Split multi-chain autocorrelation effective sample size

Inputs: $M\ge 2$ scalar chains $y_{m,1},\dots ,y_{m,L}$ of equal length $L\ge 4$. This estimator targets the functional-specific quantity in Functional-specific effective sample size from finite arrays; it does not test stationarity or a central limit theorem.

1. Split every input chain into its first and last $N=\lfloor L\mathrm{/}2\rfloor$ observations. If $L$ is odd, discard its middle observation. Denote the resulting $M^{\mathrm{\prime }}=2M$ chains by $z_{j,1},\dots ,z_{j,N}$ and set $S^{\mathrm{\prime }}=M^{\mathrm{\prime }}N$.
2. Compute their means ${\bar{z}}_j$, sample variances $s_j^2=(N-1{)}^{-1}{\sum }_{i=1}^N(z_{j,i}-{\bar{z}}_j{)}^2$, their average $W=M^{\mathrm{\prime }-1}{\sum }_j{s}_j^2$, grand mean ${\bar{z}}_{\cdot }=M^{\mathrm{\prime }-1}{\sum }_j{\bar{z}}_j$, between-chain term $B=N(M^{\mathrm{\prime }}-1{)}^{-1}{\sum }_j({\bar{z}}_j-{\bar{z}}_{\cdot }{)}^2$, and ${\widehat{V}}^{+}=(N-1)W\mathrm{/}N+B\mathrm{/}N$. If any $s_j^2=0$ or ${\widehat{V}}^{+}=0$, return not estimable.
3. For $0\le t<N$, compute the biased autocovariances ${\widehat{\gamma }}_{t,j}=N^{-1}{\sum }_{i=1}^{N-t}(z_{j,i}-{\bar{z}}_j)(z_{j,i+t}-{\bar{z}}_j)$, the within-chain autocorrelations ${\widehat{r}}_{t,j}={\widehat{\gamma }}_{t,j}\mathrm{/}{\widehat{\gamma }}_{0,j}$, and the between/within adjusted combined autocorrelation ${\widehat{\rho }}_t=1-\frac{W-M^{\mathrm{\prime }-1}{\sum }_j{s}_j^2{\widehat{r}}_{t,j}}{{\widehat{V}}^{+}}\mathrm{.}$
4. For $0\le k\le \lfloor (N-2)\mathrm{/}2\rfloor$, form $P_k={\widehat{\rho }}_{2k}+{\widehat{\rho }}_{2k+1}$. If $P_0\le 0$, return not estimable. Let $K$ be the largest $k$ for which $P_1,\dots ,P_k$ are all positive; $K=0$ if $P_1\le 0$ or no $P_1$ is available. Put ${\tilde{P}}_k={\min }_{0\le j\le k}{P}_j$ and ${\widehat{\tau }}_{\mathrm{o}\mathrm{d}\mathrm{d}}=-1+2{\sum }_{k=0}^K{\tilde{P}}_k$. If the next even lag $2K+2<N$ is available, put ${\widehat{\tau }}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}={\widehat{\tau }}_{\mathrm{o}\mathrm{d}\mathrm{d}}+2{\widehat{\rho }}_{2K+2}$; otherwise put ${\widehat{\tau }}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}={\widehat{\tau }}_{\mathrm{o}\mathrm{d}\mathrm{d}}$. Set $\widehat{\tau }=({\widehat{\tau }}_{\mathrm{o}\mathrm{d}\mathrm{d}}+{\widehat{\tau }}_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}})\mathrm{/}2$.
5. If $\widehat{\tau }\le 0$, return not estimable. Otherwise return $\stackrel{}{\mathrm{ESS}}=\min \{\frac{S^{\mathrm{\prime }}}{\widehat{\tau }},S^{\mathrm{\prime }}{\log }_{10}{S}^{\mathrm{\prime }}\}\mathrm{.}$ The second term is the paper's finite software cap for very short, strongly antithetic chains.

Negative serial correlation can make the estimate exceed the $S^{\mathrm{\prime }}$ retained split draws, subject to that software cap. The output is an estimator, not a guarantee about the target law.