Investigation of finite-sample properties of robust location and scale estimators

TL;DR

This paper studies the finite-sample properties of robust location and scale estimators, providing closed-form breakdown points, unbiasing factors, and efficiencies through simulations up to size 100, and offers predictive formulas for larger samples.

Contribution

It fills a gap by analyzing finite-sample behavior of robust estimators, offering explicit formulas and simulation results for practical sample sizes.

Findings

Finite-sample breakdown points are derived in closed form.

Unbiasing factors significantly improve estimator performance.

Predictive formulas for unbiasing factors for samples over 100 are provided.

Abstract

When the experimental data set is contaminated, we usually employ robust alternatives to common location and scale estimators such as the sample median and Hodges-Lehmann estimators for location and the sample median absolute deviation and Shamos estimators for scale. It is well known that these estimators have high positive asymptotic breakdown points and are Fisher-consistent as the sample size tends to infinity. To the best of our knowledge, the finite-sample properties of these estimators, depending on the sample size, have not well been studied in the literature. In this paper, we fill this gap by providing their closed-form finite-sample breakdown points and calculating the unbiasing factors and relative efficiencies of the robust estimators through the extensive Monte Carlo simulations up to the sample size 100. The numerical study shows that the unbiasing factor improves the finite-sample performance significantly. In addition, we provide the predicted values for the unbiasing factors obtained by using the least squares method which can be used for the case of sample size more than 100.

Tables

Table 1. Table 1: Empirical biases, variances and mean square errors of the estimates under consideration.

		standard	unbiased		unbiased		unbiased
$\delta$		deviation	std. dev.	MAD	MAD	Shamos	Shamos
		Empirical Biases
0		$-0.17971$	0.00036	$-0.53401$	0.00044	0.30333	$-0.00018$
10		0.90590	1.15528	$-0.41840$	0.14111	0.56179	0.23454
20		2.22951	2.56340	$-0.42168$	0.13713	0.54997	0.22381
30		3.57069	3.99020	$-0.42252$	0.13610	0.55349	0.22700
40		4.91276	5.41796	$-0.41812$	0.14145	0.55059	0.22436
50		6.25617	6.84714	$-0.41834$	0.14119	0.55455	0.22796
		Empirical Variances
0		0.10623	0.12023	0.21083	0.31212	0.19514	0.16093
10		0.11460	0.12970	0.23713	0.35106	0.23538	0.19411
20		0.11573	0.13098	0.23563	0.34884	0.23632	0.19489
30		0.11673	0.13211	0.23140	0.34258	0.23159	0.19099
40		0.11833	0.13392	0.23053	0.34129	0.23157	0.19098
50		0.11572	0.13097	0.23405	0.34649	0.23397	0.19295
		Empirical MSEs
0		0.13853	0.12023	0.49600	0.31212	0.28715	0.16093
10		0.93526	1.46437	0.41219	0.37097	0.55099	0.24912
20		5.08646	6.70198	0.41344	0.36764	0.53879	0.24498
30		12.86655	16.05382	0.40993	0.36110	0.53794	0.24252
40		24.25359	29.48825	0.40536	0.36130	0.53472	0.24131
50		39.25542	47.01434	0.40905	0.36643	0.54149	0.24492

Table 2. Table 2: Finite-sample breakdown points.

$n$		median and MAD	HL1 and Shamos	HL2	HL3
2		0.0000000	0.0000000	0.0000000	0.0000000
3		0.3333333	0.0000000	0.0000000	0.0000000
4		0.2500000	0.0000000	0.2500000	0.2500000
5		0.4000000	0.2000000	0.2000000	0.2000000
6		0.3333333	0.1666667	0.1666667	0.1666667
7		0.4285714	0.1428571	0.2857143	0.2857143
8		0.3750000	0.2500000	0.2500000	0.2500000
9		0.4444444	0.2222222	0.2222222	0.2222222
10		0.4000000	0.2000000	0.3000000	0.2000000
11		0.4545455	0.2727273	0.2727273	0.2727273
12		0.4166667	0.2500000	0.2500000	0.2500000
13		0.4615385	0.2307692	0.2307692	0.2307692
14		0.4285714	0.2142857	0.2857143	0.2857143
15		0.4666667	0.2666667	0.2666667	0.2666667
16		0.4375000	0.2500000	0.2500000	0.2500000
17		0.4705882	0.2352941	0.2941176	0.2352941
18		0.4444444	0.2777778	0.2777778	0.2777778
19		0.4736842	0.2631579	0.2631579	0.2631579
20		0.4500000	0.2500000	0.2500000	0.2500000
21		0.4761905	0.2380952	0.2857143	0.2857143
22		0.4545455	0.2727273	0.2727273	0.2727273
23		0.4782609	0.2608696	0.2608696	0.2608696
24		0.4583333	0.2500000	0.2916667	0.2916667
25		0.4800000	0.2800000	0.2800000	0.2800000
26		0.4615385	0.2692308	0.2692308	0.2692308
27		0.4814815	0.2592593	0.2962963	0.2592593
28		0.4642857	0.2857143	0.2857143	0.2857143
29		0.4827586	0.2758621	0.2758621	0.2758621
30		0.4666667	0.2666667	0.2666667	0.2666667
31		0.4838710	0.2580645	0.2903226	0.2903226
32		0.4687500	0.2812500	0.2812500	0.2812500
33		0.4848485	0.2727273	0.2727273	0.2727273
34		0.4705882	0.2647059	0.2941176	0.2647059
35		0.4857143	0.2857143	0.2857143	0.2857143
36		0.4722222	0.2777778	0.2777778	0.2777778
37		0.4864865	0.2702703	0.2702703	0.2702703
38		0.4736842	0.2631579	0.2894737	0.2894737
39		0.4871795	0.2820513	0.2820513	0.2820513
40		0.4750000	0.2750000	0.2750000	0.2750000
41		0.4878049	0.2682927	0.2926829	0.2926829
42		0.4761905	0.2857143	0.2857143	0.2857143
43		0.4883721	0.2790698	0.2790698	0.2790698
44		0.4772727	0.2727273	0.2954545	0.2727273
45		0.4888889	0.2888889	0.2888889	0.2888889
46		0.4782609	0.2826087	0.2826087	0.2826087
47		0.4893617	0.2765957	0.2765957	0.2765957
48		0.4791667	0.2708333	0.2916667	0.2916667
49		0.4897959	0.2857143	0.2857143	0.2857143
50		0.4800000	0.2800000	0.2800000	0.2800000

Table 3. Table 3: Empirical biases of the MAD and Shamos estimators ( n = 2 , 3 , … , 100 𝑛 2 3 … 100 n=2,3,\ldots,100 ).

$n$		MAD	Shamos		$n$		MAD	Shamos
1		NA	NA		51		$-0.0152820$	$0.0082120$
2		$-0.1633880$	$0.1831500$		52		$-0.0149951$	$0.0081874$
3		$-0.3275897$	$0.2989400$		53		$-0.0146042$	$0.0079775$
4		$-0.2648275$	$0.1582782$		54		$-0.0145007$	$0.0078126$
5		$-0.1781250$	$0.1011748$		55		$-0.0140391$	$0.0076743$
6		$-0.1594213$	$0.1005038$		56		$-0.0139674$	$0.0075212$
7		$-0.1210631$	$0.0676993$		57		$-0.0136336$	$0.0074051$
8		$-0.1131928$	$0.0609574$		58		$-0.0134819$	$0.0072528$
9		$-0.0920658$	$0.0543760$		59		$-0.0130812$	$0.0071807$
10		$-0.0874503$	$0.0476839$		60		$-0.0129708$	$0.0070617$
11		$-0.0741303$	$0.0426722$		61		$-0.0126589$	$0.0069123$
12		$-0.0711412$	$0.0385003$		62		$-0.0125598$	$0.0067833$
13		$-0.0620918$	$0.0353028$		63		$-0.0122696$	$0.0066439$
14		$-0.0600210$	$0.0323526$		64		$-0.0121523$	$0.0065821$
15		$-0.0534603$	$0.0299677$		65		$-0.0118163$	$0.0064889$
16		$-0.0519047$	$0.0280421$		66		$-0.0118244$	$0.0063844$
17		$-0.0467319$	$0.0262195$		67		$-0.0115177$	$0.0062930$
18		$-0.0455579$	$0.0247674$		68		$-0.0114479$	$0.0061910$
19		$-0.0417554$	$0.0232297$		69		$-0.0111309$	$0.0061255$
20		$-0.0408248$	$0.0220155$		70		$-0.0110816$	$0.0060681$
21		$-0.0376967$	$0.0208687$		71		$-0.0108875$	$0.0058994$
22		$-0.0368350$	$0.0199446$		72		$-0.0108319$	$0.0058235$
23		$-0.0342394$	$0.0189794$		73		$-0.0106032$	$0.0057172$
24		$-0.0335390$	$0.0182343$		74		$-0.0105424$	$0.0056805$
25		$-0.0313065$	$0.0174421$		75		$-0.0102237$	$0.0056343$
26		$-0.0309765$	$0.0166364$		76		$-0.0102132$	$0.0055605$
27		$-0.0290220$	$0.0160158$		77		$-0.0099408$	$0.0055011$
28		$-0.0287074$	$0.0153715$		78		$-0.0099776$	$0.0053872$
29		$-0.0269133$	$0.0148940$		79		$-0.0097815$	$0.0053062$
30		$-0.0265451$	$0.0144027$		80		$-0.0097399$	$0.0052348$
31		$-0.0250734$	$0.0138855$		81		$-0.0094837$	$0.0052075$
32		$-0.0248177$	$0.0134510$		82		$-0.0094713$	$0.0051173$
33		$-0.0236460$	$0.0130228$		83		$-0.0092390$	$0.0050697$
34		$-0.0232808$	$0.0127183$		84		$-0.0092875$	$0.0049805$
35		$-0.0222099$	$0.0122444$		85		$-0.0091508$	$0.0048705$
36		$-0.0220756$	$0.0118214$		86		$-0.0090145$	$0.0048695$
37		$-0.0210129$	$0.0115469$		87		$-0.0088191$	$0.0048287$
38		$-0.0207309$	$0.0113206$		88		$-0.0088205$	$0.0047315$
39		$-0.0199272$	$0.0109636$		89		$-0.0086622$	$0.0046961$
40		$-0.0197140$	$0.0106308$		90		$-0.0085714$	$0.0046698$
41		$-0.0188446$	$0.0104384$		91		$-0.0084718$	$0.0046010$
42		$-0.0188203$	$0.0100693$		92		$-0.0083861$	$0.0045544$
43		$-0.0180521$	$0.0098523$		93		$-0.0082559$	$0.0045191$
44		$-0.0178185$	$0.0096735$		94		$-0.0082650$	$0.0044245$
45		$-0.0171866$	$0.0094973$		95		$-0.0080977$	$0.0044074$
46		$-0.0170796$	$0.0092210$		96		$-0.0080708$	$0.0043579$
47		$-0.0165391$	$0.0089781$		97		$-0.0078810$	$0.0043536$
48		$-0.0163509$	$0.0088083$		98		$-0.0078492$	$0.0042874$
49		$-0.0157862$	$0.0086574$		99		$-0.0077043$	$0.0042520$
50		$-0.0157372$	$0.0084772$		100		$-0.0077614$	$0.0041864$

