zsRlm: Robust Zellner-Siow g-prior

In abnormally-distributed/Bayezilla: Bayesian Models With JAGS and Several Utilities for Data Exploration and Model Evaluation.

Description

The same model as the zsGlm Gaussian model, but with the likelihood function replaced by a "contaminated normal" model which consists of a convolution of two Gaussians with the same mean but different variances. The Gaussian with the lower variance models the "uncontaminated" data while the Gaussian with the higher variance models the "contaminated" portion of the data. Tukey, 1960; Box & Tiao, 1968; Gleason, 1993). The mixture proportions are obtained using either Huber's Psi, Hampel's Psi, or Tukey's Bisquare functions to generate weights. Information about the weight functions is given in the figure below. (Huber, 1964; Hampel et al., 1986; Hoaglin, Mosteller, & Tukey, 1983)

The Fisher information matrix used as part of the g-prior (see the documentation in the zsGlm function for more information) is calculated here using a minimum volume ellipsoid method to obtain a robust covariance matrix, which is subjected to eigendecomposition and reconstructed as an inverse using the formula L * D^-1 * L'. The matrix is then scaled according to the ratio of the trace of the non-robust Fisher information matrix to the robust matrix. Note this particular model is meant to be an empirical Bayes or frequentist method estimated using full Bayesian 'machinery' as a means to an end, and not so much a fully Bayesian model as most other functions in this package.

Usage

zsRlm(formula, data, robfun = "hampel", c = 1.345, t = 4.685,
  k = 0.9016085, log_lik = FALSE, iter = 10000, warmup = 1000,
  adapt = 5000, chains = 4, thin = 1, method = "rjparallel",
  cl = makeCluster(2), ...)

Arguments

formula	the model formula
data	a data frame
robfun	"huber" for Huber's Psi, "tukey" for Tukey's Bisquare, or "hampel" for Hampel's Psi (the default).
c	the tuning constant for the Huber weights function. Defaults to 1.345, which gives 95% the efficiency of OLS when there are no outliers.
t	the tuning constant for the Tukey's bisquare weights function. Defaults to 4.685, which gives 95% the efficiency of OLS when there are no outliers.
k	the tuning constant controlling the asymptotic relative efficiency of Hampel's psi. The default is 0.9016085, which gives 95% the efficiency of OLS when there are no outliers.
log_lik	Should the log likelihood be monitored? The default is FALSE.
iter	How many post-warmup samples? Defaults to 10000.
warmup	How many warmup samples? Defaults to 1000.
adapt	How many adaptation steps? Defaults to 2000.
chains	How many chains? Defaults to 4.
thin	Thinning interval. Defaults to 1.
method	Defaults to "rjparallel". For an alternative parallel option, choose "parallel". Otherwise, "rjags" (single core run).
cl	Use parallel::makeCluster(# clusters) to specify clusters for the parallel methods. Defaults to two cores.
...	Other arguments to run.jags.
method	Defaults to "parallel". For an alternative parallel option, choose "rjparallel". Otherwise, "rjags" (single core run).

Value

A run.jags object.

References

Box, G., & Tiao, G. (1968). "A Bayesian Approach to Some Outlier Problems."" Biometrika, 55(1), 119-129. doi:10.2307/2334456

Gleason, J. R. (1993). "Understanding Elongation: The Scale Contaminated Normal Family" JASA 88(421)

Hampel, F., E. Ronchetti, P. Rousseeuw, and W. Stahel (1986). "Robust Statistics: The Approach Based on Influence Functions."

Hoaglin, D., Mosteller, F., & Tukey, J. (1983). "Understanding Robust and Exploratory Data Analysis." John Wiley and Sons, Inc., New York.

Huber, Peter J. (1964). "Robust Estimation of a Location Parameter". Annals of Statistics. 53 (1): 73–101. doi:10.1214/aoms/1177703732

Liang, Paulo, Molina, Clyde, & Berger (2008). Mixtures of g Priors for Bayesian Variable Selection, Journal of the American Statistical Association, 103:481, 410-423, DOI: 10.1198/016214507000001337

Maronna, R. A., Martin, R. D., Yohai, V. J., & Salibian-Barrera, M. (2019). "Robust statistics: Theory and methods (with R)." Hoboken, NJ: John Wiley & Sons.

Tukey, J. W. (1960). "A Survey of Sampling from Contaminated Distributions" in I. Olkin, ed., Contributions to Probability and Statistics

Zellner, A. & Siow S. (1980). "Posterior odds ratio for selected regression hypotheses." In Bayesian statistics. Proc. 1st int. meeting (eds J. M. Bernardo, M. H. DeGroot, D. V. Lindley & A. F. M. Smith), 585?603. University Press, Valencia.

Zellner, A. (1986). "On assessing prior distributions and Bayesian regression analysis with g-prior distributions." In P. K. Goel and A. Zellner, editors, Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, 233?243.

Examples

zsRlm()

abnormally-distributed/Bayezilla documentation built on Oct. 31, 2019, 1:57 a.m.