mcmcrecord_gaussian {bbemkr} R Documentation

## 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

 `x` 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 `M` 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 `xm` 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

### 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).

### 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

### Note

Time-consuming for large iterations.

Han Lin Shang

### 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.

`logdensity_gaussian`, `logpriors_gaussian`, `loglikelihood_gaussian`, `warmup_gaussian`