mcmcrecord_gaussian: MCMC iterations

Description

Estimated averaged bandwidths of the regressors and averaged variance parameter of the normal error density

Usage

mcmcrecord_gaussian(x, inicost, mutsizp, warm = 100, M = 100, prob = 0.234,  num_batch = 10, step = 10, data_x, data_y, xm,  alpha = 0.05, prior_p = 2, prior_st = 1,  mlike = c("Chib", "Geweke", "LaplaceMetropolis", "all"))

Arguments

Log of square bandwidth

inicost

Initial cost value

mutsizp

Step size of random-walk Metropolis algorithm. At each iteration, the value of mutsizp will alter depending on acceprance or rejection. As the number of iteration increases, the final acceptance probability will converge to the optimal rate, which is 0.234 for multiple parameters

warm

Burn-in period

Number of MCMC iteration

prob

Optimal acceptance rate of random-walk Metropolis algorithm

num_batch

Number of batch samples

step

Recording value at a specific step, in order to achieve iid samples and eliminate correlation

data_x

Regressors

data_y

Response variable

Values of true regression function

alpha

Quantile of the critical value in calculating Geweke's log marginal likelihood

prior_p

Hyperparameter of inverse-gamma prior

prior_st

Hyperparameter of inverse-gamma prior

mlike

Method for calculating log marginal likelihood

Value

sum_h: Estimated parameters in an order of the bandwidths of the regressors, the variance parameter of the error density and cost value
h2: Estimated parameters in an order of the square bandwidths of the regressors, the square variance parameter of the error density
sif: Simulation inefficient factor. The small it is, the better the method is in general
mutsizp: Step size of random-walk Metropolis algroithm for each iteration of MCMCrecord
cpost: Simulation output of square bandwidths and square normal error variance obtained from MCMC
accept: Acceptance rate of random-walk Metropolis algorithm
marginalike: Log marginal likelihood
R2: R square
MSE: Mean square error

Details

Akin to the burn-in period, it determines the retained bandwidths for the regressors and the variance of the error density for finite samples. It also calculates the simulation inefficient factor (SIF) value, R square, mean square error, and log marginal density by Chib (1995), Geweke (1999) and the Laplace Metropolis method describe in Raftery (1996).

References

H. L. Shang (2013) Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density, Computational Statistics and Data Analysis, 67, 185-198.

X. Zhang and R. D. Brooks and M. L. King (2009) A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation, Journal of Econometrics, 153, 21-32.

S. Chib and I. Jeliazkov (2001) Marginal likelihood from the Metropolis-Hastings output, Journal of the American Statistical Association, 96, 453, 270-281.

S. Chib (1995) Marginal likelihood from the Gibbs output, Journal of the American Statistical Association, 90, 432, 1313-1321.

M. A. Newton and A. E. Raftery (1994) Approximate Bayesian inference by the weighted likelihood bootstrap (with discussion), Journal of the Royal Statistical Society, 56, 3-48.

J. Geweke (1998) Using simulation methods for Bayesian econometric models: inference, development, and communication, Econometric Reviews, 18(1), 1-73.

A. E. Raftery (1996) Hypothesis testing and model selection, in Markov Chain Monte Carlo In Practice by W. R. Gilks, S. Richardson and D. J. Spiegelhalter, Chapman and Hall, London.