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 $$ \lambda K $$ λ K , in which $$ \lambda $$ λ is a coefficient that leads to the minimum mean square error of the LMM, and $$ K $$ K is a genetic relationship matrix. The uncertainty regarding the use of $$ \lambda K $$ λ K 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 $$ \lambda K $$ λ K in conventional BLUP? The mathematical investigation concluded: (i) the use of a $$ \lambda K $$ λ K 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.

中文翻译：

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使用遗传关系矩阵偏向最佳线性无偏预测

几十年来，源自线性混合模型 (LMM) 的最佳线性无偏预测 (BLUP) 已被广泛用于估计动植物育种值 (BV)。传统 BLUP 有一个约束，即 BVs 是从未知 BV 之间的假设协方差估计的，即传统 BLUP 假设它的协方差矩阵是一个 $$ \lambda K $$ λ K ，其中 $$ \lambda $$ λ 是一个系数导致 LMM 的最小均方误差，$$K $$K 是遗传关系矩阵。过去的研究已经认识到在传统 BLUP 中使用 $$ \lambda K $$ λ K 的不确定性，但尚未得到充分研究。本研究旨在回答以下问题：在传统 BLUP 中使用 $$ \lambda K $$ λ K 确实合理吗？数学调查得出结论：