Table 4. Table 4: Empirical biases of the MAD and Shamos estimators along with their estimates ( n = 109 , 110 , 119 , 120 , … , 499 , 500 𝑛 109 110 119 120 … 499 500 n=109,110,119,120,\ldots,499,500 ).

$n$		MAD	$A_n$(Hayes)	$A_n$(Williams)		Shamos	$B_n$(Hayes)	$B_n$(Williams)
109		$-0.0070577$	$-0.0070648$	$-0.0070753$		0.0038374	0.0038377	0.0038425
110		$-0.0070262$	$-0.0069999$	$-0.0070104$		0.0037996	0.0038025	0.0038073
119		$-0.0064893$	$-0.0064655$	$-0.0064756$		0.0034984	0.0035124	0.0035170
120		$-0.0064342$	$-0.0064111$	$-0.0064212$		0.0034691	0.0034828	0.0034875
129		$-0.0059226$	$-0.0059599$	$-0.0059693$		0.0032441	0.0032379	0.0032422
130		$-0.0059018$	$-0.0059137$	$-0.0059230$		0.0032241	0.0032127	0.0032170
139		$-0.0054913$	$-0.0055277$	$-0.0055362$		0.0029854	0.0030031	0.0030070
140		$-0.0055067$	$-0.0054879$	$-0.0054963$		0.0029548	0.0029815	0.0029854
149		$-0.0051140$	$-0.0051539$	$-0.0051615$		0.0028230	0.0028002	0.0028036
150		$-0.0051018$	$-0.0051193$	$-0.0051267$		0.0028080	0.0027814	0.0027847
159		$-0.0047958$	$-0.0048275$	$-0.0048340$		0.0026355	0.0026229	0.0026258
160		$-0.0047790$	$-0.0047971$	$-0.0048035$		0.0026154	0.0026064	0.0026093
169		$-0.0045272$	$-0.0045399$	$-0.0045455$		0.0024503	0.0024667	0.0024692
170		$-0.0045260$	$-0.0045130$	$-0.0045185$		0.0024402	0.0024521	0.0024545
179		$-0.0042827$	$-0.0042847$	$-0.0042894$		0.0023257	0.0023281	0.0023301
180		$-0.0042517$	$-0.0042607$	$-0.0042653$		0.0023122	0.0023151	0.0023170
189		$-0.0040825$	$-0.0040566$	$-0.0040605$		0.0021780	0.0022042	0.0022058
190		$-0.0040837$	$-0.0040351$	$-0.0040389$		0.0021673	0.0021925	0.0021941
199		$-0.0038386$	$-0.0038516$	$-0.0038546$		0.0020904	0.0020928	0.0020940
200		$-0.0038277$	$-0.0038323$	$-0.0038352$		0.0020786	0.0020823	0.0020835
249		$-0.0031016$	$-0.0030747$	$-0.0030745$		0.0016628	0.0016708	0.0016704
250		$-0.0030911$	$-0.0030623$	$-0.0030621$		0.0016562	0.0016641	0.0016636
299		$-0.0025435$	$-0.0025586$	$-0.0025562$		0.0013875	0.0013904	0.0013889
300		$-0.0025436$	$-0.0025500$	$-0.0025476$		0.0013822	0.0013858	0.0013842
349		$-0.0021615$	$-0.0021908$	$-0.0021869$		0.0012033	0.0011906	0.0011884
350		$-0.0021702$	$-0.0021846$	$-0.0021806$		0.0012013	0.0011872	0.0011849
399		$-0.0019214$	$-0.0019155$	$-0.0019106$		0.0010331	0.0010410	0.0010383
400		$-0.0019273$	$-0.0019107$	$-0.0019058$		0.0010287	0.0010384	0.0010357
449		$-0.0017199$	$-0.0017017$	$-0.0016960$		0.0009196	0.0009248	0.0009217
450		$-0.0017210$	$-0.0016979$	$-0.0016922$		0.0009170	0.0009227	0.0009197
499		$-0.0015134$	$-0.0015308$	$-0.0015247$		0.0008413	0.0008319	0.0008286
500		$-0.0015061$	$-0.0015277$	$-0.0015216$		0.0008389	0.0008303	0.0008270

Table 5. Table 5: The values of n × Var ​ ( θ ^ ) 𝑛 Var ^ 𝜃 n\times\mathrm{Var}(\hat{\theta}) for the median and the Hodges-Lehmann, and the values of Var ​ ( θ ^ ) / ( 1 − c 4 ​ ( n ) 2 ) Var ^ 𝜃 1 subscript 𝑐 4 superscript 𝑛 2 \mathrm{Var}(\hat{\theta})/(1-c_{4}(n)^{2}) for the MAD and Shamos ( n = 1 , 2 , … , 50 𝑛 1 2 … 50 n=1,2,\ldots,50 ).

$n$		median	HL1	HL2	HL3		MAD	Shamos
1		1.0000	NA	1.0000	1.0000		NA	NA
2		1.0000	1.0000	1.0000	1.0000		1.1000	2.2001
3		1.3463	1.0871	1.0221	1.0871		1.4372	2.3812
4		1.1930	1.0000	1.0949	1.0949		1.1680	1.6996
5		1.4339	1.0617	1.0754	1.0754		1.9809	1.8573
6		1.2882	1.0619	1.0759	1.0602		1.6859	1.7883
7		1.4736	1.0630	1.0814	1.0756		2.2125	1.6180
8		1.3459	1.0628	1.0728	1.0705		1.9486	1.5824
9		1.4957	1.0588	1.0756	1.0678		2.3326	1.5109
10		1.3833	1.0608	1.0743	1.0641		2.1072	1.4855
11		1.5088	1.0602	1.0693	1.0649		2.4082	1.4643
12		1.4087	1.0560	1.0670	1.0614		2.2112	1.4234
13		1.5195	1.0567	1.0685	1.0629		2.4570	1.4008
14		1.4298	1.0565	1.0663	1.0603		2.2848	1.3905
15		1.5249	1.0562	1.0645	1.0603		2.4952	1.3719
16		1.4457	1.0547	1.0637	1.0590		2.3412	1.3554
17		1.5302	1.0541	1.0633	1.0587		2.5217	1.3434
18		1.4585	1.0540	1.0621	1.0574		2.3846	1.3355
19		1.5333	1.0532	1.0605	1.0567		2.5447	1.3249
20		1.4702	1.0545	1.0620	1.0581		2.4185	1.3146
21		1.5383	1.0536	1.0611	1.0573		2.5611	1.3079
22		1.4770	1.0527	1.0596	1.0557		2.4475	1.3015
23		1.5420	1.0532	1.0597	1.0564		2.5758	1.2953
24		1.4850	1.0529	1.0594	1.0560		2.4699	1.2883
25		1.5438	1.0521	1.0586	1.0553		2.5873	1.2825
26		1.4896	1.0518	1.0578	1.0545		2.4886	1.2776
27		1.5462	1.0526	1.0582	1.0553		2.5960	1.2731
28		1.4954	1.0511	1.0567	1.0538		2.5030	1.2676
29		1.5476	1.0525	1.0581	1.0552		2.6070	1.2650
30		1.5005	1.0518	1.0571	1.0543		2.5199	1.2616
31		1.5482	1.0514	1.0564	1.0538		2.6132	1.2586
32		1.5057	1.0517	1.0567	1.0541		2.5335	1.2552
33		1.5516	1.0521	1.0571	1.0545		2.6208	1.2519
34		1.5091	1.0512	1.0560	1.0534		2.5442	1.2493
35		1.5515	1.0508	1.0554	1.0530		2.6285	1.2466
36		1.5123	1.0512	1.0557	1.0534		2.5545	1.2433
37		1.5531	1.0512	1.0556	1.0534		2.6332	1.2415
38		1.5148	1.0502	1.0545	1.0522		2.5637	1.2393
39		1.5550	1.0513	1.0555	1.0533		2.6344	1.2364
40		1.5173	1.0507	1.0547	1.0526		2.5720	1.2357
41		1.5532	1.0498	1.0539	1.0518		2.6403	1.2325
42		1.5206	1.0510	1.0549	1.0528		2.5780	1.2315
43		1.5552	1.0504	1.0541	1.0522		2.6436	1.2287
44		1.5224	1.0500	1.0537	1.0518		2.5869	1.2284
45		1.5568	1.0504	1.0541	1.0522		2.6477	1.2260
46		1.5240	1.0496	1.0533	1.0514		2.5904	1.2248
47		1.5570	1.0504	1.0539	1.0521		2.6511	1.2232
48		1.5249	1.0493	1.0528	1.0510		2.5960	1.2214
49		1.5562	1.0495	1.0529	1.0512		2.6537	1.2199
50		1.5267	1.0499	1.0532	1.0514		2.6014	1.2184

