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Table 3 Expectation of elements involved in precision formulae when a uniform \( ( f ( p ) = 1 ) \) or a U shaped distribution of allelic frequencies is assumed \( \left( {f( p) = {k \mathord{\left/ {\vphantom {k {2p\left( {1\text{ - }p} \right)}}} \right. \kern-0pt} {2p\left( {1-p} \right)}}} \right) \)

From: Approximated prediction of genomic selection accuracy when reference and candidate populations are related

Element Expectation
Uniform U shaped
E[σ 2 m ] 1/3 k
E[σ 4 m ] 2/15 k/3
E[ρ m ] \( 1 - 2\frac{h}{\omega }\theta \) \( \frac{k}{\omega }\theta \)
E[ρ 2 m ] \( \left( {\frac{4\theta }{\omega } + \frac{2}{h}} \right)\left( {\frac{1 + h}{1 + 4h}} \right)^{2} - \frac{4\theta h}{\omega } \) \( \frac{k}{{\omega^{2} }}\left[ {\theta \left( {\omega - \frac{2h}{\omega }} \right) - 1} \right] \)
E[ρ 2 m /σ 2 m ] \( \frac{1}{{\omega^{2} }}\left[ {\theta \left( {\omega - \frac{2h}{\omega }} \right) - 1} \right] \) \( \frac{k}{{2\omega^{3} }}\left\{ {2\theta + \frac{\omega }{h}} \right\} \)
  1. A large effective size \( N_{e} \) of the population was assumed to make 1/N e negligible \( \theta = \log \left( {\left| {\frac{1 + \omega }{1 - \omega }} \right|} \right),\omega = \sqrt {1 + 4h} \) \( h = {{{{\uplambda }}_{{{\upbeta }}} } \mathord{\left/ {\vphantom {{{{\uplambda }}_{{{\upbeta }}} } {2n_{r} }}} \right. \kern-0pt} {2n_{r} }}, {{\uplambda }}_{{{\upbeta }}} = {{\sigma_{e}^{2} } \mathord{\left/ {\vphantom {{\sigma_{e}^{2} } {\sigma_{{{\upbeta }}}^{2} }}} \right. \kern-0pt} {\sigma_{{{\upbeta }}}^{2} }} \)