Fig. 3

Boxplots of estimated heritability (100 replicates) under different problem sizes using the proposed strategy. a s=10, m=1000, n=100000; b s=100, m=1000, n=100000; c s=1000, m=1000, n=100000; d s=10000, m=1000, n=100000. Here “fixed” refers to the estimator \(\hat {h}_{\text {fixed}}\) with \(\hat {M}=\{1,2,\cdots,n\}\), “fixed_SpaR” refers to the estimator \(\hat {h}_{\text {fixed}}\) with \(\hat {M}\) given by our sparse regularization step, and “fixed_ora” refers to the oracle estimator \(\hat {h}_{\text {fixed}}\) with \(\hat {M}={M_{0}}\). “rand.” refers to the estimator \(\hat {h}_{\text {rand.}}\) with \(\hat {M}=\{1,2,\cdots,n\}\), “rand._SpaR” refers to the estimator \(\hat {h}_{\text {rand.}}\) with \(\hat {M}\) given by our sparse regularization step, and “rand._ora” refers to the oracle estimator \(\hat {h}_{\text {rand.}}\) with \(\hat {M}={M_{0}}\). The approximation of the true heritability \(\tilde {h}^{*}\) is denoted as “approx.”. The whiskers of each boxplot are the first and third quartile