computes the first and second derivatives of incomplete Beta function with respect of Beta parameters p and or q using algorithm differentiating the aproximants of \(I_{x,p,q}\) formula in terms of forward recurrence relations where the the \(n^{th}\) approximant can be expressed as : $$ I_{x,p,q} \approx K_{x,p,q} A_n/B_n$$, \(n \geq 1\)
This technique was proposed by Moore (1982) to calculate the derivatives of incomplete gamma function.
inc.beta.deriv(
x,
p = stop("p must be specified"),
q = stop("q must be specified"),
err = .Machine$double.eps * 10000,
minapp = 2,
maxapp = 1000
)An object of class FD.inc.beta.
This object is a list containing 15 components. The first 13 components in
the list are each a vector of the same length as x (u in the model). The two
last elements are scalar terms. The output elements are:
\(I_{x,p,q}\). This equal to the output of
pbeta(x,shape1,shape2)
\(I_{x,p,q}^{p}\) denotes the first derivative of the incomplete beta function with respect to p
\(I_{x,p,q}^{pp}\) denotes the second derivative of the incomplete beta function with respect to p
\(I_{x,p,q}^{q}\) denotes the first derivative of the incomplete beta function with respect to q
\(I_{x,p,q}^{qq}\) denotes the second derivative of the incomplete beta function with respect to q
\(I_{x,p,q}^{pq}\) denotes the first derivative of the incomplete beta function with respect to p and q
\(\log[\mathrm{Beta}(p,q)]\)
\(\psi_p\)
\(\psi_p'\)
\(\psi_q\)
\(\psi_q'\)
\(\psi_{p+q}\)
\(\psi_{p+q}'\)
highest order approximant evaluated. Iteration stops if nappx>maxappx
approximate maximum absolute error of computed derivatives
vector of length k containing values to which the beta function is to be integrated
Beta shape1 parameter
Beta shape2 parameter. shape1 and shape2 can be vertors in the same dimension as x or scalars
value for error
minimal bound value
external noud value
Boik, Robert J., and James F. Robison-Cox. "Derivatives of the incomplete beta function." Journal of Statistical Software 3.1 (1998): 1-20. (arXiv)