ndtri

→Tensor

ndtri(p: Tensor)

source

Inverse standard normal CDF (probit / quantile function).

Computes $\Phi^{-1}(p)$ , the quantile function of the standard normal distribution. This is the workhorse for sampling Gaussian variates from uniforms via inverse-CDF, for probit regression, and for converting tail probabilities back to standard scores.

Parameters

pTensor

Input tensor of probabilities in

(0, 1)

; any floating-point dtype.

Returns

Tensor

$\Phi^{-1}(p)$ element-wise, same shape and dtype as p; values are unbounded reals.

Notes

The implementation uses the Beasley-Springer-Moro rational approximation. The unit interval is split into a central region $[p_{lo}, p_{hi}]$ with $p_{lo} = 0.02425$ , $p_{hi} = 1 - p_{lo}$ , in which a rational polynomial in $q = p - 0.5$ is evaluated, and two symmetric tail regions in which a rational polynomial in $r = \sqrt{-2 \log\min(p, 1-p)}$ is used:

\Phi^{-1}(p) \approx \begin{cases} q \cdot \frac{N_c(q^2)}{D_c(q^2)}, & p \in [p_{lo}, p_{hi}] \\[4pt] \pm \frac{N_t(r)}{D_t(r)}, & \text{otherwise}. \end{cases}

The approximation is accurate to roughly $1.15 \times 10^{-9}$ over its supported domain. Boundary behaviour: $\Phi^{-1}(0.5) = 0$ , and the function blows up to $\mp \infty$ at the endpoints.

Examples

>>> import lucid
>>> from lucid.special import ndtri
>>> ndtri(lucid.tensor([0.025, 0.5, 0.975]))
Tensor([-1.9600, 0.0000, 1.9600])