R: Calculate Laplace's Marginal Density Approximation

Laplace {csampling}

R Documentation

Calculate Laplace's Marginal Density Approximation

Description

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.

Usage

Laplace(which = stop("no choice made"), data = stop("data are missing"), 
        val1, idx1, val2, idx2, log.scale = TRUE)

Arguments

`which`	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).
`data`	a special conditional sampling data object. This object must be a list with the following elements: `anc` the 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. `X` the 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. `coef` the vector of true values of the regression coefficients, that is, the values used in the simulation study. `disp` the true value of the scale parameter used in the simulation study. `family` a `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. `fixed` a 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.
`val1`	sequence of values for the first MLE at which to calculate the density.
`idx1`	index of the first regression coefficient, that is, its position in the vector MLE.
`val2`	sequence of values for the second MLE at which to calculate the density.
`idx2`	index of the second regression coefficient, that is, its position in the vector MLE.
`log.scale`	logical value. If `TRUE` the approximation is calculated on the log scale. Highly recommended. The default is `TRUE`.

Details

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.

Value

Returns a Lapl.spl or Lapl.cont object with the approximate uni- or bivariate conditional distribution of one or two components of the MLE.

Demonstration

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).

References

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.