"':~
A ~10fl:TE CARLO STUDY OF ROBUST ESTI~1;UORS OF LOCATION
by
Raymond J. Carroll
and
Edlvard J. We gLla1l1
Department of Stat.istics
University of North Carolina at Chapel Hill
Institute of Statistics Ivlir1eo Series #1040
Hovember, 1975
1 Tne work of this author was sw~ported by the Air Force Office of Scientific
Research l.ll1der grant number i\FOSR-75-2840. The developnent of many of the
computer routines was also supported by AFOSR-7S-2849.
ABSTRACT
Andrews et al (1972) carried out an extensive Nonte c;arlo study of robust
estimators of location.
Their conclusions were that the hampel and the skipped
estimates, as classes, seemed to be preferable to some of the other currently
fashionable estimators.
The present study extends this
tors not previously examined.
~rork
to include
est~z-
The estimators are compared over short-tailed as
well as long-tailed alternatives and also over some dependent data generated by
first-order autoregressive schemes.
threefold.
First~
rnle conclusions of the present study are
from our lUuited study, none of the so-called robust esti-
mators are very robust over short-tailed situations.
necessary in this situation.
Second~
fibre \'lork seems to be
none of the estimators perform very well
in dependent data situations, particularly when the correlation is large and
positive.
TI1is seems to be a rather pressing problem.
tailed alternatives, the hampel
estim~tors
the hampels are the strongest classes.
Finally, for long-
and Hogg-type
~laptive
versions of
111e adaptive ha'TIpels neither unifonnly
outperfOITtl nor are they outperformed by the hampels.
However, the superiority
in terms of maximtnn relative efficiency goes to the adaptive hampels.
That is
the adaptive hampels; under their worst perfonnance ~ are superior to the usual
hampels, under their worst performance.
Key words and phrases:
M-estimators, hampel estimators, trinmled means, onestep M-estimators, adaptive estimation, rfuJUc estimators~
robustness ~ r10nte Carlo simulation, sensitivity curves,
influence curves, relative efficiency, small sample,
long-tails; short-tails, time series, first-order autoregressive, dependent data.
A.M.S. Classification Numbers:
Primary 62G35; Secondary 62COS, 62FIO, 62GOS.
1.
Introduction.
In 1972, Andrews, Bickel, Hampel, Huber, Rogers and Tukey
published a very extensive and infom.ative lIonte Carlo study of robust estimation of location in a syrnmetric probability dens itYo
68 estimators of location as well as 14 distinct
sizes used \\Tere
5? 10? 20
and
tions only the sample size of
40
This study involved sone
sml~lll1g
distributions.
SaIT;J!c
although for nost of the salTlpling situa-
20 was investigated.
For the estimators and
sampling situ<:1.tions exa'1lineu, Andrews et a1 (1972) provide a very complete and
satisfacto~/
picture.
'ire are Iilotivated, however, to supplement this study for three basic
reasons.
In Andrevls et al (1972), hereafter referred to as PRS (Princeton
Robustness
Study)~
short-tailed saw.pIing situatimls are ruled out of considera-
tion with the statement, IlRobustness for short-tailed distributions was thought
to be a rather special case, arising in situations that are usually rather
' d ill
. practIceo
,
1'1
eaSI'yI
reeovuze
'
S'Ince sorle
0f
'
,
t 11e 1ocatlOn
estImators
were
designed to protect against short-tailed alternatives as well as long-tailed
possibilities, we feel that examination of long-tailed alternatives alone
faults such estinators mfairly, rather akin to discovering a two-sided test is
not most powerful against one- sided alternatives.
Even ignoring this aspect>
the question renains, IHhat are d.esirable estimiJ.tors given short-tailed alternatives?11
A second motivation \'las prOVided by the rather disoa.l situation in the
time series
context.
The sample mean is routinely shmvn to be a consistent
estiliiator of the "'center!; of a stationar/ tll::'e series with the clear iJi1plicatioY'
I
PRS, p. 67.
2
that a time series is centered by subtracting the sample mea...'1.
If the sample
mean is lIDsatisfactory for L"'1dependent 7 but non-nonual data 7 how much worse
must it be for correlated data?
Our third motivation arises from a personal conviction that adaptive estimators, properly fOTm'Jlated, ought to be very
successfv~.
