VGAM (version 0.9-6)

pgamma.deriv: Derivatives of the Incomplete Gamma Integral

Description

The first two derivatives of the incomplete gamma integral.

Usage

pgamma.deriv(q, shape, tmax = 100)

Arguments

q, shape

As in pgamma but these must be vectors of positive values only and finite.

tmax

Maximum number of iterations allowed in the computation (per q value).

Value

• The first 5 columns, running from left to right, are the derivatives with respect to: $x$, $x^2$, $a$, $a^2$, $xa$. The 6th column is $P(a, x)$ (but it is not as accurate as calling pgamma directly).

Details

Write $x = q$ and shape = $a$. The first and second derivatives with respect to $q$ and $a$ are returned. This function is similar in spirit to pgamma; define $$P(a,x) = \frac{1}{\Gamma(a)} \int_0^x t^{a-1} e^{-t} dt$$ so that $P(a, x)$ is pgamma(x, a). Currently a 6-column matrix is returned (in the future this may change and an argument may be supplied so that only what is required by the user is computed.)

The computations use a series expansion for $a \leq x \leq 1$ or or $x < a$, else otherwise a continued fraction expansion. Machine overflow can occur for large values of $x$ when $x$ is much greater than $a$.

References

Moore, R. J. (1982) Algorithm AS 187: Derivatives of the Incomplete Gamma Integral. Journal of the Royal Statistical Society, Series C (Applied Statistics), 31(3), 330--335.

See Also

pgamma.deriv.unscaled, pgamma.

Examples

x <- seq(2, 10, length = 501)
head(ans <- pgamma.deriv(x, 2))
par(mfrow = c(2, 3))
for (jay in 1:6)
  plot(x, ans[, jay], type = "l", col = "blue", cex.lab = 1.5,
       cex.axis = 1.5, las = 1, log = "x",
       main = colnames(ans)[jay], xlab = "q", ylab = "")