mcmcse (version 1.3-2)

mcse.q: Compute Monte Carlo standard errors for quantiles.

Description

Compute Monte Carlo standard errors for quantiles.

Usage

mcse.q(x, q, size = "sqroot", g = NULL,
    method = c("bm", "obm", "sub"), warn = FALSE)

Arguments

x

a vector of values from a Markov chain.

q

the quantile of interest.

size

the batch size. The default value is “sqroot”, which uses the square root of the sample size. A numeric value may be provided if “sqroot” is not satisfactory.

g

a function such that the \(q\)th quantile of the univariate distribution function of \(g(x)\) is the quantity of interest. The default is NULL, which causes the identity function to be used.

method

the method used to compute the standard error. This is one of “bm” (batch means, the default), “obm” (overlapping batch means), or “sub” (subsampling bootstrap).

warn

a logical value indicating whether the function should issue a warning if the sample size is too small (less than 1,000).

Value

mcse.q returns a list with two elements:

est

an estimate of the \(q\)th quantile of the univariate distribution function of \(g(x)\).

se

the Monte Carlo standard error.

References

Flegal, J. M. (2012) Applicability of subsampling bootstrap methods in Markov chain Monte Carlo. In Wozniakowski, H. and Plaskota, L., editors, Monte Carlo and Quasi-Monte Carlo Methods 2010 (to appear). Springer-Verlag.

Flegal, J. M. and Jones, G. L. (2010) Batch means and spectral variance estimators in Markov chain Monte Carlo. The Annals of Statistics, 38, 1034--1070.

Flegal, J. M. and Jones, G. L. (2011) Implementing Markov chain Monte Carlo: Estimating with confidence. In Brooks, S., Gelman, A., Jones, G. L., and Meng, X., editors, Handbook of Markov Chain Monte Carlo, pages 175--197. Chapman & Hall/CRC Press.

Flegal, J. M., Jones, G. L., and Neath, R. (2012) Markov chain Monte Carlo estimation of quantiles. University of California, Riverside, Technical Report.

Jones, G. L., Haran, M., Caffo, B. S. and Neath, R. (2006) Fixed-width output analysis for Markov chain Monte Carlo. Journal of the American Statistical Association, 101, 1537--1547.

See Also

mcse.q.mat, which applies mcse.q to each column of a matrix or data frame.

mcse and mcse.mat, which compute standard errors for expectations.

Examples

# NOT RUN {
# Create 10,000 iterations of an AR(1) Markov chain with rho = 0.9.

n = 10000
x = double(n)
x[1] = 2
for (i in 1:(n - 1))
    x[i + 1] = 0.9 * x[i] + rnorm(1)

# Estimate the mean, 0.1 quantile, and 0.9 quantile with MCSEs using batch means.

mcse(x)
mcse.q(x, 0.1)
mcse.q(x, 0.9)

# Estimate the mean, 0.1 quantile, and 0.9 quantile with MCSEs using overlapping batch means.

mcse(x, method = "obm")
mcse.q(x, 0.1, method = "obm")
mcse.q(x, 0.9, method = "obm")

# Estimate E(x^2) with MCSE using spectral methods.

g = function(x) { x^2 }
mcse(x, g = g, method = "tukey")
# }