In particular, we
observe that because of the contributions of Professors Ea.'1!pel and I-luber to the
PRS,;
the so-called hampel and closely related Pl--estimators were extensively
studied.
At least 25 of the 68 estimators studied L"1volve either a hampel or
a M-estinator.
The I.1-estimators and particularly the harrrpels perform very
well both because they are innately good estimators and because they were finetuned.
111e adaptive estimators 9 hOli/ever, were not similarly fine tuned a...TJ.d do
not fare as well in the final analysis.
To state our conviction succintly, if
hanpels are good, adaptive hampels should be better.
The paper is divided into six parts.
Section 2 lists the estimators
studied in this paper and gives a short discussion of each.
Section 3 discusse
the details of the Honte Carlo aspects of this work while section 4 contains
the tables of results.
Section 5 presents stylized sensitivity curves for a
selection of the estimators and finally section 6 contains our reactions to and
conclusions about the results.
2.
The Estimators.
In choosinB
were guided by the results of the
\~1ich
PRS.
estinatoTs to exar,ine in this study, we
As
sta."ldards of cO!'1parison we chose
I'I-estimators and the related haItTpels and the trir.'ned ne3.t'1s.
standards,
we
J
Against these
measured various fonus of adaptive estimators, one-step rank
3
estimators and several miscellaneous estimators - including the Hodges-Lehmann,
Jo1ms'
~
Gastwirth' s 1 Andrews i and an estimator based on skipping.
We follow
the routine established in the PRS by listi."'113 the estimators together wi til
their mnemonic codes in the follmving table.
TIlators follows the table.
He note here that the codes describe<.1 below agree
with those in the PRS when an estJ.Jmtor is
Nur.1ber
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
•5 ~a
19%
25%
COIT,10n
to both works.
Short Description
Code
10 96
A short description of the esti-
5% symmetrically trimmed ll,ean
10% syLD':1.etrically trimmed 1'1ean
38%
13.75% symmetrically trimmed mean
25% syrlmetrical1y tr:ir.nned wean (ll1idrllcan)
37.5% synnnetrica11y tril'1IlleU mean
50%
50% symnetrically trimmed mean (median)
M real}
OM Outer mearl, meati of trimr~ings after 25% symmetrical triDming
DIO One- step Huber ~ k=1. 0, start = mediaii
PIS One- step I-iuber, k=l. 5 ~ start = median
1)20 One- step Huber, k=2. 0 ~ start = mediaD.
12A II-estimate, ljJ bends at 1.2~ 3.5;; 8.0
17A II-estirnate? ljJ bends at 1.7~ 3.4~ 8.5
21A I,i-estimate, ljJ bends at 2.1;. 4.0, J. 2
22A iI-estimate, lV bends at 2. 2 ~ 3. 7, 5.9
2SA hl-estir:J.ate, ljJ bends at 2.5, 4:;5 J 9.5
Adaptive N-estiw~te~ ljJ bends at ADA~ 4.5_ 8.0
HGI Hogg-type adaptor using trimmed mea,il.S 38%? 19% 'I rI, or'II
HG2 Hogg-tY'pe adaptor using triImned means 38%" 25%" 10%
HG.3 Hogg-type adaptor using trir~ed neans 38%, 10%, 5%
P.DA
HG4
Hogg-type adaptor using trimmed means 33%J 25%, 19%
HG5 Hogg-type adaptor usir..r, triElrred memlS 53%" 25%
HG6 Hogg-type adaptor using trimmed means 33%; 19%
1.31A Hogg-type ad.aptor using harrpels ADA, 25A~ 17A
1.31 xA Iiogg-type adaptor l1sin~ haJ:1pels 25A; 2lA) 12A
29
1.90A Hogg-type adaptor using hampels ADA" 25A, 17A
1.951. Hogg-type aCialJtor using ha.lTIpels ADA, 25A, 17A
2.00A Hogg-type adaptor using harilpels 21A) lZA
1.81D rlagg-type adaptor using I+esttnators DIG" DIS"
30
1.90D Hogg-type adaptor asing II-estimators DID') DIS,
27
28
D2D
D20
(table conit. next page)
4
(con it.)
