Correction of constants used when computing the Qn statistic, a robust scale estimator with high breakdown

In the following table you will find the list of dn constants as defined in Croux and Rousseeuw (1992). The initial values given by these authors suffered from a typographical error in equation 6; the constant 2.2219 should have been 2.2191. As a reminder, the value of the constant corresponds to:

where f -1 is the inverse standard normal distribution function. This error was reproduced in Rousseeuw and Croux (1993) and later in many subsequent articles and documents, including the ISO 13528 standard, up to version 2015. Furthermore, the initial dn constants had been estimated using a 10000 simulations. We have identified that given the variance of the estimators of the constants, the number of simulations was insufficient to obtain a good level of confidence in all digits following the first decimal place.

With respect to the improvements that we are bringing to XLSTAT:

• For n=2 the constant can be computed analytically. The digits are certain.

• The dn values for n ranging from 3 to 12 have been estimated using 109 simulations.

• The values dn values for n ranging from 13 to 40 have been estimated using 108 simulations.

• The values of dn for n ranging from 41 to 100 have been estimated using 107 simulations.

The following table lists for n in [3, 100] the estimated value of dn as well as the estimator of the variance of dn, and the confidence we can have on the fourth decimal of dn.

n	dn	stdev (dn)	Conf (%)
2	0.3994		100
3	0.9936	2.6627 x 10-5	88
4	0.5122	1.2789 x 10-4	30
5	0.8397	2.8902 x 10-5	77
6	0.6055	5.9362 x 10-5	50
7	0.8569	2.0620 x 10-5	89
8	0.6674	3.6643 x 10-5	56
9	0.8706	1.6333 x 10-5	59
10	0.7198	2.5437 x 10-5	60
11	0.8889	1.3343 x 10-5	98
12	0.7574	1.9510 x 10-5	84
13	0.9024	3.6098 x 10-5	69
14	0.7855	5.0419 x 10-5	51
15	0.9126	3.1790 x 10-5	85
16	0.8078	4.2796 x 10-5	67
17	0.9210	2.8565 x 10-5	53
18	0.8260	3.0824 x 10-5	75
19	0.9280	2.4156 x 10-5	50
20	0.8410	2.7943 x 10-5	93
21	0.9340	2.2428 x 10-5	74
22	0.8535	2.5641 x 10-5	75
23	0.9391	2.0992 x 10-5	94
24	0.8644	2.3767 x 10-5	56
25	0.9432	1.9776 x 10-5	97
26	0.8737	2.2207 x 10-5	75
27	0.9468	1.8729 x 10-5	63
28	0.8820	2.0897 x 10-5	60
29	0.9500	1.7826 x 10-5	97
30	0.8889	1.9755 x 10-5	99
31	0.9533	1.7043 x 10-5	96
32	0.8952	1.8774 x 10-5	97
33	0.9556	1.6336 x 10-5	94
34	0.9009	1.7914 x 10-5	85
35	0.9579	1.5709 x 10-5	54
36	0.9058	1.7145 x 10-5	96
37	0.9600	1.5149 x 10-5	98
38	0.9105	1.6468 x 10-5	100
39	0.9618	1.4636 x 10-5	92
40	0.9148	1.5856 x 10-5	80
41	0.9637	4.6517 x 10-5	72
42	0.9187	5.2677 x 10-5	61
43	0.9652	4.5065 x 10-5	73
44	0.9220	5.0759 x 10-5	62
45	0.9667	4.3716 x 10-5	74
46	0.9252	4.9036 x 10-5	63
47	0.9678	4.2481 x 10-5	54
48	0.9280	4.7425 x 10-5	56
49	0.9692	4.1354 x 10-5	69
50	0.9307	4.5949 x 10-5	72
51	0.9704	4.0284 x 10-5	74
52	0.9331	4.4611 x 10-5	57
53	0.9715	3.9302 x 10-5	57
54	0.9357	4.3356 x 10-5	75
55	0.9725	3.8388 x 10-5	71
56	0.9378	4.2217 x 10-5	53
57	0.9732	3.7518 x 10-5	52
58	0.9397	4.1137 x 10-5	66
59	0.9744	3.6713 x 10-5	78
60	0.9418	4.0133 x 10-5	53
61	0.9750	3.5967 x 10-5	52
62	0.9436	3.9202 x 10-5	52
63	0.9757	3.5253 x 10-5	81
64	0.9453	3.8341 x 10-5	79
65	0.9765	3.4565 x 10-5	85
66	0.9470	3.7503 x 10-5	56
67	0.9770	3.3937 x 10-5	86
68	0.9483	3.6736 x 10-5	52
69	0.9776	3.3343 x 10-5	72
70	0.9496	3.5991 x 10-5	57
71	0.9782	3.2757 x 10-5	78
72	0.9510	3.5309 x 10-5	63
73	0.9788	3.2222 x 10-5	76
74	0.9524	3.4649 x 10-5	60
75	0.9794	3.1677 x 10-5	56
76	0.9534	3.4031 x 10-5	86
77	0.9800	3.1214 x 10-5	85
78	0.9546	3.3438 x 10-5	67
79	0.9804	3.0715 x 10-5	57
80	0.9558	3.2884 x 10-5	53
81	0.9809	3.0261 x 10-5	85
82	0.9568	3.2341 x 10-5	87
83	0.9815	2.9837 x 10-5	78
84	0.9578	3.1834 x 10-5	79
85	0.9819	2.9412 x 10-5	91
86	0.9587	3.1337 x 10-5	85
87	0.9821	2.9017 x 10-5	89
88	0.9597	3.0871 x 10-5	55
89	0.9826	2.8634 x 10-5	92
90	0.9604	3.0417 x 10-5	90
91	0.9831	2.8246 x 10-5	91
92	0.9614	2.9994 x 10-5	70
93	0.9832	2.7912 x 10-5	73
94	0.9622	2.9560 x 10-5	76
95	0.9837	2.7551 x 10-5	62
96	0.9628	2.9175 x 10-5	86
97	0.9842	2.7220 x 10-5	88
98	0.9638	2.8795 x 10-5	66
99	0.9842	2.6903 x 10-5	91
100	0.9644	2.8415 x 10-5	92

For higher numbers of n, the dn constants are estimated as follows:

For n>100 where n is odd,

For n>100 where n is even,

References:
Croux C. and Rousseeuw P. J. (1992): Time-Efficient Algorithms for Two Highly Robust Estimators of Scale. Computational Statistics, 1, 411-428.
ISO 13528, second edition 2015-08-01. Statistical methods for use in proficiency testing by interlaboratory comparison. Reference number ISO 13528:2015(E)
Rousseeuw P.J. and Croux C. (1993). Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association, 88, 1273-1283.