pbetaDiff function

Differentiate Regularized Incomplete Beta Function.

Calculate derivatives of the regularized incomplete beta function that is a cumulative distribution function of the beta distribution.

pbetaDiff(x, p = 10, q = 0.5, n = 10L, is_validation = TRUE, control = NULL)

Arguments

• x: numeric vector of values between 0 and 1. It is similar to q argument of pbeta function.
• p: similar to shape1 argument of pbeta function.
• q: similar to shape2 argument of pbeta function.
• n: positive integer representing the number of iterations used to calculate the derivatives. Greater values provide higher accuracy by the cost of more computational resources.
• is_validation: logical; if TRUE then input arguments are validated. Set to FALSE to slightly increase the performance of the function.
• control: list of control parameters. Currently not intended for the users.

Returns

The function returns a list which has the following elements:

• dx - numeric vector of derivatives respect to each element of x.
• dp - numeric vector of derivatives respect to p for each element of x.
• dq - numeric vector of derivatives respect to q for each element of x.

Details

The function implements differentiation algorithm of R. Boik and J. Robinson-Cox (1998). Currently only first-order derivatives are considered.

Examples

# Some values from Table 1 of R. Boik and J. Robinson-Cox (1998)
pbetaDiff(x = 0.001,  p = 1.5, q = 11)
pbetaDiff(x = 0.5,  p = 1.5, q = 11)

# Compare analytical and numeric derivatives
delta <- 1e-6
x <- c(0.01, 0.25, 0.5, 0.75, 0.99)
p <- 5
q <- 10
out <- pbetaDiff(x = x, p = p, q = q)
p0 <- pbeta(q = x, shape1 = p, shape2 = q)

# Derivatives respect to x
p1 <- pbeta(q = x + delta, shape1 = p, shape2 = q)
data.frame(numeric = (p1 - p0) / delta, analytical = out$dx)
  
# Derivatives respect to p
p1 <- pbeta(q = x, shape1 = p + delta, shape2 = q)
data.frame(numeric = (p1 - p0) / delta, analytical = out$dp)

# Derivatives respect to q
p1 <- pbeta(q = x, shape1 = p, shape2 = q + delta)
data.frame(numeric = (p1 - p0) / delta, analytical = out$dq)

References

Boik, R. J. and Robinson-Cox, J. F. (1998). Derivatives of the Incomplete Beta Function. Journal of Statistical Software, 3 (1), pages 1-20.