Table 1.
Theoretical ARE [see (51)] of β̂WQAE(ω∗) compared to β̂LS, Zou and Yuan (2008)’s CQR estimator β̂CQR, and β̂ (τ), using 9 quantiles τj = j/10, j = 1, …, 9. Mixture 1: 0.5N(0,1)+0.5N(0,0.56); Mixture 2: 0.5N(−2,1)+0.5N(2,1). For Student-t1, t2, LS is not applicable. [Numbers ≥ 1 indicate better performance of β̂WQAE(ω∗)].
| ε | β̂(τ) with τ = | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| distribution | β̂LS | β̂CQR | τ1 | τ2 | τ3 | τ4 | τ5 | τ6 | τ7 | τ8 | τ9 |
| Student-t1 | NA | 1.58 | 47.12 | 6.40 | 2.34 | 1.40 | 1.19 | 1.40 | 2.34 | 6.40 | 47.12 |
| Student-t2 | NA | 1.12 | 9.01 | 2.85 | 1.65 | 1.27 | 1.17 | 1.27 | 1.65 | 2.85 | 9.01 |
| N(0,1) | 0.96 | 1.03 | 2.80 | 1.96 | 1.67 | 1.54 | 1.51 | 1.54 | 1.67 | 1.96 | 2.80 |
| Mixture 1 | 10.17 | 2.43 | 91.95 | 32.64 | 3.26 | 1.80 | 1.55 | 1.80 | 3.26 | 32.64 | 91.95 |
| Mixture 2 | 3.28 | 2.80 | 3.01 | 2.81 | 3.68 | 7.84 | 56.24 | 7.84 | 3.68 | 2.81 | 3.01 |
| Laplace | 2.00 | 1.32 | 9.00 | 4.00 | 2.33 | 1.50 | 1.00 | 1.50 | 2.33 | 4.00 | 9.00 |
| Gamma(1) | 9.00 | 3.67 | 1.00 | 2.25 | 3.86 | 6.00 | 9.00 | 13.50 | 21.00 | 36.00 | 81.00 |
| Gamma(2) | 2.44 | 1.73 | 1.12 | 1.49 | 1.91 | 2.42 | 3.10 | 4.08 | 5.65 | 8.68 | 17.34 |
| Gamma(3) | 1.71 | 1.41 | 1.26 | 1.42 | 1.64 | 1.94 | 2.35 | 2.94 | 3.88 | 5.68 | 10.75 |
| Beta(1,1) | 1.67 | 2.04 | 1.80 | 3.20 | 4.20 | 4.80 | 5.00 | 4.80 | 4.20 | 3.20 | 1.80 |
| Beta(1,2) | 2.35 | 2.33 | 1.05 | 2.11 | 3.16 | 4.22 | 5.27 | 6.33 | 7.38 | 8.44 | 9.49 |
| Beta(1,3) | 3.31 | 2.71 | 1.02 | 2.12 | 3.32 | 4.66 | 6.19 | 8.00 | 10.27 | 13.33 | 19.04 |