Calculates the Laplace approximation to the uni- and bivariate marginal densities of components of the MLE in a regression-scale model. The reference distribution is the conditional distribution given the ancillary.
Laplace(which = stop("no choice made"), data = stop("data are missing"),
val1, idx1, val2, idx2, log.scale = TRUE)the kind of marginal density that should be approximated.
Possible choices are c (univariate: regression
coefficient),
s (univariate: scale parameter), cc (bivariate: two
regression coefficients) and cs (bivariate: regression
coefficient and scale parameter).
a special conditional sampling data object. This object must be a list with the following elements:
ancthe vector containing the values of the ancillary; usually the Pearson residuals. It has to be of the same length than the number of observations in the linear regression model.
Xthe model matrix. It may be obtained applying
model.matrix to the fitted rsm
object of interest. The number of observations has to be the
same than the dimension of the ancillary, and the number of
covariates must correspond to the number of regression
coefficients defined in the coef component.
coefthe vector of true values of the regression coefficients, that is, the values used in the simulation study.
dispthe true value of the scale parameter used in the simulation study.
familya family.rsm object characterizing the error
distribution of the linear regression model. The following
generator functions are available in the marg
package of the R package bundle hoa:
student (Student's t), extreme (Gumbel or extreme
value), logistic, logWeibull,
logExponential, logRayleigh and Huber
(Huber's least favourable). The demonstration file
margdemo.R that accompanies the marg package shows
how to create a new generator function.
fixeda logical value. If TRUE the scale parameter is known.
The make.sample.data function can be used
to create this data object from a fitted rsm model.
sequence of values for the first MLE at which to calculate the density.
index of the first regression coefficient, that is, its position in the vector MLE.
sequence of values for the second MLE at which to calculate the density.
index of the second regression coefficient, that is, its position in the vector MLE.
logical value. If TRUE the approximation is calculated on
the log scale. Highly recommended. The default is TRUE.
Returns a Lapl.spl or Lapl.cont object with the
approximate uni- or bivariate conditional distribution of one or two
components of the MLE.
The file csamplingdemo.R contains code that can be used to
run a conditional simulation study similar to the one described
in Brazzale (2000, Section 7.3) using the data given in
Example 3 of DiCiccio, Field and Fraser (1990).
Laplace's integral approximation method is used in order to avoid
multi-dimensional numerical integration. The uni- and bivariate
approximations to the marginal distributions give insight into how
the multivariate conditional distribution of the MLE
vector is structured. Methods are available to plot them. They
help in choosing a suitable candidate generation density to be used
in the rsm.sample function.
All information is supplied through the data argument. Note
that the user has to keep to the structure described above. If a
conditional simulation is to be performed for a fitted rsm
object, the make.sample.data function can be
used to generate this special object. The logical switch
fixed in the conditional sampling data object must be
specified.
Brazzale, A. R. (2000) Practical Small-Sample Parametric Inference. Ph.D. Thesis N. 2230, Department of Mathematics, Swiss Federal Institute of Technology Lausanne.
DiCiccio, T. J., Field, C. A. and Fraser, D. A. S. (1990) Approximations of marginal tail probabilities and inference for scalar parameters. Biometrika, 77, 77--95.