https://github.com/cran/sn
Tip revision: bc33612e6cc33fcf28f50655cab5f1931985ccde authored by Adelchi Azzalini on 04 April 2023, 17:10:02 UTC
version 2.1.1
version 2.1.1
Tip revision: bc33612
zeta.Rd
% file sn/man/zeta.Rd
% This file is a component of the package 'sn' for R
% copyright (C) 1998,2013 Adelchi Azzalini
%---------------------
\name{zeta}
\alias{zeta}
\title{Function \eqn{\log(2\,\Phi(x))}{log(2 Phi(x))} and its derivatives}
\description{The function \code{log(2*pnorm(x))} and its derivatives,
including inverse Mills ratio.}
\usage{zeta(k, x)}
\arguments{
\item{k}{an integer number between 0 and 5.}
\item{x}{a numeric vector. Missing values (\code{NA}s) and \code{Inf}s are
allowed.}
}
\value{a vector representing the \code{k}-th order derivative evaluated at
\code{x}.}
\details{
For \code{k} between 0 and 5, the derivative of order \code{k} of
\eqn{\log(2\,\Phi(x))}{log(2\Phi(x))} is evaluated, where \eqn{\Phi(x)}
denotes the \eqn{N(0,1)} cumulative distribution function.
The derivative of order \code{k=0} refers to the function itself.
If \code{k} is not an integer within \code{0,..., 5}, \code{NULL} is returned.
}
\section{Background}{
The computation for \code{k>1} is reduced to the case \code{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 \code{zeta(1,x)} when
\code{x < -50} makes use of the asymptotic expansion (26.2.13) of
Abramowitz and Stegun (1964).
\code{zeta(1,-x)} equals \code{dnorm(x)/pnorm(-x)} (in principle, apart from
the above-mentioned asymptotic expansion), called the
\emph{inverse Mills ratio}.
}
\references{
Abramowitz, M. and Stegun, I. A., editors (1964).
\emph{Handbook of Mathematical Functions}.
Dover Publications.
Azzalini, A. and Capitanio, A. (1999).
Statistical applications of the multivariate skew normal distribution.
\emph{J.Roy.Statist.Soc. B} \bold{61}, 579--602. Full-length version
available at \url{https://arXiv.org/abs/0911.2093}
Azzalini, A. with the collaboration of Capitanio, A. (2014).
\emph{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)
}
\keyword{math}
\concept{Mills ratio}