Gradients of common density functions in their standard forms, i.e.,
with zero location (mean) and unit scale. These are implemented in C
for speed and care is taken that the correct results are provided for
the argument being NA
, NaN
, Inf
, -Inf
or
just extremely small or large.
gnorm(x)glogis(x)
gcauchy(x)
a numeric vector of gradients.
numeric vector of quantiles.
Rune Haubo B Christensen
The gradients are given by:
gnorm: If \(f(x)\) is the normal density with mean 0 and spread 1, then the gradient is $$f'(x) = -x f(x)$$
glogis: If \(f(x)\) is the logistic density with mean 0 and scale 1, then the gradient is $$f'(x) = 2 \exp(-x)^2 (1 + \exp(-x))^{-3} - \exp(-x)(1+\exp(-x))^{-2}$$
pcauchy: If \(f(x) = [\pi(1 + x^2)^2]^{-1}\) is the cauchy density with mean 0 and scale 1, then the gradient is $$f'(x) = -2x [\pi(1 + x^2)^2]^{-1}$$
These gradients are used in the Newton-Raphson algorithms in fitting
cumulative link models with clm
and cumulative link
mixed models with clmm
.
Gradients of densities are also implemented for the extreme value
distribtion (gumbel
) and the the log-gamma distribution
(log-gamma
).
x <- -5:5
gnorm(x)
glogis(x)
gcauchy(x)
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