scipy.special.stdtrit

scipy.special.stdtrit(df, p, out=None) = <ufunc 'stdtrit'>

The p-th quantile of the student t distribution.

This function is the inverse of the student t distribution cumulative distribution function (CDF), returning t such that stdtr(df, t) = p.

Returns the argument t such that stdtr(df, t) is equal to p.

Parameters:

dfarray_like

Degrees of freedom

parray_like

Probability

outndarray, optional

Optional output array for the function results

Returns:

tscalar or ndarray

Value of t such that stdtr(df, t) == p

See also

stdtr

Student t CDF

stdtridf

inverse of stdtr with respect to df

scipy.stats.t

Student t distribution

Notes

The student t distribution is also available as scipy.stats.t. Calling stdtrit directly can improve performance compared to the ppf method of scipy.stats.t (see last example below).

The function is computed using the Boost Math library [1], which relies on the incomplete beta function.

Array API Standard Support

stdtrit has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	✅
PyTorch	✅	⛔
JAX	⛔	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. stdtrit does not currently support out for the CuPy and PyTorch backends.

See Support for the array API standard for more information.

References

[1]

Boost C++ Libraries, https://www.boost.org/

Examples

Try it in your browser!

stdtrit represents the inverse of the student t distribution CDF which is available as stdtr. Here, we calculate the CDF for df at x=1. stdtrit then returns 1 up to floating point errors given the same value for df and the computed CDF value.

>>> import numpy as np
>>> from scipy.special import stdtr, stdtrit
>>> import matplotlib.pyplot as plt
>>> df = 3
>>> x = 1
>>> cdf_value = stdtr(df, x)
>>> stdtrit(df, cdf_value)
1.0000000000000007

Plot the function for three different degrees of freedom.

>>> x = np.linspace(0, 1, 1000)
>>> parameters = [(1, "solid"), (2, "dashed"), (5, "dotted")]
>>> fig, ax = plt.subplots()
>>> for (df, linestyle) in parameters:
...     ax.plot(x, stdtrit(df, x), ls=linestyle, label=f"$df={df}$")
>>> ax.legend()
>>> ax.set_ylim(-10, 10)
>>> ax.set_title("Student t distribution quantile function")
>>> plt.show()

The function can be computed for several degrees of freedom at the same time by providing a NumPy array or list for df:

>>> stdtrit([1, 2, 3], 0.7)
array([0.72654253, 0.6172134 , 0.58438973])

It is possible to calculate the function at several points for several different degrees of freedom simultaneously by providing arrays for df and p with shapes compatible for broadcasting. Compute stdtrit at 4 points for 3 degrees of freedom resulting in an array of shape 3x4.

>>> dfs = np.array([[1], [2], [3]])
>>> p = np.array([0.2, 0.4, 0.7, 0.8])
>>> dfs.shape, p.shape
((3, 1), (4,))

>>> stdtrit(dfs, p)
array([[-1.37638192, -0.3249197 ,  0.72654253,  1.37638192],
       [-1.06066017, -0.28867513,  0.6172134 ,  1.06066017],
       [-0.97847231, -0.27667066,  0.58438973,  0.97847231]])

The t distribution is also available as scipy.stats.t. Calling stdtrit directly can be much faster than calling the ppf method of scipy.stats.t. To get the same results, one must use the following parametrization: scipy.stats.t(df).ppf(x) = stdtrit(df, x).

>>> from scipy.stats import t
>>> df, x = 3, 0.5
>>> stdtrit_result = stdtrit(df, x)  # this can be faster than below
>>> stats_result = t(df).ppf(x)
>>> stats_result == stdtrit_result  # test that results are equal
True

Go BackOpen In Tab