zeta {sn}R Documentation

Function ‘log(2*pnorm(x))’ and its derivatives

Description

The function log(2*(pnorm(x)) and its derivatives, including inverse Mills ratio.

Usage

zeta(k, x)

Arguments

k an integer scalar between 0 and 5.
x a vector. Missing values (NAs) and Infs are allowed

Details

For k between 0 and 5, the derivative of order k of log(2*pnorm(x)) is evaluated; the derivative of order k=0 refers to the function itself. If k is not integer, it is converted to integer and a warning message is generated. If k<0 or k>5, NULL is returned.

The computation for k>1 is reduced to the case k=1, making use of expressions given by Azzalini and Capitanio (1999). For numerical stability, the evaluation of zeta(1,x) when x < -50 makes use of the asymptotic expansion (26.2.13) in Abramowitz and Stegun (1964).

zeta(1,-x) equals dnorm(x)/pnorm(-x) (in principle, apart from the asymptotic expansion mentioned above), called the inverse Mills ratio.

Value

a vector giving the k-th order derivative evaluated at x

References

Abramowitz, M. and Stegun, I. A., editors (1964). Handbook of Mathematical Functions. Dover Publications.

Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution. Technical report available at http://azzalini.stat.unipd.it/SN. An abriged version is published in J.Roy.Statist.Soc. B 61, 579–602.

Examples

y <- zeta(2,seq(-20,20,by=0.5))
#
for(k in 0:5) curve(zeta(k,x), from=-1.5, to=5, col = k+2, add = k > 0)
legend(3.5, -0.5, legend=as.character(0:5), col=2:7, lty=1)

[Package sn version 0.4-14 Index]