Number
Code
31
32
1.95D
1.60D
1.75D
33
34
35
36
37
,3'"'
0
39
Short Description
Hogg-type adaptor using N-estir.lators DIO, ilU,
Hogg-type adaptor usiuZ r'l-estll',1ators DlO:; DIS,
Hogg-type adaptor using i'll-estimators DID ~ DIS,
Gistwirth's estllJator
Pndrew's estim~tor
GAS
AMI'
D20
D20
D20
H/L Hodges- Lehmann estimator
RN I~ormal-scores rank estiQator
: RN50 One-step, nonlnl-scores~ rank estll~~to4 8=50%
RNA One- step:; nOITlal- scores ~ rfulk estimator, e=the adaptive estimatm:
RUSO One- step" unifonn- scores ran.1'-: estimator? 8=50%
RUA One-step, lIDiform-scores rank estir.lator~ e = the adaptive estima1
J0A Jo1m is adaptive estimator
5T4 r"hltiply-skipped mean, max(51~,2):s.6N deleted
40
41
42
43
TABLE 2.1.
A brief description of the 43
IIU1emonic code for each.
estL~tors
of location together with a short
Trimmed I1eans
It is well-kno\'ffi that the usual arithmetic mea'1. is non-robust in the presence of extrene outliers.
A simple scheme for robustifying the mean is to
eliminate "extreme" obsenrations.
carets the
[(N+l)a.]
largest and
The
a.(lOO)%
s)'lTI1'1etric trimmed mean dis-
snallest observations from the
[0,J+1)0.]
sample al1d computes the arithnetic average of the rerr:aining observations.
Here
[0]
is the greatest integer fDllction.
outer meaTl is sometimes calculated.
with 0.:-::.25.
J\1ore
observations and
precisely~
L(a.)
mator in its
lO%~
19%,
O\'ID
25%~
if D(a)
is the
and~
r.le~m
[U~+l)a]
In this study~ the
right as well as in HGl.
38%, 50%
situations~
the
The outer mean is the mean of the trimmin[.!'
is the mea.Tl of the
then Oli = (U(.25)+L(.25))/2.
For short-tailed
on
of the
smallest
0%
largest
observations~
is used. as a robust esti-
11~ trim~ed-De&'1.
of course, with
[(H+l)a]
triE1Eling,
estimators are
11.
Adaptive
5
estirn.ators based on trimmed means include HGI ~ HG2) HG3 ~ HG4, HGS
and HG6.
Huber's M-estimatoTs
T ~ of a"1 equation of the form
i'.l-estinators of location are solutions,
nL l/J [x._J--TJ
s
. I
J=
where
l/J
is an odd function and
==
0,
is a scale estimate which is either esti-
s
mated ill.dependently or froD act equation of the fonn
.I
x(Xi -'1']
==
O.
)=1
Here, of course,
xl' ..
"~
are the observations.
}tuber (1964) proposed
l/J
of the form
1jJ(x;k) ==
r~.
'.
Hampel (1974) proposed using 1/J
x<k
-k<x<k
x>k
as above but with
s estimated by
:'!e refer to this latter quantity as the hampel scale estimator.
111e Huber 1.1- estm.ators exanined in this \Vork are one- step estirlators (cf.
Bickel (1975)). TIUlt is, rather
tha.~
selecting T as the exact root of the
equation
we perfonn one iteration of the
as the starting value.
NeHton-E~aphson method
using 50% (the median)
Thus the estimator T is given by
6
- 1. \'il
,
r =
n L.i=lW
(50%) + s
0
n-l\,~
;c.1 - (SO%~
s
[X.SO'i}-)
W' ~1~
_
£1=1
TIle scale
s
S
was chosen accordine to tvlO a1sorithr:1s
Hampel scale and also the scale
5
0
[<,Te investizateo. the
= lQ:{jl.18DS. Res:.llts based on HarJpel
scale \,rere shular and [;enerally sonm'!hat superior to those based on the interquartile range.
section 40
We will report only the results based on Hampel scale in
k=l.O, 105
have
DIO~ Dl5~
The Huber estir..ators discussed include
a~d
2.0
17A,
21A~
respectively.
