Skip to main content
. Author manuscript; available in PMC: 2018 Mar 30.
Published in final edited form as: J Am Stat Assoc. 2017 Mar 30;112(518):613–622. doi: 10.1080/01621459.2016.1149070

Table 2.

Estimation of conditional quantile functions

t = 0.2 t = 0.5 t = 0.8
sample size: 200
Orig B 7 −23 24 1 −18 10 −4 −20 −7
SD 472 466 488 364 359 388 333 344 356
SE 508 496 510 378 374 396 337 347 368
CI 93.9 94.0 94.9 93.6 94.2 93.9 93.8 93.7 94.7
AI-𝕏
 MAE 0.84 0.84 0.84 B 13 −20 41 6 −14 17 9 −7 −11
RMISE 0.95 0.95 0.95 SD 449 439 462 364 354 385 316 328 344
BRK 0.10 CI 94.5 95.7 95.6 93.7 94.3 94.1 95.1 94.2 95.1
AI-{x} MAE 0.91 0.91 0.91 B −45 −74 −19 2 −18 14 49 33 40
RMISE 0.98 0.98 0.98 SD 462 452 478 364 356 386 325 338 352
BRK 0.41 0.41 0.47 CI 94.9 94.9 95.5 93.6 94.4 93.9 93.9 93.7 94.7
RA MAE 0.91 0.91 0.93 B 10 −14 30 0 −16 12 −4 −19 −9
RMISE 0.97 0.97 0.98 SD 463 455 480 360 352 385 321 334 352
BRK 1.00 1.00 1.00 CI 94.5 94.7 95.2 94.0 94.5 94.5 94.7 94.4 94.8
sample size: 400
Orig B 10 −3 5 4 3 12 −2 −3 −4
SD 335 323 342 252 250 268 236 240 255
SE 348 342 359 263 261 279 235 244 256
CI 94.5 95.1 95.3 95.0 94.4 94.6 92.8 93.7 93.4
AI-𝕏
 MAE 0.88 0.88 0.88 B 15 0 14 6 4 13 5 −1 −8
RMISE 0.97 0.97 0.97 SD 326 314 334 252 250 267 231 237 250
BRK 0.10 CI 94.9 95.9 95.8 94.9 94.7 94.8 92.8 94.4 94.1
AI-{x} MAE 0.93 0.94 0.94 B −14 −31 −17 5 3 12 26 22 21
RMISE 0.99 0.99 0.99 SD 332 318 342 252 250 267 233 237 254
BRK 0.40 0.41 0.46 CI 94.9 95.7 95.7 94.9 94.4 94.9 92.3 94.2 93.5
RA MAE 0.94 0.94 0.95 B 10 −3 6 4 3 12 −2 −4 −4
RMISE 0.99 0.99 0.99 SD 333 319 340 251 248 268 233 238 253
BRK 1.00 1.00 1.00 CI 94.7 95.6 95.4 94.9 94.3 94.7 93.3 93.9 93.5

Orig: the original estimator; AI-𝕏S: the adaptive interpolation estimator with 𝕏S and τ ≏ 0.5; RA: the rearranged estimator.

MAE: maximum absolute error over [0.1, 0.9]; RMISE: root mean integrated squared error over [0.1, 0.9]; BRK: number of knots or breakpoints. All are average measures, reported as relative to the original estimator. B: Empirical bias (×1000); SD: Empirical standard deviation (×1000); SE: Average standard error (×1000); CI: Empirical coverage (%) of 95% Wald confidence interval.

Three columns as grouped correspond to the estimated conditional quantile functions with non-constant covariate vectors (1, 0)⊤, (0, 1)⊤, and (1, 1)⊤.