Compute approximations to the quantile (i.e., inverse cumulative) function of the Gamma distribution.
qgammaAppr(p, shape, lower.tail = TRUE, log.p = FALSE,
tol = 5e-07)
qgamma.R (p, alpha, scale = 1, lower.tail = TRUE, log.p = FALSE,
EPS1 = 0.01, EPS2 = 5e-07, epsN = 1e-15, maxit = 1000,
pMin = 1e-100, pMax = (1 - 1e-14),
verbose = getOption("verbose"))qgammaApprKG(p, shape, lower.tail = TRUE, log.p = FALSE)
qgammaApprSmallP(p, shape, lower.tail = TRUE, log.p = FALSE)
numeric vector (possibly log tranformed) probabilities.
shape parameter, non-negative.
scale parameter, non-negative, see qgamma.
logical, see, e.g., qgamma(); must
have length 1.
tolerance of maximal approximation error.
small positive number. ...
small positive number. ...
small positive number. ...
maximal number of iterations. ...
boundaries for p. ...
logical indicating if the algorithm should produce “monitoring” information.
numeric
qgammaApprSmallP(p, a) should be a good approximation in the
following situation when both p and shape \(= \alpha =:
a\) are small :
If we look at Abramowitz&Stegun \(gamma*(a,x) = x^-a * P(a,x)\) and its series \(g*(a,x) = 1/gamma(a) * (1/a - 1/(a+1) * x + ...)\),
then the first order approximation \(P(a,x) = x^a * g*(a,x) ~= x^a/gamma(a+1)\) and hence its inverse \(x = qgamma(p, a) ~= (p * gamma(a+1)) ^ (1/a)\) should be good as soon as \(1/a >> 1/(a+1) * x\)
<==> x << (a+1)/a = (1 + 1/a)
<==> x < eps *(a+1)/a
<==> log(x) < log(eps) + log( (a+1)/a ) = log(eps) + log((a+1)/a) ~ -36 - log(a) where log(x) ~= log(p * gamma(a+1)) / a = (log(p) + lgamma1p(a))/a
such that the above
<==> (log(p) + lgamma1p(a))/a < log(eps) + log((a+1)/a)
<==> log(p) + lgamma1p(a) < a*(-log(a)+ log(eps) + log1p(a))
<==> log(p) < a*(-log(a)+ log(eps) + log1p(a)) - lgamma1p(a) =: bnd(a)
Note that qgammaApprSmallP() indeed also builds on lgamma1p().
.qgammaApprBnd(a) provides this bound \(bnd(a)\);
it is simply a*(logEps + log1p(a) - log(a)) - lgamma1p(a), where
logEps is \(\log(\epsilon)\) = log(eps) where eps <-
.Machine$double.eps, i.e. typically (always?) logEps\(= \log
\epsilon = -52 * \log(2) = -36.04365\).
Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. https://en.wikipedia.org/wiki/Abramowitz_and_Stegun provides links to the full text which is in public domain.
qgamma for R's Gamma distribution functions.
# NOT RUN {
## TODO : Move some of the curve()s from ../tests/qgamma-ex.R !!
# }
Run the code above in your browser using DataCamp Workspace