Closely related are the
22A, 2SA ffild P0A,
estL~ators
12A~
1. 60D
and
1. 75D
1. 9SA
and
20 20A are adaptive fl...arJ:pels.
t/J(x)
D20
which
tillm~el-t)~)e
Also I.BID, 1.90D; 1.95D
are adaptive H-estinators &"'1d
is an l1-estinate \'1ith l/J
and
j
2. OOA~ 1. 8IA;; 1. 81 *A) L gOA,
Finally AYff, the Andrews estll1ator"
chosen by
= {Sil1(X/201)
o
Ixl<Z.l'IT
othenlise
0
Hampels
The llary;?e1 estinators are i'luber-type "]-estir.1ators \;Tith 1)J
Given by
O::;lxl<a
a::;lxl<b
b::;lxl<c
Ixl~c
The parameters 9
a, b and
c
are given below together witll the code syr:JDol.
7
code
12ft.
l7A
2lA
22A
25A
ADA
The parameter,
tl~
ADA~
a
b
1.2 3.5
1.7 3,4
2.1 4.0
2,2 3.7
2.5 4.5
ADA 4.5
c
B.O
8.5
8.2
5.9
9.5
3.0
is chosen as a ftnlction of the estinated tail length of
distribution which is estimated by
Here s is the Hampel scale estiraator. The parameter ADA is defined by
L:o;.44
1.0
ADA = (75L-25)/8 •44<L:o;.6
{
2.5
.6~L
All of the r~el estLmators are one-step M-estlinators computed by the algorit.hr.l discussed above.
Hampel estimators include
l2A~
l7A,2lA, 22A, 25A a"'ld
ADA and adaptive Hampel-type estimators 1.81A; 1.8l*A, 1.90A) L9SA and
2. aDA.
Related are the Huber II-estimators and the pJld.rew's estimator already
mentioned.
Hogg-Type Adaptors
Hogg (1974) has suggested an ad2ptation procedure which chooses among 2
or more estimators of the center depending on the value of some statistic which
measures the length of the tail.
of tail length.
and
Eogg first suggested the Kurtosis as a measure
More recent suggestions include
8
_ U(.20)-L(.ZO)
Q2 - U(.SO)-L(.SO)
where
U and L are defined as in the section on trimmed means.
The exact
formulation of the various adaptors is given below.
Code
Formulation
38%
HG1
T = 19%
Q1>3.2
2.6<Ql~3.2
2.6<Ql~2.6
Ql:>;2.0
HGZ
HG3
HG4
HGS
HG6
38%
T = 25%
{
10%
T
T
T
T
=
{
87
QZ~1.8l
87
38%
Q~>l.
10%
5%
Q2~1.8l
"
1.8l<QZ:>;l.87
Q >1. 80
2
25%
19%
1.5S<QZ~1.30
QZ:>;1.55
= {38%
QZ>2.Z0
25%
QZ~2.20
= {38%
QZ>2.20
19%
Q2~2.20
=
{
length~
_ADA
T
1.81<Q2~1.
38%
Using Q as a measure of tail
Z
also constructed.
1.BlA
QZ>l. 87
=
{
25A
17A
several adaptive Hampel estimators were
1. 81<QZ:O;l.87
QZ:O;1.8l
QZ>L87
9
1. 81*A
I.90A
l.9SA
T
T
=
=
ZlA
ZSA
1. 81<Q2::;1.87
Q2::;1.81
lZA
Q2>1. 87
ADA
1.90<Q2::;Z,OS
Q2::;1.90
QZ>Z.OS
{
Z5A
{17A
T ={:
17A
Z.OOA
T ::
{ZlA
lZA
1. 95<QZ::;2. 10
QZ::;I.95
QZ>Z.10
Q2::;Z,00
Q2>2.00 •
Also included in this study are Hogg-type adaptors using Huber
l.81D
DIO
T = DI5
{DZO
DIO
I.90D
T = DIS
{ D20
D10
1.9SD
T
= DIS
{
D20
DI0
1.60D
I.7SD
T
=
{
QZ>l. 87
1. 81<QZ::;I. 87
Qz::;I.81
QZ>2.05
1. 90<Qz::;Z, 05
Q2::;1.90
Qz>Z.10
1. 9S<Q2::;2. 10
O~
Q2-<1 . .-v
DIS
Q >1. 70
2
1. 60<Q2::;1. 70
DZO
QZ::;1.60
D10
T = DIS
{DZO
QZ>1. 75
1. 70 <QZ::;1. 75
Q2::;1.70 •
~l-estirnators.