Table 6. Table 6: The values of n × Var ​ ( θ ^ ) 𝑛 Var ^ 𝜃 n\times\mathrm{Var}(\hat{\theta}) for the median and the Hodges-Lehmann, and the values of Var ​ ( θ ^ ) / ( 1 − c 4 ​ ( n ) 2 ) Var ^ 𝜃 1 subscript 𝑐 4 superscript 𝑛 2 \mathrm{Var}(\hat{\theta})/(1-c_{4}(n)^{2}) for the MAD and Shamos ( n = 51 , 52 , … , 100 𝑛 51 52 … 100 n=51,52,\ldots,100 ).

$n$		median	HL1	HL2	HL3		MAD	Shamos
51		1.5583	1.0502	1.0534	1.0517		2.6577	1.2199
52		1.5298	1.0499	1.0532	1.0515		2.6053	1.2174
53		1.5592	1.0501	1.0533	1.0517		2.6568	1.2160
54		1.5298	1.0489	1.0519	1.0503		2.6125	1.2156
55		1.5584	1.0493	1.0523	1.0508		2.6631	1.2144
56		1.5330	1.0497	1.0527	1.0512		2.6139	1.2132
57		1.5589	1.0496	1.0526	1.0510		2.6649	1.2126
58		1.5337	1.0495	1.0524	1.0509		2.6161	1.2098
59		1.5598	1.0501	1.0530	1.0515		2.6671	1.2095
60		1.5349	1.0489	1.0517	1.0503		2.6219	1.2095
61		1.5594	1.0492	1.0519	1.0505		2.6667	1.2073
62		1.5361	1.0492	1.0520	1.0505		2.6235	1.2071
63		1.5594	1.0485	1.0512	1.0498		2.6695	1.2064
64		1.5373	1.0494	1.0521	1.0507		2.6260	1.2050
65		1.5598	1.0488	1.0514	1.0500		2.6731	1.2067
66		1.5380	1.0496	1.0521	1.0508		2.6297	1.2036
67		1.5606	1.0494	1.0519	1.0506		2.6722	1.2034
68		1.5389	1.0491	1.0516	1.0503		2.6341	1.2030
69		1.5607	1.0479	1.0504	1.0491		2.6748	1.2025
70		1.5399	1.0490	1.0514	1.0502		2.6351	1.2016
71		1.5595	1.0482	1.0506	1.0494		2.6738	1.2005
72		1.5410	1.0491	1.0515	1.0503		2.6351	1.1993
73		1.5622	1.0492	1.0515	1.0503		2.6754	1.1993
74		1.5426	1.0498	1.0521	1.0510		2.6395	1.1990
75		1.5619	1.0489	1.0512	1.0500		2.6763	1.1985
76		1.5415	1.0486	1.0509	1.0497		2.6411	1.1975
77		1.5616	1.0485	1.0508	1.0496		2.6780	1.1975
78		1.5434	1.0494	1.0516	1.0505		2.6453	1.1971
79		1.5639	1.0493	1.0515	1.0504		2.6794	1.1968
80		1.5445	1.0497	1.0519	1.0508		2.6453	1.1958
81		1.5612	1.0486	1.0507	1.0496		2.6815	1.1960
82		1.5444	1.0494	1.0515	1.0504		2.6472	1.1947
83		1.5626	1.0484	1.0505	1.0494		2.6815	1.1947
84		1.5449	1.0490	1.0511	1.0500		2.6475	1.1939
85		1.5630	1.0484	1.0504	1.0494		2.6831	1.1938
86		1.5441	1.0479	1.0499	1.0489		2.6505	1.1931
87		1.5643	1.0495	1.0514	1.0504		2.6830	1.1923
88		1.5448	1.0478	1.0497	1.0487		2.6535	1.1929
89		1.5640	1.0487	1.0506	1.0496		2.6857	1.1931
90		1.5463	1.0483	1.0503	1.0493		2.6562	1.1920
91		1.5634	1.0486	1.0505	1.0495		2.6853	1.1914
92		1.5477	1.0491	1.0509	1.0500		2.6567	1.1913
93		1.5631	1.0481	1.0500	1.0490		2.6859	1.1906
94		1.5482	1.0488	1.0507	1.0497		2.6584	1.1907
95		1.5629	1.0481	1.0499	1.0490		2.6878	1.1905
96		1.5466	1.0477	1.0495	1.0486		2.6576	1.1894
97		1.5636	1.0480	1.0498	1.0489		2.6881	1.1895
98		1.5477	1.0477	1.0495	1.0486		2.6613	1.1899
99		1.5642	1.0483	1.0501	1.0492		2.6888	1.1887
100		1.5484	1.0481	1.0498	1.0489		2.6604	1.1874

Table 7. Table 7: The values of n × Var ​ ( θ ^ ) 𝑛 Var ^ 𝜃 n\times\mathrm{Var}(\hat{\theta}) for the median and the Hodges-Lehmann, and the values of Var ​ ( θ ^ ) / ( 1 − c 4 ​ ( n ) 2 ) Var ^ 𝜃 1 subscript 𝑐 4 superscript 𝑛 2 \mathrm{Var}(\hat{\theta})/(1-c_{4}(n)^{2}) for the MAD and Shamos ( n = 109 , 110 , 119 , 120 , … , 499 , 500 𝑛 109 110 119 120 … 499 500 n=109,110,119,120,\ldots,499,500 ).

$n$		median	HL1	HL2	HL3		MAD	Shamos
109		1.5655	1.0490	1.0506	1.0498		2.6889	1.1857
110		1.5508	1.0484	1.0500	1.0492		2.6657	1.1856
119		1.5651	1.0478	1.0492	1.0485		2.6936	1.1830
120		1.5526	1.0478	1.0493	1.0486		2.6717	1.1836
129		1.5661	1.0477	1.0490	1.0483		2.6953	1.1809
130		1.5541	1.0478	1.0492	1.0485		2.6727	1.1804
139		1.5671	1.0491	1.0503	1.0497		2.6963	1.1792
140		1.5567	1.0495	1.0508	1.0502		2.6770	1.1794
149		1.5666	1.0484	1.0496	1.0490		2.7008	1.1789
150		1.5566	1.0484	1.0496	1.0490		2.6815	1.1788
159		1.5673	1.0484	1.0495	1.0490		2.7006	1.1768
160		1.5584	1.0485	1.0495	1.0490		2.6827	1.1765
169		1.5661	1.0474	1.0485	1.0479		2.7012	1.1757
170		1.5578	1.0476	1.0486	1.0481		2.6861	1.1755
179		1.5676	1.0477	1.0487	1.0482		2.7038	1.1750
180		1.5590	1.0480	1.0490	1.0485		2.6889	1.1750
189		1.5663	1.0473	1.0483	1.0478		2.7043	1.1743
190		1.5584	1.0473	1.0482	1.0478		2.6903	1.1741
199		1.5681	1.0481	1.0490	1.0486		2.7049	1.1741
200		1.5608	1.0482	1.0491	1.0486		2.6904	1.1732
249		1.5679	1.0472	1.0479	1.0476		2.7083	1.1705
250		1.5623	1.0479	1.0486	1.0483		2.6977	1.1709
299		1.5689	1.0477	1.0483	1.0480		2.7084	1.1673
300		1.5642	1.0479	1.0485	1.0482		2.6986	1.1670
349		1.5700	1.0479	1.0484	1.0481		2.7131	1.1673
350		1.5654	1.0479	1.0484	1.0481		2.7049	1.1675
399		1.5691	1.0475	1.0479	1.0477		2.7126	1.1650
400		1.5646	1.0469	1.0474	1.0472		2.7072	1.1651
449		1.5694	1.0474	1.0478	1.0476		2.7125	1.1639
450		1.5659	1.0475	1.0479	1.0477		2.7056	1.1645
499		1.5701	1.0475	1.0479	1.0477		2.7147	1.1637
500		1.5674	1.0482	1.0486	1.0484		2.7101	1.1646

Table 8. Table 8: Relative efficiencies of the median and Hodges-Lehmann estimators to the sample mean and those of the MAD and Shamos estimators to the sample standard deviation under the normal distribution ( n = 1 , 2 , … , 50 𝑛 1 2 … 50 n=1,2,\ldots,50 ).

