Beta inverse cumulative distribution function

`x = betaincinv(y,z,w)`

x = betaincinv(y,z,w,tail)

`x = betaincinv(y,z,w)`

computes the inverse
incomplete beta function for corresponding elements of `y`

, `z`

,
and `w`

, such that `y = betainc(x,z,w)`

.
The elements of `y`

must be in the closed interval
[0,1], and those of `z`

and `w`

must
be nonnegative. `y`

, `z`

, and `w`

must
all be real and the same size (or any of them can be scalar).

`x = betaincinv(y,z,w,tail)`

specifies the
tail of the incomplete beta function. Choices are `'lower'`

(the
default) to use the integral from 0 to `x`

, or `'upper'`

to
use the integral from `x`

to 1. These two choices
are related as follows: `betaincinv(y,z,w,'upper') = betaincinv(1-y,z,w,'lower')`

.
When `y`

is close to 0, the `'upper'`

option
provides a way to compute `x`

more accurately than
by subtracting `y`

from 1.

The incomplete beta function is defined as

$${I}_{x}(z,w)=\frac{1}{\beta (z,w)}{\displaystyle \underset{0}{\overset{x}{\int}}{t}^{(z-1)}{(1-t)}^{(w-1)}dt}$$

`betaincinv`

computes the inverse of the
incomplete beta function with respect to the integration limit `x`

using
Newton's method.

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