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, ...)Nonexceedance probability (\(F\)) for \(x\).
A real value vector.
The parameters from pargdd or vec2par.
A logical controlling whether the parameters are checked for validity.
The argument of silent for the try() operation wrapped on integrate().
Additional argument to pass.
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, tools:::Rd_expr_doi("10.1080/00949655.2014.996566").
pdfgdd, quagdd, lmomgdd, pargdd