$n$		median	HL1	HL2	HL3		MAD	Shamos
1		1.0000	NA	1.0000	1.0000		NA	NA
2		1.0000	1.0000	1.0000	1.0000		0.9091	0.4545
3		0.7427	0.9199	0.9784	0.9199		0.6958	0.4199
4		0.8382	1.0000	0.9133	0.9133		0.8562	0.5884
5		0.6974	0.9419	0.9299	0.9299		0.5048	0.5384
6		0.7763	0.9417	0.9295	0.9432		0.5932	0.5592
7		0.6786	0.9407	0.9248	0.9297		0.4520	0.6180
8		0.7430	0.9409	0.9322	0.9342		0.5132	0.6320
9		0.6686	0.9445	0.9297	0.9365		0.4287	0.6618
10		0.7229	0.9426	0.9308	0.9398		0.4746	0.6732
11		0.6628	0.9432	0.9352	0.9391		0.4153	0.6829
12		0.7098	0.9470	0.9372	0.9422		0.4522	0.7026
13		0.6581	0.9464	0.9359	0.9408		0.4070	0.7139
14		0.6994	0.9465	0.9378	0.9432		0.4377	0.7192
15		0.6558	0.9468	0.9394	0.9432		0.4008	0.7289
16		0.6917	0.9482	0.9402	0.9443		0.4271	0.7378
17		0.6535	0.9486	0.9405	0.9445		0.3966	0.7444
18		0.6856	0.9487	0.9415	0.9457		0.4194	0.7488
19		0.6522	0.9495	0.9430	0.9463		0.3930	0.7547
20		0.6802	0.9483	0.9416	0.9451		0.4135	0.7607
21		0.6501	0.9491	0.9424	0.9458		0.3905	0.7646
22		0.6770	0.9499	0.9437	0.9472		0.4086	0.7684
23		0.6485	0.9495	0.9437	0.9466		0.3882	0.7720
24		0.6734	0.9498	0.9440	0.9470		0.4049	0.7762
25		0.6478	0.9504	0.9447	0.9476		0.3865	0.7798
26		0.6713	0.9507	0.9454	0.9483		0.4018	0.7827
27		0.6468	0.9501	0.9450	0.9476		0.3852	0.7855
28		0.6687	0.9514	0.9463	0.9490		0.3995	0.7889
29		0.6462	0.9501	0.9451	0.9477		0.3836	0.7905
30		0.6664	0.9507	0.9459	0.9485		0.3968	0.7926
31		0.6459	0.9511	0.9466	0.9489		0.3827	0.7945
32		0.6641	0.9508	0.9463	0.9486		0.3947	0.7967
33		0.6445	0.9505	0.9460	0.9483		0.3816	0.7988
34		0.6627	0.9513	0.9470	0.9493		0.3931	0.8004
35		0.6445	0.9516	0.9475	0.9496		0.3804	0.8022
36		0.6612	0.9513	0.9472	0.9493		0.3915	0.8043
37		0.6439	0.9513	0.9473	0.9493		0.3798	0.8055
38		0.6601	0.9522	0.9483	0.9504		0.3901	0.8069
39		0.6431	0.9512	0.9475	0.9494		0.3796	0.8088
40		0.6591	0.9518	0.9481	0.9500		0.3888	0.8093
41		0.6438	0.9525	0.9489	0.9507		0.3787	0.8113
42		0.6576	0.9515	0.9479	0.9498		0.3879	0.8120
43		0.6430	0.9521	0.9486	0.9504		0.3783	0.8138
44		0.6569	0.9524	0.9490	0.9508		0.3866	0.8141
45		0.6423	0.9520	0.9487	0.9504		0.3777	0.8157
46		0.6562	0.9527	0.9494	0.9511		0.3860	0.8164
47		0.6422	0.9520	0.9489	0.9505		0.3772	0.8175
48		0.6558	0.9530	0.9499	0.9515		0.3852	0.8187
49		0.6426	0.9529	0.9497	0.9513		0.3768	0.8197
50		0.6550	0.9525	0.9495	0.9511		0.3844	0.8208

Table 9. Table 9: Relative efficiencies of the median and Hodges-Lehmann estimators to the sample mean and those of the MAD and Shamos estimators to the sample standard deviation under the normal distribution ( n = 51 , 52 , … , 100 𝑛 51 52 … 100 n=51,52,\ldots,100 ).

$n$		median	HL1	HL2	HL3		MAD	Shamos
51		0.6417	0.9522	0.9493	0.9508		0.3763	0.8197
52		0.6537	0.9525	0.9495	0.9510		0.3838	0.8214
53		0.6414	0.9523	0.9494	0.9509		0.3764	0.8223
54		0.6537	0.9534	0.9506	0.9521		0.3828	0.8226
55		0.6417	0.9530	0.9503	0.9517		0.3755	0.8234
56		0.6523	0.9527	0.9499	0.9513		0.3826	0.8243
57		0.6415	0.9528	0.9501	0.9514		0.3752	0.8247
58		0.6520	0.9528	0.9502	0.9515		0.3822	0.8266
59		0.6411	0.9523	0.9497	0.9510		0.3749	0.8268
60		0.6515	0.9534	0.9508	0.9521		0.3814	0.8268
61		0.6413	0.9531	0.9506	0.9519		0.3750	0.8283
62		0.6510	0.9531	0.9506	0.9519		0.3812	0.8284
63		0.6413	0.9538	0.9513	0.9526		0.3746	0.8289
64		0.6505	0.9529	0.9505	0.9517		0.3808	0.8299
65		0.6411	0.9535	0.9511	0.9523		0.3741	0.8287
66		0.6502	0.9528	0.9505	0.9516		0.3803	0.8308
67		0.6408	0.9529	0.9507	0.9518		0.3742	0.8310
68		0.6498	0.9532	0.9509	0.9521		0.3796	0.8312
69		0.6408	0.9543	0.9520	0.9532		0.3739	0.8316
70		0.6494	0.9533	0.9511	0.9522		0.3795	0.8323
71		0.6412	0.9540	0.9518	0.9529		0.3740	0.8330
72		0.6489	0.9532	0.9510	0.9521		0.3795	0.8338
73		0.6401	0.9532	0.9510	0.9521		0.3738	0.8338
74		0.6483	0.9525	0.9504	0.9515		0.3789	0.8341
75		0.6402	0.9534	0.9513	0.9524		0.3736	0.8344
76		0.6487	0.9536	0.9516	0.9526		0.3786	0.8351
77		0.6404	0.9537	0.9517	0.9527		0.3734	0.8351
78		0.6479	0.9529	0.9509	0.9520		0.3780	0.8353
79		0.6394	0.9530	0.9510	0.9520		0.3732	0.8355
80		0.6475	0.9526	0.9507	0.9517		0.3780	0.8363
81		0.6405	0.9537	0.9517	0.9527		0.3729	0.8361
82		0.6475	0.9529	0.9510	0.9520		0.3778	0.8370
83		0.6400	0.9538	0.9520	0.9529		0.3729	0.8370
84		0.6473	0.9533	0.9514	0.9523		0.3777	0.8376
85		0.6398	0.9539	0.9520	0.9529		0.3727	0.8376
86		0.6476	0.9543	0.9525	0.9534		0.3773	0.8381
87		0.6393	0.9529	0.9511	0.9520		0.3727	0.8387
88		0.6473	0.9544	0.9526	0.9535		0.3769	0.8383
89		0.6394	0.9536	0.9518	0.9527		0.3723	0.8382
90		0.6467	0.9539	0.9521	0.9530		0.3765	0.8389
91		0.6396	0.9536	0.9519	0.9528		0.3724	0.8394
92		0.6461	0.9532	0.9515	0.9524		0.3764	0.8394
93		0.6398	0.9541	0.9524	0.9533		0.3723	0.8399
94		0.6459	0.9535	0.9518	0.9526		0.3762	0.8399
95		0.6398	0.9541	0.9524	0.9533		0.3721	0.8400
96		0.6466	0.9545	0.9529	0.9537		0.3763	0.8407
97		0.6396	0.9542	0.9526	0.9534		0.3720	0.8407
98		0.6461	0.9545	0.9529	0.9537		0.3757	0.8404
99		0.6393	0.9539	0.9523	0.9531		0.3719	0.8412
100		0.6458	0.9541	0.9525	0.9533		0.3759	0.8422

