Table 1.

Regression coefficient estimation under a quantile regression model

				t = 0.2			t = 0.5			t = 0.8
sample size: 200
Orig			B	−40	47	17	−27	28	9	−17	13	−3
			SD	466	631	611	343	470	480	308	440	435
			SE	485	659	657	347	491	493	314	448	451
			CI	93.4	95.1	95.9	94.3	94.7	94.8	94.3	94.6	95.2
IO	MAE	0.99	B	−30	49	19	−22	31	11	−11	15	1
	RMISE	1.00	SD	466	630	609	341	467	479	307	438	436
SW	MAE	1.02	B	−181	26	−2	−120	−5	−14	−62	−41	−63
	RMISE	1.05	SD	516	668	653	353	480	482	314	337	439
$\text{AI-}\widehat{𝕏}$	MAE	0.87	B	−45	60	26	−26	32	12	7	−1	−12
	RMISE	0.96	SD	440	598	578	339	464	478	299	422	421
	BRK	0.11	CI	94.9	95.8	97.3	94.5	94.8	94.7	95.0	94.9	95.2
$\text{AI-}𝕏$	MAE	0.86	B	−48	60	28	−26	32	12	13	−4	−20
	RMISE	0.95	SD	438	593	573	339	464	478	298	422	416
	BRK	0.10	CI	95.1	95.7	97.1	94.4	94.8	94.7	95.6	94.9	95.9
$\text{AI-}\widehat{𝕏}$	MAE	0.87	B	−46	61	26	−37	41	31	−14	15	0
τ ≏ 0.8	RMISE	0.96	SD	440	598	578	333	457	465	306	437	435
	BRK	0.11	CI	94.9	95.8	97.3	95.3	94.7	95.2	94.8	95.0	94.8
sample size: 400
Orig			B	2	8	−5	−4	8	7	−1	−1	−2
			SD	330	429	446	241	338	320	215	314	304
			SE	338	456	457	242	345	343	219	314	313
			CI	94.2	95.0	94.2	92.9	94.5	96.1	94.2	93.9	94.1
IO	MAE	0.99	B	8	8	−5	−1	9	7	2	1	−1
	RMISE	1.00	SD	330	429	445	240	338	320	214	314	302
SW	MAE	0.99	B	−68	−7	−9	−51	−9	−3	−25	−30	−43
	RMISE	1.03	SD	346	437	458	246	344	320	217	314	310
$\text{AI-}\widehat{𝕏}$	MAE	0.90	B	1	12	0	−3	9	7	11	−6	−12
	RMISE	0.98	SD	324	420	434	240	339	320	210	307	299
	BRK	0.10	CI	94.8	95.1	94.4	92.8	94.5	96.0	94.3	94.7	95.0
$\text{AI-}𝕏$	MAE	0.90	B	1	14	−1	−3	9	7	12	−7	−13
	RMISE	0.97	SD	322	419	432	240	339	319	210	307	298
	BRK	0.10	CI	94.9	95.2	94.5	92.8	94.5	96.1	94.3	95.1	95.2
$\text{AI-}\widehat{𝕏}$	MAE	0.91	B	1	12	0	−11	19	18	0	1	−1
τ ≏ 0.8	RMISE	0.97	SD	324	420	434	239	335	316	213	313	301
	BRK	0.10	CI	94.8	95.1	94.4	93.3	94.7	96.0	94.2	93.6	94.6

Orig: the original estimator; IO: the interpolated original estimator; SW: the stairwise monotonicity-respecting estimator; $\text{AI-}{𝕏}_S$: the proposed adaptive interpolation estimator with ${𝕏}_S$ and, unless otherwise specified, τ ≏ 0.5.

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 under each t value correspond to the estimated intercept and two slopes.