10
Rank Estimators
In what fo110\'/s ~ let Ri (8)
we rank xl -8, x Z-8 ~ •••
R. (8)
1
~~ -8
according to magnitude (but not sign) and
is the product of sgn(x.
-8) and rank of (x. -8).
1 1
h
= ~-l(l+U)
-Z- ~ were
J()
u
'¥
~.1S some e strl
li'b
'
fun'
utlon
. etlon.
'i'
taken to be nomal or miform.
as the root ~
That is ~
denote the signed ra11ks of xi -8.
Further let us defi.'11:
T
' 11y ~
yplca
~
.
lS
The rank statistic with nonnal scores is taken
T ~ of the equation
J*(T)-J
=0
where
and
-1 n
J=n
I J(i/n+l).
i=l
I*,
The symbol ~
indicates the swnmation over all positive ran.1cs.
Roots were
found by the method of bisection using a maximum of twelve iterations.
is taken as
uniform~
If
~
the resulting rank statistic is asymptotically equivalent
to the Hodges-Lehmann estimator - namely the median of all pairwise means of
the sample (c.f. Hodges-Lehmarm (1963)).
This proved to be a very close approx··
imation so that in this study we report only the Hodges-Lehmann estimator and
not the uniform-scores rank estimator.
Just as with the Huber-tYrle I;-estimators ~ one may calculate one-step rank
estimators based on one iteration of a rJmvton-Raphson method (Kraft and Van
Eeden(1969)).
becomes
In our previous
notation~
the one-step rank estimator,
T,
11
J*(8) -J;J
B:-"
T=e+--~
0
e is chosen as an initial robust estimator of location and B* is an estimator of B =
f J'(F(x))f2(x)dx~
where, of course F and f are respectively
the distribution and density of the sample.
In this study 1 we employed a nonparametric estimate of B proposed by
Schweder (1974)
B*
where
x(l)~ .•• ~x(n)
=
are the ordered observations.
B*~
This estimate,
is
based on the standard kernel estimator of a probability density function (c.f.
Parzen~
l~e
(1962)) with tmiform kernel ~
K.
chose £20 = 1.60, although T seems relatively insensitive to the
choice of
unifonn on
E.
In the study of one-step rank estimators 1 we chose
(-1,1), nonna1 1 logistic ~ exponential alld triangular.
<p
to be
The logistic-
scores) exponential-scores and the triangular-scores estlinators are not reported
since they were typically bettered by one of the others.
6 was chosen according to two schemes.
median.
The initial estimator
The simpler choice was 6=50%, the
The more complex choice was an adaptive estimator given by
e = {21A
50%
Estimators based on ranks include H/L 1 Rl.IJ, RNSO, PJ\IA, RU50 and RUA.
Other estimators
JOMS (1974) provides an adaptive estimator which can be \\lTitten as a
linear combination of two trimmed means
12
where
and
n/2
T2 -- n-2r-2s
1
~
[+
]
L.
x(]") x (n-J" +1)
j=r+s+l
with c1 and
C
determined from the sample by
z
and
c1
ex:
n1Z (2s(n+r+2s))
12r+sJ(n-Zr)
1
c
ex:
Z
1 e(n-Zr-2s))
;Z
n-2r
Z
The Di
in the
Z n-2r-2s
- D DZ n-Zr
1
__4_ (_S_) .
DIDZ n-Zr
are proportional to the lengths of the segments of the sample entering
Ti ' Values of r
and s for n=20 were r=l and s=4.
. gram to compute this estimator was adapted directly from PRS.
details see Johns (1974).
Johns' estimator is coded as
The pro-
For further
J~H.
A skipping procedure is used to identify and delete extreme points.
For
n=20, let
= h -1.5(hZ-h1) and t z = hZ+1.5(h2-hl ). A singly skipped estimate deletes
1
l
all observations smaller than t 1 and larger than t Z ' If a single skipping
deletes a total of k points, the est~~ator 5T4 deletes a further i points
t
13
from each end where
• ( nmx (12k)
• 6n:-k)
".
= mIn
~ -,
2 .,
n
N
and the mean of the remaining sample is used as the estimator.
was fonnulated in the
PRS.
Finally, we included
T = .3x
3.
3J:1.
estimator proposed by Gastwirth (1966), defined by
([~lj)
Monte Carlo Details.
investigated in the
PRS,
+
.4(50%) + .3x
n
(n-[3])
In addition to the long-tailed sampling situations
we also investigated short-tailed situations and
situations involving dependent data.