Equations

$${\mathbf{x}}_{(m)}^{\ast }=\{x_1^{\ast },x_2^{\ast },\dots ,x_m^{\ast },x_{m+1},\dots ,x_n\}.$$

$$\mathrm{Bias}(\hat{\theta };{\mathbf{x}}_{(m)}^{\ast },\mathbf{x})=\sup |\hat{\theta }({\mathbf{x}}_{(m)}^{\ast })-\hat{\theta }(\mathbf{x})|,$$

$$\epsilon_{n}=\frac{1}{n}\max\big{\{}m:\mathrm{Bias}(\hat{\theta};\mathbf{x}_{(m)}^{*},\mathbf{x})<\infty\big{\}}.$$

$$\hat{\theta }(X_1+b,X_2+b,\dots ,X_n+b)=\hat{\theta }(X_1,X_2,\dots ,X_n)+b.$$

$${ϵ}_n=\frac{\lfloor (n-1)/2\rfloor }{n}.$$

$${ϵ}_n=\frac{\lfloor (n-1)/2\rfloor }{n}=\frac{1}{2}-\frac{1}{2n}-\frac{\delta }{n}.$$

$$\mathop{\mathrm{median}}\Big{(}\frac{X_{i}+X_{j}}{2}\Big{)}.$$

$$\mathrm{HL1}=\mathop{\mathrm{median}}_{i<j}\Big{(}\frac{X_{i}+X_{j}}{2}\Big{)},\quad\mathrm{HL2}=\mathop{\mathrm{median}}_{i\leq j}\Big{(}\frac{X_{i}+X_{j}}{2}\Big{)},\quad\mathrm{HL3}=\mathop{\mathrm{median}}_{\forall(i,j)}\Big{(}\frac{X_{i}+X_{j}}{2}\Big{)},$$

$$\frac{n^2-(n-k{)}^2}{n^2}\le \frac{\lfloor (n^2-1)/2\rfloor }{n^2},$$

$$k^{*}=\max\Big{\{}k:k^{2}-2nk+\lfloor(n^{2}-1)/2\rfloor\geq 0\textrm{~{}and~{}}k=0,1,2,\ldots,n\Big{\}}.$$

$$x_1=n-\sqrt{n^2-\lfloor (n^2-1)/2\rfloor }\text{ and }{x}_2=n+\sqrt{n^2-\lfloor (n^2-1)/2\rfloor }.$$

$$\epsilon_{n}=\frac{\Big{\lfloor}n-\sqrt{n^{2}-\lfloor(n^{2}-1)/2\rfloor}\Big{\rfloor}}{n}.$$

$${ϵ}_n=\frac{n-\sqrt{n^2-(n^2-1)/2+{\delta }_1}-{\delta }_2}{n},$$

$$\frac{n(n-1)/2-(n-k)(n-k-1)/2}{n(n-1)/2}\leq\frac{1}{n(n-1)/2}\Big{\lfloor}\frac{n(n-1)/2-1}{2}\Big{\rfloor},$$

$$\epsilon_{n}=\frac{\Big{\lfloor}n-1/2-\sqrt{(n-1/2)^{2}-2\lfloor(n^{2}-n-2)/4\rfloor}\Big{\rfloor}}{n}.$$

$$\epsilon_{n}=\frac{\Big{\lfloor}n+1/2-\sqrt{(n+1/2)^{2}-2\lfloor(n^{2}+n-2)/4\rfloor}\Big{\rfloor}}{n}.$$

$${ϵ}_n(\text{HL2})\ge {ϵ}_n(\text{HL3})\ge {ϵ}_n(\text{HL1}).$$

$$\mathrm{MAD}=\frac{\mathrm{missing}{median}_{1\le i\le n}|X_i-\tilde{\mu }|}{{\Phi }^{-1}(3/4)}\approx 1.4826\cdot \mathrm{missing}{median}_{1\le i\le n}|X_i-\tilde{\mu }|,$$

$$\mathrm{Shamos}=\frac{\displaystyle\mathop{\mathrm{median}}_{i<j}\big{(}|X_{i}-X_{j}|\big{)}}{\sqrt{2}\,\Phi^{-1}(3/4)}\approx 1.048358\cdot\displaystyle\mathop{\mathrm{median}}_{i<j}\big{(}|X_{i}-X_{j}|\big{)},$$

$$\hat{\theta }(a{X}_1+b,a{X}_2+b,\dots ,a{X}_n+b)=|a|\cdot \hat{\theta }(X_1,X_2,\dots ,X_n).$$

$$\frac{\mathrm{MAD}}{1+A_n},$$

$$\frac{\mathrm{Shamos}}{1+B_n},$$

$$A_n=-\frac{0.76213}{n}-\frac{0.86413}{n^2},$$

$$A_n=-0.804168866\cdot n^{-1.008922},$$

$$B_n$$

$$c_4(n)=\sqrt{\frac{2}{n-1}}\cdot \frac{\Gamma (n/2)}{\Gamma (n/2-1/2)}.$$

$$\frac{\mathrm{MAD}}{c_5(n)}\text{ and }\frac{\mathrm{Shamos}}{c_6(n)},$$

$$\mathrm{RE}({\hat{\theta }}_2|{\hat{\theta }}_1)=\frac{\mathrm{Var}({\hat{\theta }}_1)}{\mathrm{Var}({\hat{\theta }}_2)}.$$

$$\mathrm{ARE}({\hat{\theta }}_2|{\hat{\theta }}_1)=\underset{n\to \infty }{\lim }\mathrm{RE}({\hat{\theta }}_2|{\hat{\theta }}_1).$$

$$n\mathrm{Var}(\mathrm{median})$$

Full text

In this paper, we fill this gap by providing their closed-form finite-sample breakdown points and calculating the unbiasing factors and relative efficiencies of the robust estimators through the extensive Monte Carlo simulations up to the sample size 100. The numerical study shows that the unbiasing factor improves the finite-sample performance significantly. In addition, we provide the predicted values for the unbiasing factors obtained by using the least squares method which can be used for the case of sample size more than 100.

KEYWORDS: breakdown, unbiasedness, robustness, relative efficiency

1 Introduction

Estimation of the location and scale parameters of a distribution, such as the mean and standard deviation of a normal population, is a common and important problem in the various branches of science and engineering including: biomedical, chemical, materials, mechanical and industrial engineering, etc. The quality of the data plays an important role in estimating these parameters. In practice, the experimental data are often contaminated due to the measurement errors, the volatile operating conditions, etc. Thus, robust estimations are advocated as alternatives to commonly used location and scale estimators (e.g., the sample mean and sample standard deviation) for estimating these parameters. For example, for the case where some of the observations are contaminated by outliers, we usually adopt the sample median and Hodges-Lehmann (Hodges and Lehmann,, 1963) estimators for location and the sample median absolute deviation (Hampel,, 1974) and Shamos (Shamos,, 1976) estimators for scale, because these estimators have a large breakdown point and perform well in either the presence or absence of outliers.

The breakdown point is a common criterion for measuring the robustness of an estimator. The larger the breakdown point of an estimator, the more robust it is. The finite-sample breakdown point based on the idea of Hodges, Jr., (1967) is defined as the maximum proportion of incorrect or arbitrarily observations that an estimator can deal with without making an egregiously incorrect value. The breakdown point of an estimator is a measure of its resistance to data contamination (Wilcox,, 2016). For example, the breakdown points of the sample mean and the sample median are 0 and $1/2$, respectively. Given that the breakdown point can generally be written as a function of the sample size, we provide the finite-sample breakdown points for the various location and scale estimators mentioned above. It is shown that when the sample sizes are small, they are noticeably different from the asymptotic breakdown point, which is the limit of the finite-sample breakdown point when the sample size tends to infinity.

It deserves mentioning that for a robust scale estimation, the sample median absolute deviation (MAD) and Shamos estimators not only have positive asymptotic breakdown points, but also are Fisher-consistent (Fisher,, 1922) as the sample size goes to infinity. However, when the sample size is small, they have serious biases and thus provide inappropriate estimation. Some bias-correction techniques are commonly adopted to improve the finite-sample performance of these estimators. For instance, Williams, (2011) studied the finite-sample correction factors through computer simulations for several simple robust estimators of the standard deviation of a normal population, which include the MAD, interquartile range, shortest half interval, and median moving range. Later on, under the normal distribution, Hayes, (2014) obtained the finite-sample bias-correction factors of the MAD for scale. They have shown that finite-sample bias-correction factors can significantly eliminate systematic biases of these robust estimators, especially when the sample size is small.

To the best of our knowledge, finite-sample properties of the MAD and Shamos estimators have received little attention in the literature except for some references covering topics for small sample sizes. This observation motivates us to employ the extensive Monte Carlo simulations to obtain the empirical biases of these estimators. Given that the empirical variance of an estimator is one of the important metrics for evaluating an estimator, we also obtain the values of the finite-sample variances of the median, Hodges-Lehmann (HL), MAD and Shamos estimators under the standard normal distribution, which are not fully provided in the statistics literature. Numerical results show that the unbiasing factor improves the finite-sample performance of the estimator significantly. In addition, we provide the predicted values for the unbiasing factors obtained by the least squares method which can be used for the case of the sample size more than 100.

The rest of this paper is organized as follows. In Section 2, we derive the finite-sample breakdown points for robust location estimators and robust scale estimators, respectively. Through the extensive Monte Carlo simulations, we calculate the empirical biases of the MAD and Shamos estimators in Section 3 and the finite-sample variances of the median, HL, MAD, and Shamos estimators in Section 4. An example is provided to illustrate the application of the proposed methods to statistical quality control charts for averages and standard deviations in Section 5. Some concluding remarks are given in Section 6.

2 Finite-sample breakdown point

In this section, we introduce the definitions of the bias and breakdown of an estimator $\hat{\theta}$ using Definition 1.6.1 of Hettmansperger and McKean, (2010) and Section 2 of Hampel et al., (1986), which is based on the idea of Hodges, Jr., (1967). According to these definitions, we derive the explicit finite-sample breakdown points of the robust estimators for location and scale parameters.

