curesurv (version 0.1.1)

inc.beta.deriv: inc_beta_deriv function

Description

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 : $$I_{x,p,q}\approx K_{x,p,q}{A}_n/B_n$$, $n\ge 1$

This technique was proposed by Moore (1982) to calculate the derivatives of incomplete gamma function.

Usage

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
)

Value

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

$I_{x,p,q}$. This equal to the output of pbeta(x,shape1,shape2)

Ip

$I_{x,p,q}^p$ denotes the first derivative of the incomplete beta function with respect to p

Ipp

$I_{x,p,q}^{pp}$ denotes the second derivative of the incomplete beta function with respect to p

Iq

$I_{x,p,q}^q$ denotes the first derivative of the incomplete beta function with respect to q

Iqq

$I_{x,p,q}^{qq}$ denotes the second derivative of the incomplete beta function with respect to q

Ipq

$I_{x,p,q}^{pq}$ denotes the first derivative of the incomplete beta function with respect to p and q

log.Beta

$\log [\mathrm{Beta}(p,q)]$

digamma.p

${\psi }_p$

trigamma.p

${\psi }_p^{\prime }$

digamma.q

${\psi }_q$

trigamma.q

${\psi }_q^{\prime }$

digamma.pq

${\psi }_{p+q}$

trigamma.pq

${\psi }_{p+q}^{\prime }$

nappx

highest order approximant evaluated. Iteration stops if nappx>maxappx

errapx

approximate maximum absolute error of computed derivatives

Arguments

x

vector of length k containing values to which the beta function is to be integrated

p

Beta shape1 parameter

q

Beta shape2 parameter. shape1 and shape2 can be vertors in the same dimension as x or scalars

err

value for error

minapp

minimal bound value

maxapp

external noud value

References

Boik, Robert J., and James F. Robison-Cox. "Derivatives of the incomplete beta function." Journal of Statistical Software 3.1 (1998): 1-20. (arXiv)