Editing Markov chain Monte Carlo (section)

==== Stationary Test ====

Let <math>\{X_t\}_{t=1}^n</math> be the output of an MCMC simulation for a scalar function <math>g(X_t)</math>, and <math>g_1,g_2,\dots,g_n</math> the evaluations of the function <math>g</math> over the chain. Define the standardized cumulative sum process:

:<math>
B_n(t) = \dfrac{\sum_{i=1}^{\text{round}(nt)} g_i - \text{round}(nt) \bar{g}_n}{\sqrt{n\hat{S}(0)}},\;\;\; t\in[0,1]
</math> 
where <math>\bar{g}_n = \frac{1}{n}\sum_{i=1}^n g_i</math> is the sample mean and <math>\hat{S}(0)</math> is an estimate of the spectral density at frequency zero.

Under the null hypothesis of convergence, the process <math>B_n(t)</math> converges in distribution to a [[Brownian bridge]]. The following [[Cramér–von Mises criterion|Cramér-von Mises statistic]] is used to test for stationarity:

:<math>
C_n = \int_0^1 B_n(t)^2 dt.
</math>

This statistic is compared against known critical values from the Brownian bridge distribution. If the null hypothesis is rejected, the first 10% of the samples are discarded and the test can be repeated on the remaining chain until either stationarity is accepted or 50% of the chain is discarded.