Definition 1** (Bias).**

Let $\mathbf{x}=\{x_{1},x_{2},\ldots,x_{n}\}$ be a realization of a sample $\mathbf{X}=\{X_{1},X_{2},\ldots,X_{n}\}$. We denote the corrupted sample of any $m$ of the $n$ observations as

[TABLE]

The bias of an estimator $\hat{\theta}$ is defined as

[TABLE]

where the $\sup$ is taken over all possible corrected observations $x_{i}^{*}$ with $x_{i}$ being fixed at their original observations in the realization of $\mathbf{x}_{(m)}^{*}$.

If $\mathrm{Bias}(\hat{\theta};\mathbf{x}_{(m)}^{*},\mathbf{x})$ makes an arbitrarily large value (say, $\infty$), we say that the estimator $\hat{\theta}$ has broken down.

Definition 2** (Breakdown).**

We define the finite-sample breakdown as

[TABLE]

This approach to breakdown is called $\epsilon$-replacement breakdown because observations are replaced by corrupted values. For more details, readers are referred to (Donoho and Huber,, 1983), Subsection 2.2a of Hampel et al., (1986), and Subsection 1.6.1 of Hettmansperger and McKean, (2010). The sample median and HL estimators for robust location estimation are considered in Subsection 2.1 and the MAD and Shamos estimators for robust scale estimation are in Subsection 2.2.

2.1 Robust location estimators

It is well known that the asymptotic breakdown point of the sample median is 1/2; see, for example, Example 2 in Section 2.2 of Hampel et al., (1986). For the HL estimator, the asymptotic breakdown point is given by $1-1/\sqrt{2}\approx 0.293$; see Example 8 of Section 2.3 of Hampel et al., (1986) for more details. Note that the sample median and HL estimators are in closed-forms and are location-equivariant in the sense that

[TABLE]

However, in many cases, the finite-sample breakdown points can be noticeably different from the asymptotic breakdown point, especially when the sample size $n$ is small. For instance, when $n=10$, we observe from Equation (2) that the finite-sample breakdown point for the median is 0.4, which is different from its asymptotic breakdown point of 0.5. This motivates the need for further investigating the finite-sample properties of these robust estimators.

Suppose that we have a sample, $\{X_{1},X_{2},\ldots,X_{n}\}$. If the sample size $n$ is given by an odd number (say, $n=2k+1$), the median can be resistant to $k$ corrupted observations. Thus, the finite-sample breakdown point of the median with $n=2k+1$ is given by $\epsilon_{n}=k/n$. Next, if the sample size $n$ is given by an even number (say, $n=2k$), then the median can be resistant to $k-1$ corrupted observations which gives $\epsilon_{n}=(k-1)/n$. Let $\lfloor\cdot\rfloor$ be the floor function. That is, $\lfloor x\rfloor$ is the largest integer not exceeding $x$. Then we have $\lfloor(n-1)/2\rfloor=k$ for $n=2k+1$ and $\lfloor(n-1)/2\rfloor=k-1$ for $n=2k$ so that the finite-sample breakdown point can be written by the single formula $\lfloor(n-1)/2\rfloor/n$ for both odd and even cases. The finite-sample breakdown point of the median is thus given by

[TABLE]

It should be noted that Equation (1.6.1) of Hettmansperger and McKean, (2010) defines the finite-sample breakdown point to be \epsilon_{n}^{\star}=\min\big{\{}m/n:\mathrm{Bias}(\hat{\theta};\mathbf{x}_{(m)}^{*},\mathbf{x})=\infty\big{\}}. According to this definition, the finite-sample breakdown point of the median is given by $\epsilon_{n}^{\star}={\lfloor(n+1)/2\rfloor}/{n}$ which is always greater than $1/2$ for any odd value of $n$. For example, $\epsilon_{n}^{\star}=2/3$ with $n=3$. Also, for the sample mean, $\epsilon_{n}^{\star}$ is always $1/n$, not zero. For example, $\epsilon_{n}^{\star}=1/2$ with $n=2$ although its asymptotic breakdown point is zero. On the other hand, based on the finite-sample breakdown point in Equation (1), we have $\epsilon_{n}=1/3$ with $n=3$ for the median and $\epsilon_{n}=0$ always for the sample mean. It deserves mentioning that there are several variants of the breakdown point. We refer the interested readers to Remark 1 in Section 2.2 of Hampel et al., (1986) and Chapter 11 of Huber and Ronchetti, (2009) for more details. In this paper, we follow Equation (1) in Definition 2 to define the finite-sample breakdown.

Using the fact that $\lfloor x\rfloor$ can be rewritten as $\lfloor x\rfloor=x-\delta$ where $0\leq\delta<1$, we have

[TABLE]

Thus, the asymptotic breakdown point of the median is obtained by taking the limit of the finite-sample breakdown point as $n\to\infty$, which provides that $\epsilon=\lim_{n\to\infty}\epsilon_{n}=1/2$.

The HL estimator (Hodges and Lehmann,, 1963) is defined as the median of all pairwise averages of the sample observations and is given by

[TABLE]

Note that the median of all pairwise averages can be calculated for the three cases: (i) $i<j$, (ii) $i\leq j$, and (iii) $\forall(i,j)$, where $i,j=1,2,\ldots,n$. We denoted these three versions as

[TABLE]

respectively. Note that each of the paired averages $(X_{i}+X_{j})/2$ where $i\leq j$ and $i,j=1,2,\ldots,n$ is called a Walsh average (Walsh,, 1949). In what follows, we first derive the finite-sample breakdown point for the $\mathrm{HL3}$ and then use a similar approach to derive the finite-sample breakdown points for $\mathrm{HL1}$ and $\mathrm{HL2}$. It is noteworthy that all three versions are asymptotically equivalent as mentioned in Example 3.7 of Huber and Ronchetti, (2009) although they are practically different with a sample of finite size.

The basic idea of how to derive the finite-sample breakdown point for the HL estimators was proposed by Ouyang et al., (2019), but they did not provide explicit formulae. In this paper, we provide the method of how to derive the explicit formulae. Suppose that we make $k$ of the $n$ observations arbitrarily large, where $k=0,1,2,\ldots,n$. Notice that there are $n\times n$ paired average terms in the HL3 estimator: $(X_{i}+X_{j})/2$, where $i,j=1,2,\ldots,n$. Because the HL3 estimator is the median of the $n\times n$ values, the finite-sample breakdown point cannot be greater than $\lfloor(n^{2}-1)/2\rfloor/n^{2}$ due to Equation (2). If we make $k$ of the $n$ observations arbitrarily large, then the number of arbitrarily large paired averages becomes $n^{2}-(n-k)^{2}$. These two facts provide the following relationship

[TABLE]

which is equivalent to $k^{2}-2nk+\lfloor(n^{2}-1)/2\rfloor\geq 0$. The finite-sample breakdown point of the $\mathrm{HL3}$ is then given by $\epsilon_{n}=k^{*}/n$, where

[TABLE]

To obtain an explicit formula for Equation (3), we let $f(x)=x^{2}-2nx+\lfloor(n^{2}-1)/2\rfloor$. Since $f^{\prime}(x)=2(x-n)$, $f(x)$ is decreasing for $x<n$. The roots of $f(x)=0$ are given by

[TABLE]

Since $k$ is an integer and $k\leq n$, we have $k^{*}=\lfloor x_{1}\rfloor$, that is, k^{*}=\Big{\lfloor}n-\sqrt{n^{2}-\lfloor(n^{2}-1)/2\rfloor}\Big{\rfloor}. Then we have the closed-form finite-sample breakdown point of the $\mathrm{HL3}$

[TABLE]

The asymptotic breakdown point of $\mathrm{HL3}$ is given by $\epsilon=\lim_{n\to\infty}\epsilon_{n}$. Using $\lfloor x\rfloor=x-\delta$ where $0\leq\delta<1$, we can rewrite Equation (4) as

[TABLE]

where $0\leq\delta_{1}<1$ and $0\leq\delta_{2}<1$. Thus, we have $\epsilon=1-1/\sqrt{2}\approx 29.3\%$.

In the case of the $\mathrm{HL1}$ estimator, there are $n(n-1)/2$ Walsh averages. Since the $\mathrm{HL1}$ estimator is the median of the $n(n-1)/2$ Walsh averages, the finite-sample breakdown point cannot be greater than $\lfloor\{n(n-1)/2-1\}/2\rfloor/\{n(n-1)/2\}$ due to Equation (2) again. If we make $k$ observations arbitrarily large with $k\leq n-1$, then there are $n(n-1)/2-(n-k)(n-k-1)/2$ arbitrarily large Walsh averages. Thus, the following inequality holds

[TABLE]

which is equivalent to $k^{2}-(2n-1)k+2\lfloor(n^{2}-n-2)/4\rfloor\geq 0.$ In a similar way as done for Equation (3), we let $k^{*}$ be the largest integer $k$ satisfying the above, where $k=0,1,2,\ldots,n-1$. For convenience, we let $f(x)=x^{2}-(2n-1)x+2\lfloor(n^{2}-n-2)/4\rfloor$. Then $f(x)$ is decreasing for $x<n-1/2$ due to $f^{\prime}(x)=2x-(2n-1)$ and the roots of $f(x)=0$ are given by $n-1/2\pm\sqrt{(n-1/2)^{2}-2\lfloor(n^{2}-n-2)/4\rfloor}$. Thus, using the similar argument to that used for the $\mathrm{HL3}$ case, we have k^{*}=\Big{\lfloor}n-1/2-\sqrt{(n-1/2)^{2}-2\lfloor(n^{2}-n-2)/4\rfloor}\Big{\rfloor}. Then we have the closed-form finite-sample breakdown point of the $\mathrm{HL1}$

[TABLE]

It should be noted that we also have $\epsilon=\lim_{n\to\infty}\epsilon_{n}=1-1/\sqrt{2}$.

