cdfgdd | R Documentation |
This function computes the cumulative probability or nonexceedance probability of the Gamma Difference distribution (Klar, 2015) given parameters (\alpha_1 > 0
, \beta_1 > 0
, \alpha_2 > 0
, \beta_2 > 0
) computed by pargdd
. The cumulative distribution function is complex and numerical integration is used.
F(x) = \frac{\beta_2^{\alpha_2}}{\Gamma(\alpha_1)\Gamma(\alpha_2)} \int_{\mathrm{max}\{0, -t\}}^\infty \!\!\!\!\!\!\!\!x^{\alpha_2 - 1} e^{-\beta_2x}\gamma\bigl(\alpha_1, \beta_1(x+t)\bigr)\,\mathrm{d}x\mbox{,}
where F(x)
is the nonexceedance probability for quantile x \in (-\infty, \infty)
, \Gamma(y)
is the complete gamma function, and \gamma(a, y)
is the lower incomplete gamma function
\gamma(a, y) = \int_0^y t^{a-1}e^{-t}\,\mathrm{d}t\mbox{.}
The so-called Gamma Difference distribution is the distribution for the difference of two Gamma random variables X_1 \sim \Gamma(\alpha_1, \beta_1)
and X_1 \sim \Gamma(\alpha_2, \beta_2)
; X = X_1 - X_2
is a Gamma Difference random variable. The distribution has other names in the literature.
cdfgdd(x, para, paracheck=TRUE, silent=TRUE, ...)
x |
A real value vector. |
para |
The parameters from |
paracheck |
A logical controlling whether the parameters are checked for validity. |
silent |
The argument of |
... |
Additional argument to pass. |
Nonexceedance probability (F
) for x
.
W.H. Asquith
Klar, B., 2015, A note on gamma difference distributions: Journal of Statistical Computation and Simulation v. 85, no. 18, pp. 1–8, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00949655.2014.996566")}.
pdfgdd
, quagdd
, lmomgdd
, pargdd
## Not run:
x <- seq(-5, 7, by=0.01)
para <- list(para=c(3, 1, 1, 1), type="gdd")
plot(x, cdfgdd(x, para), type="l", xlim=c(-5,7), ylim=c(0, 1),
xlab="x", ylab="distribution function of gamma difference distribution")
para <- list(para=c(2, 1, 1, 1), type="gdd")
lines(x, cdfgdd(x, para), lty=2)
para <- list(para=c(1, 1, 1, 1), type="gdd")
lines(x, cdfgdd(x, para), lty=3)
para <- list(para=c(0.5, 1, 1, 1), type="gdd")
lines(x, cdfgdd(x, para), lty=4) #
## End(Not run)
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