An adaptive lack of fit test for big data

Figures & data

Table 1. Empirical size, normal errors.

		DM	GL			ZH
		K	c
N	α	40 (100)	1	1.5	2	h1	h2	h3
	1.00	1.28	6.19	1.64	0.79	0.99	0.81	0.81
20,000 (4800, 7400)	5.00	6.08	9.77	5.30	4.55	4.68	4.82	4.49
	10.00	11.10	14.31	10.14	9.44	9.72	9.88	9.14
	1.00	1.23	6.22	1.52	0.72	0.97	0.99	0.63
40,000 (6200, 9500)	5.00	5.57	9.77	5.08	4.37	4.80	5.04	4.55
	10.00	10.69	14.37	9.96	9.35	10.11	9.91	9.33

Percentages of rejection at 1%, 5%, 10% nominal levels.

Table 2. Empirical size, Student errors.

		DM	GL			ZH
		K	c
N	α	40 (100)	1	1.5	2	h1	h2	h3
	1.00	1.54	6.55	1.68	0.83	0.99	1.04	0.97
20,000 (4800, 7400)	5.00	5.65	9.81	4.87	4.07	5.09	5.27	4.45
	10.00	10.95	14.66	9.97	9.27	10.36	10.38	9.35
	1.00	1.00	6.65	1.65	0.83	0.96	0.89	0.85
40,000 (6200, 9500)	5.00	5.01	10.10	5.30	4.59	4.75	4.61	4.17
	10.00	9.54	14.93	10.38	9.72	9.61	9.06	9.11

Percentages of rejection at 1%, 5%, 10% nominal levels.

Figure 1. Comparison of power curves under model 1 of two different error terms at significant level α = 0.05. We set (N, K) = (20,000, 40), (40,000, 100) for DM test. N = 4800, 6200 for GL test. N = 7400, 9500 for ZH test.

Figure 2. Comparison of power curves under low frequency of model 2 based on two different error terms at significant level α = 0.05.

Figure 3. Comparison of power curves under high frequency of model 2 based on two different error terms at significant level α = 0.05.

Table 3. Empirical power of DM test and ZH test based on the whole dataset when the error distribution is normal.

	DM			ZH
	K			h
b	20	40	50	h1	h2	h3
0.1	46.1	32.8	26.8	52.1	95.1	100
0.15	99.4	94.0	89.3	98.9	100	100
0.2	100	100	100	100	100	100
0.25	100	100	100	100	100	100
0.3	100	100	100	100	100	100

Percentages of rejection at 5% nominal levels.

Table 4. Empirical power of DM test and ZH test based on the whole dataset when the error distribution is Student.

	DM			ZH
	K			h
b	20	40	50	h1	h2	h3
0.1	20.7	16.1	13.1	22.6	59.9	96.1
0.15	72.9	55.0	47.0	70.4	100	100
0.2	99.5	96.1	93.7	100	100	100
0.25	100	100	99.9	100	100	100
0.3	100	100	100	100	100	100

Percentages of rejection at 5% nominal levels.