# Expected and observed Fisher information for SN and ST distributions

### Description

Computes Fisher information for parameters of simple sample having
skew-normal (SN) or skew-*t* (ST) distribution or
for a regression model with errors term having such distributions, in the
DP and CP parametrizations.

### Usage

1 2 3 4 5 6 7 8 9 | ```
sn.infoUv(dp=NULL, cp=NULL, x=NULL, y, w, penalty=NULL, norm2.tol=1e-06)
sn.infoMv(dp, x=NULL, y, w, penalty=NULL, norm2.tol=1e-06)
st.infoUv(dp = NULL, cp = NULL, x = NULL, y, w, fixed.nu = NULL,
symmetr = FALSE, penalty = NULL, norm2.tol = 1e-06)
st.infoMv(dp, x = NULL, y, w, fixed.nu = NULL, symmetr = FALSE,
penalty = NULL, norm2.tol = 1e-06)
``` |

### Arguments

`dp, cp` |
direct or centred parameters, respectively; one of the two
vectors must be supplied, but not both. For the univariate SN
distribution, |

`x` |
an optional matrix which represents the design matrix of a regression model |

`y` |
a numeric vector (for |

`w` |
an optional vector of weights; if missing, a vector of 1's is generated. |

`fixed.nu` |
an optional numeric value which declared a fixed value of the
degrees of freedom, |

`symmetr` |
a logical flag which indicates whether a symmetry condition of
the distribution is being imposed; default is |

`penalty` |
a optional character string with the name of the penalty function
used in the call to |

`norm2.tol` |
for the observed information case, the Mahalanobis squared
distance of the score 0 is evaluated; if it exceeds |

### Value

a list containing the following components:

`dp, cp` |
one of the two arguments is the one supplied on input; the other one matches the previous one in the alternative parametrization. |

`type` |
the type of information matrix: "observed" or "expected". |

`info.dp, info.cp` |
matrices of Fisher (observed or expected) information in the two parametrizations. |

`asyvar.dp, asyvar.cp` |
inverse matrices of Fisher information in the two parametrizations, when available; See ‘Details’ for additional information. |

`aux` |
a list containing auxiliary elements, depending of the selected function and the type of computation. |

### Details

In the univariate SN case, when `x`

is not set, then a simple
random sample is assumed and a matrix `x`

with a single column of all
1's is constructed; in this case, the supplied vector `dp`

or `cp`

must have length 3. If `x`

is set, then the supplied vector of parameters,
`dp`

or `cp`

, must have length `ncol(x)+2`

.
In the multivariate case, a direct extension of this scheme applies.

If the observed information matrix is required, `dp`

or `dp`

should
represent the maximum likelihood estimates (MLE) for the given `y`

,
otherwise the information matrix may fail to be positive-definite. Therefore,
the squared Mahalobis norm of the score vector is evaluated and compared with
`norm2.tol`

. If it exceeds this threshold, it is taken as an indication
that the parameter is not at the MLE and a warning message is issued.
The returned list still includes `info.dp`

and `info.cp`

, but in
this case these represent merely the matrices of second derivatives;
`asyvar.dp`

and `asyvar.cp`

are set to `NULL`

.

### Background

The information matrix for the the univariate SN distribution in the two stated parameterizations in discussed in Sections 3.1.3–4 of Azzalini and Capitanio (2014). For the multivariate distribution, Section 5.2.2 of this monograph summarizes briefly the findings of Arellano-Valle and Azzalini (2008).

For ST distributions, only the observed information matrix is provided currently. Computation for the univariate case is based on DiCiccio and Monti (2011). For the multivariate case, the score function is computed using expression of Arellano-Valle (2010) followed by numerical differentiation.

### References

Arellano-Valle, R. B. (2010).
The information matrix of the multivariate skew-$t$ distribution.
*Metron*, **LXVIII**, 371–386.

Arellano-Valle, R. B., and Azzalini, A. (2008).
The centred parametrization for the multivariate skew-normal distribution.
*J. Multiv. Anal.* **99**, 1362–1382.
Corrigendum: vol.\,100 (2009), p.\,816.

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

DiCiccio, T. J. and Monti, A. C. (2011).
Inferential aspects of the skew *t*-distribution.
*Quaderni di Statistica* **13**, 1–21.

### See Also

`dsn`

, `dmsn`

, `dp2cp`

### Examples

1 2 3 4 5 6 7 8 9 10 | ```
infoE <- sn.infoUv(dp=c(0,1,5)) # expected information
set.seed(1); rnd <- rsn(100, dp=c(0, 1, 3))
fit <- selm(rnd~1, family="SN")
infoO <- sn.infoUv(cp=coef(fit), y=rnd) # observed information
#
data(wines)
X <- model.matrix(~ pH + wine, data=wines)
fit <- sn.mple(x=X, y=wines$alcohol)
infoE <- sn.infoUv(cp=fit$cp, x=X)
infoO <- sn.infoUv(cp=fit$cp, x=X, y=wines$alcohol)
``` |