\text{,}$$

where

$$Z_i\left(X_i\right)=\frac{1}{\sqrt{2\pi {\sigma }_i^2}}{e}^{-{(X_i-{\mu }_i)}^2/\left(2{\sigma }_i^2\right)}\text{,}-\infty <X_i<\infty \text{.}$$

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

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

   $$z_i=s_i\frac{A_i\left(s_i^2\right)}{B_i\left(s_i^2\right)}\text{,}$$

   where $s_i=\sqrt{2\pi }{q}_i$ and $A_i$, $B_i$ are polynomials of degree $7$.

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

   $$z_i=sign\left(q_i\right)\left(\frac{C_i\left(t_i\right)}{D_i\left(t_i\right)}\right)\text{,}$$

   where $t_i=\left|q_i\right|-0.3$ and $C_i$, $D_i$ are polynomials of degree $5$.

3. If $\left|q_i\right|>0.42$, $z_i$ is computed as

   $$z_i=sign\left(q_i\right)[\left(\frac{E_i\left(u_i\right)}{F_i\left(u_i\right)}\right)+u_i]\text{,}$$

   where $u_i=\sqrt{-2\times \log \left(min(p_i,1-p_i)\right)}$ and $E_i$, $F_i$ are polynomials of degree $6$.

$x_{p_i}$ is then calculated from $z_i$, using the relationsship $z_{p_i}=\frac{x_i-{\mu }_i}{{\sigma }_i}$.

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

The input arrays to this function are designed to allow maximum flexibility in the supply of vector arguments by re-using elements of any arrays that are shorter than the total number of evaluations required. See the G01 Introduction for further information.

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