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

k 
an integer number between 0 and 5. 
x 
a numeric vector. Missing values ( 
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.
a vector representing the k
th order derivative evaluated at x
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 fulllength 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 abovementioned 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. Fulllength version available at http://arXiv.org/abs/0911.2093
Azzalini, A. with the collaboration of Capitanio, A. (2014). The SkewNormal and Related Families. Cambridge University Press, IMS Monographs series.
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