Power-expected-posterior prior Bayes factor consistency for nested linear models with increasing dimensions

Figures & data

Table 1. Consistency of $B{F}_{p\ell }^{PEP}$ when model $M_{\ell }$ has dimension $\mathrm{d}\mathrm{i}\mathrm{m}\left(M_{\ell }\right)=i=O\left(1\right)$ and ${\delta }_{\ell }\in \{n,n-p\}$.

	$M_{\ell }$ is correct	$M_p$ is correct
$p=O\left(1\right)$	Consistent	Consistent
$p=O\left(n\right)$	Consistent	Consistent if $\underset{n\to +\mathrm{\infty }}{lim}{ϵ}_{p\ell }\ge {ϵ}_p^2\left(r\right)$ for large r

Table 2. Consistency of $B{F}_{p\ell }^{PEP}$ when model $M_{\ell }$ has dimension $\mathrm{d}\mathrm{i}\mathrm{m}\left(M_{\ell }\right)=i=O\left(1\right)$ and ${\delta }_{\ell }=p$.

	$M_{\ell }$ is correct	$M_p$ is correct
$p=O\left(1\right)$	Inconsistent	Consistent
$p=O\left(n\right)$	Consistent	Consistent if $\underset{n\to +\mathrm{\infty }}{lim}{ϵ}_{p\ell }\ge {ϵ}_p^1\left(r\right)$ for large r

Table 3. Consistency of $B{F}_{p\ell }^{PEP}$ when model $M_{\ell }$ has dimension $\mathrm{d}\mathrm{i}\mathrm{m}\left(M_{\ell }\right)=i=O\left(1\right)$ and ${\delta }_{\ell }=\delta >0$.

	$M_{\ell }$ is correct	$M_p$ is correct
$p=O\left(1\right)$	Consistent if δ is large	Consistent
$p=O\left(n\right)$	Consistent if $\delta >{\text{β}}_1\left(r\right)$	Consistent if $\delta <{\text{β}}_2\left(r\right)$ or if $\delta ={\text{β}}_2\left(r\right)$ and large