burr12_sf_alt

mdtools.functions.burr12_sf_alt(t, tau=1, beta=1, d=1)[source]

Survival function of the Burr Type XII distribution function:

\[S(t) = \frac{ 1 }{ \left[ 1 + \left( \frac{t}{\tau} \right)^\beta \right]^{\frac{d}{\beta}} }\]

This function uses an alternative parameterization compared to burr12_sf() by setting \(d = \beta \delta\). The advantage is that one can know specify bounds for \(\beta \delta\) when fitting data with this function. One might want to do this, because the n-th raw moment of the Burr Type XII distribution, which is

\[\langle t^n \rangle = \tau^n \frac{ \Gamma\left( \delta - \frac{n}{\beta}\right) \Gamma\left( 1 + \frac{b}{\beta}\right) }{ \Gamma(\delta) },\]

only exists if \(\beta \delta > n\).

Parameters:

t (scalar or array_like) – Value(s) at which to evaluate \(f(t)\).
tau (scalar or array_like, optional) – Scale parameter(s).
beta (scalar or array_like, optional) – Shape parameter(s).
d (scalar or array_like, optional) – Shape parameter(s).

Returns:

sf (scalar or numpy.ndarray) – The outcome of \(S(t)\). \(S(t)\) is evaluated element-wise if at least one of the input arguments is an array.