Pseudo-random numbers were generated by
the familiar multiplicative congruential generator.
tion and cautions in the use of these generators).
o
This estimator
(See the
PRS
for descrip-
The output, scaled bet,,,een
and 1, form the basis for all subsequent calculations since they approxi -
mate a uniformly distributed sample.
Normal pseudo-random deviates were generated according to the well-knmvn Box-Muller transform 2 and Cauchy pseudo-random
deviates were generated by the arctangent transform.
These basic three dis-
tributions, or mixtures thereof, form the basis of all results reported in
this work.
The basic sampling situations are surrnnarized in Table 3.1.
2 This transform is widely attributed to Box and lviuller who published their
,,,ork in 1959. The result is ~ however, pointed out by Paley and Wiener in
their 1934 book (pages 146-147).
14
Table 3.1.
Sampling situations for independent observations. All had sample size 20.
N/U represents a variate formed by dividing a Normal
variate unifonn on the interval
(0,1)
variate by a
(0,1).
Distributions
UnifoID, mean 0, variance 1
97% Uniform (0,1) + 3% Nonnal (0,1) contamination
90% Unifonn (0,1) + 10% Nonnal (0,1) contamination
50% Unifom (0,1) + 50% NOTI:Jal (0,1) contamination
Normal, mean 0, variance 1
99% Nonna1 (0,1) + 1% Normal (0)9) contamination
97.5% rJormal (0,1) + 2.5% Normal (0,9) contamination
95% Nonial (0,1) + 5% Normal (0,9) contamination
90% Normal (0,1) + 10% Normal (0,9) contamination
85% Nom.al (0,1) + 15% Normal (0,9) contamination
75% Normal (0,1) + 25% Normal (0,9) contamination
95% Norrral (0~1) + 5% Normal (0,100) contamination
90% Nom~l (0,1) + 10% Norrr~l (0,100) contamination
75% Normal (0,1) + 25% Nom.al (0,100) contamination
97% Normal (0,1) + 3% Cauchy contamination
90% NOTw.a1 (0,1) + 10% Cauchy contamination
50% Normal (0,1) + 50% Cauchy contamination
Cauchy, median =
90% Normal (0,1) + 10% H/U
75% Normal (0,1) + 25% N/U
°
u
U + .03N(O,1)
U + .10N(O,l)
U + .5m~(0,1)
N(O,l) = N
N + .OlN(0,9)
N + .025H(O,9)
N + • 5N (0 ,9)
N + .10N(0,9)
N + .15N(0,9)
1'1 + .25N(0,9)
H + .05N(0,lOO)
N + .10N(0,100)
N + .25N(0,lOO)
°
N + .03C
N + .10C
N + .50C
C
N + .10N/U
N + .25N/U
In addition to the sets of independent observations discussed above, \',re
generated observations according to a first order autoregressive schene,
x. = px. 1 + £., J" = 1,2, .•• )20. The shock, £J"' ~~s cllosen according to
J
JJ
either N~ U or C and the correlation coefficient, p, as either .2,.5
or
.9.
The initial value,
discussed in section 2
tions.
wer~
xO' was chosen as
O.
All of the 43 estimators
investigated in the 20 independent sampling situa-
Only the non-adaptive estimators were studied in the 9 time-series
sampling schemes.
IS
Tne results we report a.re based on 1000 nonte Carlo replications.
The
sample variances of the estimators were calculated and then scaled by a factor
of 20 (the sample size) to make the results comparable to those in the
PRS.
Assuming approximate nonnality for the estimators ~ the variance of 20x sample
4
.7992aT
where aT2 is the true variance of the estiFor example 1of T = -X, t hen a'l'2. is .fur and the variance of 20
variance would be about
mator 1 T•
times the sample variance is
.001998.
Thus one may expect about one decimal
place of accuracy with correspondingly less significance as the value a~
cl~bs.
We report two decimal places, in most
were calculated over the same samples.
these
l~ithout
cases~
because all estimators
attempting inferences outside
smaples, the extra decimal place(s) are meaningful.
We also report pseudo-variances as discussed in the
distributions~
PRS.