Similar to the case of the $\mathrm{HL1}$, we obtain that the closed-form finite-sample breakdown point of the $\mathrm{HL2}$ estimator is given by

[TABLE]

In Table 2, we provide the finite-sample breakdown points of the three HL estimators along with the median, the MAD and the Shamos estimators, which will be covered in the following subsection. Also we provide the plot of these values in Figure 1. By comparing these values, we observe that

[TABLE]

2.2 Robust scale estimators

For robust scale estimation, we consider the MAD (Hampel,, 1974) and the Shamos estimator (Shamos,, 1976). The MAD is given by

[TABLE]

where $\tilde{\mu}=\mathrm{median}(X_{i})$ and $\Phi^{-1}({3}/{4})$ is needed to make this estimator Fisher-consistent (Fisher,, 1922) for the standard deviation under the normal distribution. For more details, see Example 4 in Section 2.3 of Hampel et al., (1986) and Rousseeuw and Croux, (1992, 1993). This resembles the median and its finite-sample breakdown point is the same as that of the median in (2). The Shamos estimator is given by

[TABLE]

where $\sqrt{2}\,\Phi^{-1}(3/4)$ is needed to make this estimator Fisher-consistent for the standard deviation under the normal distribution (Lèvy-Leduc et al.,, 2011).

Of particular note is that the Shamos estimator resembles the HL1 estimator by replacing the Walsh averages by pairwise differences. Thus, its finite-sample breakdown point is the same as that of the HL1 estimator in (5). In the case of the HL estimator, the median is calculated for $i<j$, $i\leq j$, and $\forall(i,j)$, but the median in the Shamos estimator is calculated only for $i<j$ because $|X_{i}-X_{j}|=0$ for $i=j$. Note that the MAD and Shamos estimators are in a closed-form and are scale-equivariant in the sense that

[TABLE]

As afore-mentioned, we also provide the values the finite-sample breakdown points of the Shamos estimator in Table 2 and the plot of these values in Figure 1.

3 Empirical biases

As mentioned above, the MAD in (7) and the Shamos estimator in (8) are Fisher-consistent for the standard deviation under the normal distribution, that is, as the sample size goes to infinity, it converges to the standard deviation $\sigma$ under the normal distribution, $N(\mu,\sigma)$. However, due to the cost of sampling and measurements in practical applications, we often need to find parameter estimates from data with small sample sizes; see, for example, Williams, (2011), whereas these estimates have serious biases, especially when the sample sizes are small. To the best of our knowledge, an empirical investigation of finite sample correction factors of these estimates has not been well studied in the literature. This observation motivates us to obtain the unbiasing factors for the MAD and Shamos estimators through the Monte Carlo simulation methods (Rousseeuw and Croux,, 1992). In what follows, we discuss how to conduct simulations under two cases: $2\leq n\leq 100$ and $n>100$.

For the case when $2\leq n\leq 100$, we generated a sample of size $n$ from the standard normal distribution, $N(0,1)$, and calculated the MAD and Shamos estimates. We repeated this simulation ten million times ($I=10^{7}$) to obtain the empirical biases of these two estimators. These values are provided in Table 3 and can also be seen in Figure 2. It is clear that an empirically unbiased MAD is given by

[TABLE]

where $A_{n}$ is the empirical bias of the $\mathrm{MAD}$ under the standard normal distribution. Similarly, an empirically unbiased Shamos estimator is given by

[TABLE]

where $B_{n}$ is the empirical bias of the Shamos estimator under the standard normal distribution.

For the case when $n>100$, we suggested to use the methods proposed by Hayes, (2014) and Williams, (2011). To be more specific, Hayes, (2014) suggested the use of $A_{n}\approx{a_{1}}/{n}+{a_{2}}/{n^{2}}$ and Williams, (2011) suggested the use of $A_{n}\approx{a_{3}}{n}^{-a_{4}}$. Similarly, we can estimate $B_{n}$ using $B_{n}\approx{b_{1}}/{n}+{b_{2}}/{n^{2}}$ and $B_{n}\approx{b_{3}}{n}^{-b_{4}}$. These suggestions are quite reasonable, because the MAD in (7) and Shamos in (8) are Fisher-consistent, and $A_{n}$ and $B_{n}$ converge to zero as $n$ goes to infinity.

Using the methods proposed by Hayes, (2014) and Williams, (2011), we obtained the least squares estimate $A_{n}$ as follows:

[TABLE]

and

[TABLE]

respectively. In a similar fashion, the least squares estimate $B_{n}$ using Hayes, (2014) and Williams, (2011) is given by

[TABLE]

respectively. The empirical biases of the MAD and Shamos estimators are provided in Table 4 for $n=109,110,119,120,\ldots,499,500$. We observed that the estimated biases are very accurate up to the fourth decimal point. Note that Hayes, (2014) and Williams, (2011) estimated $A_{n}$ for the case of odd and even values of $n$ separately. However, as one expects, for a large value of $n$, the gain in precision may not be noticeable as shown in Figure 2.

It is well known that the sample standard deviation $S_{n}$ is not unbiased under the normal distribution. To make it unbiased, the unbiasing factor $c_{4}(n)$ is widely used in engineering statistics including control charting so that $S_{n}/c_{4}(n)$ is unbiased, where $c_{4}(n)$ is given by

[TABLE]

It is worth mentioning that this $c_{4}$ notation was originally used in ASTM E11, (1976) and it has since then been a standard notation in the engineering statistics literature.

We suggest to use $c_{5}(n)$ and $c_{6}(n)$ notations for the unbiasing factors of the MAD and Shamos estimators, respectively. Consequently, we can obtain the unbiased MAD and Shamos estimators for any value of $n$ given by

[TABLE]

where $c_{5}(n)=1+A_{n}$ and $c_{6}(n)=1+B_{n}$.

4 Empirical variances and relative efficiencies

In this section, through the extensive Monte Carlo simulations, we calculate the finite-sample variances of the median, Hodges-Lehmann (HL1, HL2, HL3), MAD and Shamos estimators under the standard normal distribution. We generated a sample of size $n$ from the standard normal distribution and calculated their empirical variances for a given value of $n$. We repeated this simulation ten million times ($I=10^{7}$) to obtain the empirical variance for each of $n=2,3,\ldots,100$.

It should be noted that the values of the asymptotic relative efficiency (ARE) of various estimators are known. Here, the ARE is defined as the limit of the relative efficiency (RE) as $n$ goes to infinity. The RE of $\hat{\theta}_{2}$ with respect to $\hat{\theta}_{1}$ is defined as

[TABLE]

where $\hat{\theta}_{1}$ is often a reference or baseline estimator. Then the ARE is given by

[TABLE]

For example, under the normal distribution, we have $\mathrm{ARE}(\mathrm{median}|\bar{X})={2}/{\pi}\approx 0.6366$, $\mathrm{ARE}(\mathrm{HL}|\bar{X})={3}/{\pi}\approx 0.9549$, $\mathrm{ARE}(\mathrm{MAD}|S_{n})=0.37$, and $\mathrm{ARE}(\mathrm{Shamos}|S_{n})=0.863$, where $\bar{X}$ is the sample mean and $S_{n}$ is the sample standard deviation. For more details, see Serfling, (2011) and Lèvy-Leduc et al., (2011).

Note that with a random sample of size $n$ from the standard normal distribution, we have $\mathrm{Var}(\bar{X})=1/n$ and $\mathrm{Var}(S_{n})=1-c_{4}(n)^{2}$, where $c_{4}(n)$ is given in (9). Thus, we have

[TABLE]

for a large value of $n$. We provided these values for each of $n$ in Tables 5 and 6 and also plotted these values in Figure 4.

For the case when $n>100$, we suggest to estimate these values based on Hayes, (2014) or Williams, (2011) as we did the biases in the previous section. We suggest the following models to obtain these values for $n>100$:

[TABLE]

One can also use the method based on Williams, (2011). For brevity, we used the method based on Hayes, (2014). To estimate these values for $n>100$, we obtained the empirical REs in Table 7 for $n=109$, $110$, $119$, $120$, $\ldots$, $499$, $500$. We observe from Figure 4 that it is reasonable to estimate the values for the median and MAD in the case of odd and even values of $n$ separately. Using the large values of $n$, we can estimate the above coefficients. For this reason, we use the simulation results in Tables 6 and 7. Then the least squares estimates based on the method of Hayes, (2014) are given by

[TABLE]

In Tables 8 and 9, we also calculated the REs of the afore-mentioned estimators for $n=1,2,\ldots,100$ using the above empirical variances. For $n>100$, we can easily obtain the REs using the above estimated variances. It should be noted that the REs of the median, HL2 and HL3 are exactly one for $n=1,2$. When $n=1,2$, the median, HL2 and HL3 are essentially the same as the sample mean.

