Gradients of common densities

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.

Usage

gnorm(x)

glogis(x)

gcauchy(x)

Arguments

x: numeric vector of quantiles.

Details

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.

Value

a numeric vector of gradients.

Author

Rune Haubo B Christensen

Examples


x <- -5:5
gnorm(x)
#>  [1]  7.433598e-06  5.353209e-04  1.329555e-02  1.079819e-01  2.419707e-01
#>  [6]  0.000000e+00 -2.419707e-01 -1.079819e-01 -1.329555e-02 -5.353209e-04
#> [11] -7.433598e-06
glogis(x)
#>  [1]  0.006559068  0.017027336  0.040891575  0.079962501  0.090857748
#>  [6]  0.000000000 -0.090857748 -0.079962501 -0.040891575 -0.017027336
#> [11] -0.006559068
gcauchy(x)
#>  [1]  0.004708726  0.008811346  0.019098593  0.050929582  0.159154943
#>  [6]  0.000000000 -0.159154943 -0.050929582 -0.019098593 -0.008811346
#> [11] -0.004708726