sn (version 0.4-18)

zeta: 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

Value

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

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< code=""> 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.

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)