gesm: Generalized Elliptical Semi-parametric Models
In BayesGESM: Bayesian Analysis of Generalized Elliptical Semi-Parametric Models and Flexible Measurement Error Models

Description Usage Arguments Details Value Author(s) References See Also Examples

gesm is used to obtain the statistical inference based on the Bayesian approach for regression models under the assumption that independent additive errors follow a scale mixtures of normal distribution (i.e., normal, Student-t, slash, contaminated normal, Laplace and symmetric hyperbolic distribution), where both location and dispersion parameters of the response variable distribution include nonparametric additive components described by B-splines.

1	gesm(formula, data, family, eta, burn.in, post.sam.s, thin)

`formula`	a symbolic description of the systematic component of the model to be fitted. This description allows parametric and nonparametric functions in the locacion and dispersion parameters. See details for further information.
`data`	an optional data frame, list or environment containing the variables in the model.
`family`	a description of the error distribution to be used in the model. Supported distributions include Normal, Student-t, Slash, Hyperbolic, Laplace and ContNormal, which correspond to normal, Student-t, slash, symmetric hyperbolic, Laplace and contaminated normal distributions, respectively.
`eta`	(optional) a numeric value or numeric vector that represents the extra parameter of the specified error distribution. This parameter can be assumed known or unknown.
`burn.in`	the number of burn-in iterations for the MCMC algorithm.
`post.sam.s`	the required size for the posterior sample of interest parameters.
`thin`	(optional) the thinning interval used in the simulation to obtain the required size for the posterior sample.

The argument formula comprises three parts, namely: (i) observed response variable; (ii) covariates for the location parameter including the nonparametric components; and (iii) covariates for the dispersion parameter including the nonparametric components. The first two parts are separated by the symbol "~" and the second and third parts are separated by the symbol "|". Furthemore, the nonparametric components can be specified by using the function bsp() in the second and third parts of the argument formula.

We implemented an efficient MCMC algorithm by combining Gibbs sampler and Metropolis-Hastings algorithm, which is mainly based on the ability of the B-splines to be expressed linearly and on the fact that the distribution of the model error can be obtained as scale mixture of normal distributions. We assume that a priori, the four parameters vectors (parametric and nonparametric components on location and dispersion submodels) are independent and normally distributed. The considered values for hyperparameters enable a direct comparison of the results with those obtained under the classical approach.

`chains`	A matrix that contains the posterior sample of interest parameters. Each column represents the marginal posterior sample of each parameter.
`res`	a vector of quantile residuals, proposed by Dunn and Smyth (1996) in the context of classical inference, but suited here to the Bayesian case.
`K-L`	a vector of case-deletion influence measures based on the Kullback-Leibler divergence.
`X_2`	a vector of case-deletion influence measures based on the X2-Distance divergence.
`DIC`	DIC criterion for model selection.
`LMPL`	Log-marginal pseudo-likelihood for model selection.

Luz Marina Rondon <lumarp@gmail.com> and Heleno Bolfarine

Rondon, L.M. and Bolfarine, H. (2015) Bayesian Analysis of Generalized Elliptical Semi-parametric Models. (submitted).

Dunn, P.K. e Smyth, G.K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics. 5, 236-244.

bsp

##########   European Rabbit   ##########
#library(ssym)
#data(Erabbits)
#Erabbits2 <- Erabbits[order(Erabbits$age,Erabbits$wlens),]
#attach(Erabbits2)

#fit <- gesm(wlens ~ bsp(age) | bsp(age), family= "ContNormal", eta=c(0.8,0.9),
#				 burn.in=1000, post.sam.s=5000, thin=10)			 
#summary(fit)

##########   Plot nonparametric components for the location and dispersion parameters 
#par(mfrow=c(1,2))
#bsp.graph.gesm(fit, which=1, age, xlab="Rabbit age", ylab="f(age)", main="Location")
#bsp.graph.gesm(fit, which=2, age, xlab="Rabbit age", ylab="g(age)", main="Dispersion")
									
##########   Residual plot
#par(mfrow=c(1,2))
#plot(fit$res, ylim=c(-2,2), xlab="Index", ylab="", main="Residuals", cex=0.3,
#	 type="p", lwd=3)
#abline(h=0,lty=3)
#qqnorm(fit$res, xlim=c(-2,2), ylim=c(-2,2), xlab="Quantile", ylab="Residuals",
#	   cex=0.3, type="p", lwd=3)
#abline(0,1,lty=3)

##########   Influence measures plot
#par(mfrow=c(1,2))
#plot(fit$KL, xlab="Index", ylab="", main="K-L divergence", cex=0.3, type="p", lwd=3)
#abline(h=3*mean(fit$KL))
#plot(fit$X_2, xlab="Index", ylab="", main="X2 divergence", cex=0.3, type="p", lwd=3)
#abline(h=3*mean(fit$X_2))