actuar (version 3.3-4)

betaint: The “Beta Integral”

Description

The “beta integral” which is just a multiple of the non regularized incomplete beta function. This function merely provides an R interface to the C level routine. It is not exported by the package.

Usage

betaint(x, a, b)

Value

The value of the integral.

Invalid arguments will result in return value NaN, with a warning.

Arguments

x

vector of quantiles.

a, b

parameters. See Details for admissible values.

Author

Vincent Goulet

Details

Function betaint computes the “beta integral” $$ B(a, b; x) = \Gamma(a + b) \int_0^x t^{a-1} (1-t)^{b-1} dt$$ for \(a > 0\), \(b \neq -1, -2, \ldots\) and \(0 < x < 1\). (Here \(\Gamma(\alpha)\) is the function implemented by R's gamma() and defined in its help.) When \(b > 0\), $$ B(a, b; x) = \Gamma(a) \Gamma(b) I_x(a, b),$$ where \(I_x(a, b)\) is pbeta(x, a, b). When \(b < 0\), \(b \neq -1, -2, \ldots\), and \(a > 1 + [-b]\), $$% \begin{array}{rcl} B(a, b; x) &=& \displaystyle -\Gamma(a + b) \left[ \frac{x^{a-1} (1-x)^b}{b} + \frac{(a-1) x^{a-2} (1-x)^{b+1}}{b (b+1)} \right. \\ & & \displaystyle\left. + \cdots + \frac{(a-1) \cdots (a-r) x^{a-r-1} (1-x)^{b+r}}{b (b+1) \cdots (b+r)} \right] \\ & & \displaystyle + \frac{(a-1) \cdots (a-r-1)}{b (b+1) \cdots (b+r)} \Gamma(a-r-1) \\ & & \times \Gamma(b+r+1) I_x(a-r-1, b+r+1), \end{array}$$ where \(r = [-b]\).

This function is used (at the C level) to compute the limited expected value for distributions of the transformed beta family; see, for example, levtrbeta.

References

Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, Dover.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Examples

x <- 0.3
a <- 7

## case with b > 0
b <- 2
actuar:::betaint(x, a, b)
gamma(a) * gamma(b) * pbeta(x, a, b)    # same

## case with b < 0
b <- -2.2
r <- floor(-b)        # r = 2
actuar:::betaint(x, a, b)

## "manual" calculation
s <- (x^(a-1) * (1-x)^b)/b +
    ((a-1) * x^(a-2) * (1-x)^(b+1))/(b * (b+1)) +
    ((a-1) * (a-2) * x^(a-3) * (1-x)^(b+2))/(b * (b+1) * (b+2))
-gamma(a+b) * s +
    (a-1)*(a-2)*(a-3) * gamma(a-r-1)/(b*(b+1)*(b+2)) *
    gamma(b+r+1)*pbeta(x, a-r-1, b+r+1)