# Function log(2*Phi(x)) and its derivatives

### Description

The function `log(2*(pnorm(x))`

and its derivatives,
including inverse Mills ratio.

### Usage

1 | ```
zeta(k, x)
``` |

### Arguments

`k` |
an integer number between 0 and 5. |

`x` |
a numeric vector. Missing values ( |

### 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

1 2 3 4 |