The use of a genetic relationship matrix biases the best linear unbiased prediction

The best linear unbiased prediction (BLUP), derived from the linear mixed model (LMM), has been popularly used to estimate animal and plant breeding values (BVs) for a few decades. Conventional BLUP has a constraint that BVs are estimated from the assumed covariance among unknown BVs, namely conventional BLUP assumes that its covariance matrix is a λK, in which λ is a coefficient that leads to the minimum mean square error of the LMM, and K is a genetic relationship matrix. The uncertainty regarding the use of λK in conventional BLUP was recognized by past studies, but it has not been sufficiently investigated. This study was motivated to answer the following question: is it indeed reasonable to use a λK in conventional BLUP? The mathematical investigation concluded: (i) the use of a λK in conventional BLUP biases the estimated BVs, and (ii) the objective BLUP, mathematically derived from the LMM, has the same representation as the least squares.