Table 1:
Simulation I results: the means of PEs and MSEs for regularized low-rank (RLR), OLS, Lasso, fused Lasso (Fused) and tensor envelope (Envelope) estimates and their associated standard errors in the parentheses. For each case, 100 simulated data sets are used.
| (n, ) | Method | MSE(B1) | MSE(B2) | MSE(B3) | MSE(B4) | PE |
|---|---|---|---|---|---|---|
| (100, 1) | RLR | 11.67(0.21) | 9.96(0.22) | 43.21(0.43) | 44.88(0.52) | 1.03(0.0002) |
| OLS | 58.08(0.83) | 72.38(1.08) | 71.66(1.01) | 58.00(0.92) | 1.05(0.0004) | |
| Lasso | 42.96(0.79) | 53.79(1.08) | 53.21(0.98) | 44.87(0.75) | 1.04(0.0004) | |
| Fused | 11.85(0.20) | 11.25(0.22) | 13.61(0.23) | 17.87(0.26) | 1.02(0.0002) | |
| Envelope | 21.20(0.34) | 24.95(0.36) | 51.14(0.49) | 55.62(0.67) | 1.04(0.0002) | |
| (200, 1) | RLR | 7.27(0.09) | 6.73(0.10) | 23.77(0.20) | 23.08(0.23) | 1.02(0.0001) |
| OLS | 28.61(0.30) | 34.88(0.36) | 34.85(0.38) | 28.11(0.32) | 1.03(0.0001) | |
| Lasso | 19.29(0.38) | 23.86(0.43) | 23.73(0.41) | 20.30(0.29) | 1.02(0.0002) | |
| Fused | 5.93(0.09) | 5.62(0.08) | 6.59(0.10) | 8.63(0.10) | 1.01(0.0001) | |
| Envelope | 11.21(0.16) | 13.13(0.14) | 38.40(0.30) | 43.33(0.35) | 1.03(0.0001) | |
| (500, 1) | RLR | 3.46(0.03) | 3.53(0.03) | 10.54(0.06) | 9.75(0.06) | 1.006(0.00003) |
| OLS | 11.01(0.08) | 13.87(0.10) | 13.88(0.09) | 11.04(0.07) | 1.009(0.00003) | |
| Lasso | 5.93(0.17) | 7.89(0.17) | 7.78(0.18) | 6.70(0.13) | 1.007(0.00009) | |
| Fused | 2.36(0.03) | 2.37(0.02) | 2.73(0.03) | 3.29(0.03) | 1.004(0.00002) | |
| Envelope | 5.44(0.07) | 6.60(0.07) | 31.72(0.21) | 38.29(0.30) | 1.02(0.0001) | |
| (100, 25) | RLR | 121.61(1.69) | 119.58(2.37) | 227.58(2.01) | 263.90(2.77) | 25.37(0.0027) |
| OLS | 1451.95(20.64) | 1809.40(27.01) | 1791.53(25.36) | 1450.10(23.10) | 26.27(0.0099) | |
| Lasso | 1360.23(19.03) | 1683.95(24.74) | 1669.88(23.97) | 1367.71(21.34) | 26.22(0.0093) | |
| Fused | 238.87(3.04) | 254.78(4.36) | 290.50(4.47) | 283.07(3.89) | 25.42(0.0031) | |
| Envelope | 175.09(1.57) | 139.95(2.71) | 259.48(2.18) | 286.98(1.81) | 25.39(0.0023) | |
| (200, 25) | RLR | 79.44(1.01) | 71.27(1.25) | 171.12(1.21) | 201.43(1.63) | 25.26(0.0013) |
| OLS | 715.28(7.49) | 872.02(8.93) | 871.33(9.41) | 702.70(8.00) | 25.66(0.0037) | |
| Lasso | 657.54(7.19) | 798.43(8.58) | 798.01(9.04) | 652.91(7.14) | 25.62(0.0035) | |
| Fused | 156.75(2.00) | 162.27(1.95) | 174.79(2.34) | 175.61(2.20) | 25.27(0.0016) | |
| Envelope | 151.68(1.55) | 105.18(1.61) | 202.96(2.78) | 230.24(2.63) | 25.29(0.0016) | |
| (500, 25) | RLR | 42.17(0.50) | 39.70(0.59) | 110.16(0.79) | 125.08(0.75) | 25.10(0.0005) |
| OLS | 275.31(2.05) | 346.69(2.54) | 346.93(2.36) | 276.05(1.83) | 25.24(0.0008) | |
| Lasso | 238.43(2.33) | 299.22(2.99) | 298.81(2.90) | 243.44(1.95) | 25.22(0.0011) | |
| Fused | 80.31(0.84) | 89.14(0.79) | 93.22(0.90) | 89.8(0.83) | 25.10(0.0006) | |
| Envelope | 95.49(0.99) | 75.41(0.99) | 142.24(1.51) | 171.34(1.31) | 25.14(0.0008) |