The deviate,
, associated with the lower tail probability,
, of the beta distribution with parameters
and
is defined as the solution to
The algorithm is a modified version of the Newton–Raphson method, following closely that of
Cran et al. (1977).
An initial approximation,
, to
is found (see
Cran et al. (1977)), and the Newton–Raphson iteration
where
is used, with modifications to ensure that
remains in the range
.
Cran G W, Martin K J and Thomas G E (1977) Algorithm AS 109. Inverse of the incomplete beta function ratio Appl. Statist. 26 111–114
The required precision, given by
tol, should be achieved in most circumstances.
Background information to multithreading can be found in the
Multithreading documentation.
The typical timing will be several times that of
g01eec and will be very dependent on the input argument values. See
g01eec for further comments on timings.
This example reads lower tail probabilities for several beta distributions and calculates and prints the corresponding deviates until the end of data is reached.