log(2*(pnorm(x)) and its derivatives,
including inverse Mills ratio.
an integer number between 0 and 5.
a numeric vector. Missing values (
k between 0 and 5, the derivative of order
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.
k is not integer, it is converted to integer and a warning
message is generated.
NULL is returned.
a vector representing the
k-th order derivative evaluated at
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
x < -50 makes use of the asymptotic expansion (26.2.13) of
Abramowitz and Stegun (1964).
dnorm(x)/pnorm(-x) (in principle, apart from
the above-mentioned asymptotic expansion), called the
inverse Mills ratio.
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.
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