Derivatives of the Incomplete Gamma Integral (Unscaled Version)
The first two derivatives of the incomplete gamma integral with scaling.
pgamma.deriv.unscaled(q, shape)
q, shape |
As in |
Define
G(a,x) = integral_0^x t^(a-1) exp(-t) dt
so that
G(x, a) is pgamma(x, a) * gamma(a).
Write x = q and shape = a.
The 0th and first and second derivatives with respect to a
of G are returned. This function is similar in spirit to
pgamma.deriv
but here there is no gamma function to scale things.
Currently a 3-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.)
This function is based on Wingo (1989).
The 3 columns, running from left to right, are the 0:2th derivatives
with respect to a.
These function seems inaccurate for q = 1 and q = 2;
see the plot below.
T. W. Yee.
See truncweibull.
x <- 3; aa <- seq(0.3, 04, by = 0.01)
ans.u <- pgamma.deriv.unscaled(x, aa)
head(ans.u)
## Not run: par(mfrow = c(1, 3))
for (jay in 1:3) {
plot(aa, ans.u[, jay], type = "l", col = "blue", cex.lab = 1.5,
cex.axis = 1.5, las = 1, main = colnames(ans.u)[jay],
log = "", xlab = "shape", ylab = "")
abline(h = 0, v = 1:2, lty = "dashed", col = "gray") # Inaccurate at 1 and 2
}
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.