R: Function log(2*Phi(x)) and its derivatives

zeta {sn} | R Documentation

## Function log(2*Phi(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 number between 0 and 5.
`x` | a numeric vector. Missing values (`NA`s) and `Inf`s are allowed.

### Details

For `k` between 0 and 5, the derivative of order `k` of log(2Φ(x)) is evaluated, where Φ(x) denotes the N(0,1) cumulative distribution function. 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.

### Value

a vector representing the `k`-th order derivative evaluated at `x`.

### Background

The computation for `k>1` is reduced to the case `k=1`, making use of expressions given by Azzalini and Capitanio (1999); see especially Section 4 of the full-length version of the paper. The main facts are summarized in Section 2.1.4 of Azzalini and Capitanio (2014).

For numerical stability, the evaluation of `zeta(1,x)` when `x < -50` makes use of the asymptotic expansion (26.2.13) of Abramowitz and Stegun (1964).

`zeta(1,-x)` equals `dnorm(x)/pnorm(-x)` (in principle, apart from the above-mentioned asymptotic expansion), 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. J.Roy.Statist.Soc. B 61, 579–602. Full-length version available at http://arXiv.org/abs/0911.2093

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.

### 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 1.5-3 Index]