Computational Statistics in Data Science

Typically, MCMC algorithms exhibit positive correlation implying that is larger . This naturally implies that MCMC simulations require more samples than IID simulations. Using Theorem 1to assess the simulation reliability requires estimation of and , which we describe in Section 4.

Let

An asymptotic distribution for sample quantiles is available under both IID Monte Carlo and MCMC.

Let be absolutely continuous, twice differentiable with density , and let be bounded within some neighborhood of .

1 IID. Let , then

2 MCMC. [11] If the Markov chain is polynomially ergodic of order and , then

The density value, , can be estimated using a Gaussian kernel density estimator. In addition, is replaced with , the univariate version of for . We present methods of estimating in Section 4.

For many estimators, a delta method argument can yield a limiting normal distribution. For example, a CLT for and a delta method argument yields an elementwise asymptotic distribution of . Let denote the th element of . If and denote the components of and , respectively, then the th diagonal of is

We obtain the asymptotic distribution of . A similar argument can be made for the off‐diagonals of . Under the conditions of Theorem 1,

where is

Under IID sampling, the infinite sum above reduces to

Applying the delta method for function , we obtain

Suppose that is an estimate of the limiting Monte Carlo variance–covariance matrix, for IID sampling, and for MCMC sampling. Let be the ‐quantile of a distribution. The CLT yields a large‐sample confidence region around as