Another noticeable result is that the RE of the HL1 is exactly one especially for $n=4$. When $n=4$, the HL1 is the median of $(X_{1}+X_{2})/2$, $(X_{1}+X_{3})/2$, $(X_{1}+X_{4})/2$, $(X_{2}+X_{3})/2$, $(X_{2}+X_{4})/2$ and $(X_{3}+X_{4})/2$. Then this is the same as the median of $(X_{(1)}+X_{(2)})/2$, $(X_{(1)}+X_{(3)})/2$, $(X_{(1)}+X_{(4)})/2$, $(X_{(2)}+X_{(3)})/2$, $(X_{(2)}+X_{(4)})/2$ and $(X_{(3)}+X_{(4)})/2$, where $X_{(i)}$ are order statistics. Because

[TABLE]

we have

[TABLE]

Thus, the RE of the HL1 should be exactly one. In this case, as expected, the finite-sample breakdown is zero as provided in Table 2.

It should be noted that the $\mathrm{MAD}/c_{5}(n)$ and $\mathrm{Shamos}/c_{6}(n)$ are unbiased for $\sigma$ under the normal distribution, but their square values are not unbiased for $\sigma^{2}$. Using the empirical and estimated variances, we can obtain the unbiased versions as follows. For convenience, we denote $v_{5}(n)=\mathrm{Var}(\mathrm{MAD})$ and $v_{6}(n)=\mathrm{Var}(\mathrm{Shamos})$, where the variances are obtained using a sample of size $n$ from the standard normal distribution $N(0,1)$ as mentioned earlier. Since the MAD and Shamos estimators are scale-equivariant, we have $\mathrm{Var}(\mathrm{MAD})=v_{5}(n)\sigma^{2}$ and $\mathrm{Var}(\mathrm{Shamos})=v_{6}(n)\sigma^{2}$ with a sample from the normal distribution $N(\mu,\sigma^{2})$. It is immediate from Equation (10) that $E(\mathrm{MAD})=c_{5}(n)\sigma$ and $E(\mathrm{Shamos})=c_{6}(n)\sigma$. Considering $E(\hat{\theta}^{2})=\mathrm{Var}(\hat{\theta})+E(\hat{\theta})^{2}$, we have

[TABLE]

Thus, the following estimators are unbiased for $\sigma^{2}$ under the normal distribution

[TABLE]

5 Example

In this section, we provide an example to illustrate the application of the proposed methods to statistical quality control charts (Shewhart,, 1926) for averages and standard deviations. For more details, see Section 3.8 of ASTM E11, (2018). Three-sigma control limits are most widely used to set the upper and lower control limits in statistical control charts due to Shewhart, (1931).

We briefly introduce how to construct the statistical control charts and then propose robust alternatives using the proposed methods. In general, the construction of statistical quality control charts is involved with two phases: Phase-I and Phase-II (Vining,, 2009; Montgomery,, 2013). In Phase-I, the goal is to establish reliable control limits with a clean set of process data. In Phase-II, using the control limits obtained in Phase-I, we monitor the process by comparing the statistic for each successive subgroup as future observations are collected. The Phase-II control charts are based on the estimates in Phase-I and the performance of the Phase-II control chart relies on the accuracy of the Phase-I estimate. Thus, if the Phase-I control limits are estimated with a corrupted data set, this can produce an adverse effect in Phase-II. It is well known that the sample mean and standard deviations are highly sensitive to data contamination and can even be destroyed by a single corrupted value (Rousseeuw,, 1991). Thus, the conventional control charts based on these statistics are very sensitive to contamination. This problem can be easily overcome by using the proposed robust methods.

As an illustration, we deal with the control limits of the conventional control charts for averages and standard deviations. Suppose that the mean and standard deviation are given by $\mu$ and $\sigma$ with a sample of size $n$. Then the parameters of the control limits of the $\bar{x}$ chart are given by $\mu+3\sigma/\sqrt{n}$ (UCL) and $\mu-3\sigma/\sqrt{n}$ (LCL). We assume that we have $k$ subgroups which is of size $n$. Let $X_{ij}$ be the $i$th sample, where $i=1,2,\ldots,k$ and $j=1,2,\ldots,n$ and $x_{ij}$ be its realization. Let $\bar{x}_{i}$ and $s_{i}$ be the sample mean and standard deviation of the $i$th sample, respectively. Then the estimates of $\mu\pm 3\sigma/\sqrt{n}$ are given by $\bar{\bar{x}}\pm A_{3}(n)\bar{s}$, where $\bar{\bar{x}}=\sum_{i=1}^{k}\bar{x}_{i}/k$, $\bar{s}=\sum_{i=1}^{k}s_{i}/k$, and $A_{3}(n)=3/c_{4}(n)\sqrt{n}$; see ASQC and ANSI, (1972) for the notation $A_{3}(n)$. Note that an estimate of $3\sigma$ is given by $\sqrt{n}A_{3}(n)\bar{s}$ since $A_{3}(n)\bar{s}$ is an estimate of $3\sigma/\sqrt{n}$.

As obtained in (10), the unbiased MAD and Shamos estimators for any value of $n$ are given by

[TABLE]

respectively. For convenience, we let $A_{5}(n)=3/c_{5}(n)\sqrt{n}$ and $A_{6}(n)=3/c_{6}(n)\sqrt{n}$. Let $\mathrm{MAD}_{i}$ and $\mathrm{SH}_{i}$ be the MAD and Shamos estimates of the $i$th sample, respectively. Then robust analogues of $3\sigma$ can be obtained as

[TABLE]

where $\overline{\mathrm{MAD}}=\sum_{i=1}^{k}\mathrm{MAD}_{i}/k$ and $\overline{\mathrm{SH}}=\sum_{i=1}^{k}\mathrm{SH}_{i}/k$.

As mentioned above, the conventional limits are very sensitive to data contamination. For example, when there are outlying observations, the control limits are moved outward from the center line, which makes it difficult to monitor any out-of-control signal. Since the three-sigma plays an important role in constructing the charts, we investigate the estimates of $3\sigma$. To this end, we carry out a simulation. We assume that $X_{ij}$ are normally distributed with $N(\mu,\sigma^{2})$ where $\mu=5$ and $\sigma=1$ and there are $k=10$ subgroups which are all of size $n=5$. We repeated this simulation ten thousand times.

To investigate the robust properties of the proposed methods, we corrupted the observation $X_{11}$ by adding $\delta$ to $X_{11}$, where $\delta=0,1,2,\ldots,50$. We provide the empirical biases, variances and mean square errors of the estimates of $3\sigma$ in Table 1 and Figure 3. For brevity, we provide the results only for $\delta=0,10,20,30,40,50$ in the table. In the figure, the solid lines denote the unbiased estimates after the finite-sample correction. From the results, if there is no contamination ($\delta=0$), the unbiased/original standard deviations clearly perform the best. The unbiased version is slightly better than the original standard deviation. Thus, the unbiasing factor, $c_{4}(n)$, improves the bias and MSE. However, if there exists data contamination, the unbiased/original standard deviations started to break down as the corruption level $\delta$ increases. We also observe the unbiasing factor, $c_{4}(n)$, does not improve either the bias or the MSE unless $\delta$ is very small.

On the other hand, the robust alternatives (MAD and Shamos) do not break down with the considered corruption as expected. We also note that the unbiasing factors, $c_{5}(n)$ and $c_{6}(n)$, could help the bias and MSE decrease even when there is contamination. Especially, the original Shamos estimator is much improved with the unbiasing factor $c_{6}(n)$. Note that the MSE of the unbiased standard deviation is $0.12023$, while that of the unbiased Shamos is $0.16093$ when there is no contamination. This shows that the two estimators are quite comparable even when there is no contamination. In summary, the unbiased Shamos is the best choice in this illustrative example.

6 Concluding remarks

In this paper, we studied the finite-sample properties of the sample median and Hodges-Lehmann estimators for location and the MAD and Shamos estimators for scale. We obtained the closed-form finite-sample breakdown points of these estimators for the population parameters. We also calculated the unbiasing factors and relative efficiencies of the MAD and the Shamos estimators for the scale parameter through the extensive Monte Carlo simulations up to the sample size of 100. The numerical study showed that the unbiasing factor significantly improves the finite-sample performance of these estimators under consideration. In addition, we provided the predicted values for the unbiasing factors obtained by using the least squares method which can be used for the case of sample size more than 100. To facilitate the implementation of the proposed methods into various application fields, Park and Wang, (2019) developed the R package library which is published in the CRAN (R Core Team,, 2019).

According to the application of the proposed methods to statistical quality control charts for averages and standard deviations, it would be useful to develop robustified alternatives to traditional Shewhart-type control charts because its control limits are also constructed based on the unbiased estimators under the normal distribution. We obtained the empirical and estimated biases of several commonly used robust estimators under the normal distribution, which can be widely used for bias-corrections of the estimates of the parameters. For other parametric models, one can obtain empirical and estimated biases with appropriate robust estimators as suggested in this paper. Given that the breakdown point used in this paper does not depend on a probability model as pointed out in Subsection 2.2a of Hampel et al., (1986), it is of particular interest to obtain biases and relative efficiencies under other parametric models.

Acknowledgment

The authors greatly appreciate the reviewers for their thoughtful comments and suggestions, which have significantly improved the quality of this manuscript. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. NRF-2017R1A2B4004169).

Appendix: Tables and Figures