For long-tailed
the variance is essentially determined by a few observations in
Thus if the distribution of the estirr.ator ~ T~ is itself
the extreme tails.
long-tailed, the sample variance of T would itself not be a robust estimate
of the scale of T.
As in the
PRS, we
compute the 2.5% - ile point, square this
quantity and divide by the normalizing constant
(1.96) 2 . This pseudo-variance
is further rescaled by multiplying by 20 (the sample size) to yield quantities
comparable to those in the
4.
The Results:
3 tables.
PRS.
The main results of the liIonte Carlo sttldyare summarized ill.
Table 4.1 is a table of sample variances (over the 1000 I/Zonte Carlo
replications) for the 20 independent sampling schemes and 43 estimators of
location.
Table 4.2 is a table of pseudo-variances (based on the 2.5%-ile
16
point) for the same 20 independent sampling schemes and 43 esti.-rnators.
Table 4.3 is a table of biases and variances for the 9 sampling schemes
invo1vll1g first order autoregressive variates.
None of the Hogg-type adaptive
estimators are listed for the two-fold reason that the choice of Ql and QZ
is predicated on independent observations and that tIle basic estimators tri:mrned means, ha.1Ipels, etc. - perfonn poorly. Some preliminary calculations in'
dicate that Hogg-type adaptors perform similarly lIDder the time series alternatives.
l1e withhold discussion of these results to section 6.
17
~
~
0
0::
w
~
~
.....
8
z
M
-
~
0
0
~
~
~
Z
+0
~
Z
+0
:::l Ln
0
:::>~
19%
25%
38\
1.18
1.34
1.50
'1.82
2.30
1.15
1.32
1.49
1.80
2.08
1.21
1.37
1.53
1.83
2.41
50\
M
CJ.!
DIO
DI5
2.48
1.01
.61
1.95
1. 55
2.43
.98
.60
2.10
1.66
D20
1ZA
17A
ZlA
2ZA
1.23
1.82
1.41
1.20
1.17
1.32
1.97
1.52
1.30
1.29
1.08
1.02
1.11
1.13
1.14
1. 22
1. 23
1.41
1.61 1 1 . 36
1.55
1.82
2.51
1.05
.94
LOS
1.25
.67 I .84
1.16
1.97
1.47
1.07
1.27
1.58
1.04
1.11
1.29
1.14
1.42
1.85
1.06
1. 22
1.46
LOS
1.11
1.26
1.08
1.13
1.25
2SA
ADA
HG1
HG2
HG3
1.08
1.19
.80
1.39
1.21
1.17
1.31
.80
1.37
1.20
1.14
1.28
.88
1.41
1.t5
1.04
1.11
.96
1.19
1.08
21
HG4
22
23
HG5
HG6
l.8lA
l.81*A
1.77
1.82
1.50
1.10
1.11
1.72
1.80
1.49
1.20
1.21
1.78
1.83
1.53
1.17
1.17
1.40
1.41
1.22
1.07
1.08
1.05
1.90A
I.9SA
2.00A
1.810
1.900
1.09
1.08
1.20
1.25
1.24
1.18
1.18
1.30
1.34
1.32
1.15
1.14
1.26
1.31
1.29
1.05
1.04
1.11
1.14
1.12
1.03
1.03
1.05
1.04
1.04
1.95D
1.60D
1. 75D
GAS
AMT
1.23
1.49
1.34
1.90
1.22
1.32
1.58
1.44
1.96
1.31
1.11
1.30
1.22
1.44
HlL
1.21.
.89
1.41
1.01
2.47
-r:28
.94
1.29
1.54
1.41
1.93
1.27
·1.25
1.09
2.60
1.19
.88
1.22
1.30
.88
1.28
1
Z
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
24
25
26
27
. 28
29
.30
31
32
33
34
lS
36
37
38
39
40
41
42
43
5\
10~
RN
RN50
RNA
RUSO
RUA
J~
5T4
:::>
0
:z
Z
+5
0
~
LCl
N
+0
-
0\
0
0
~
~
:z
z
Z
0
LCl
+0
~
0\
Ln
+~
+~
:z
:z:
:z:
:z:
1.09
1.18
1. 32
1.11
1.13
1.16
1.23
1.36
1. 25
1.24
1.26
1.35
1.50
1.42
1.32
1.31
1.43
1.56
1.63
1. 51
1.49
1. 56
1.66
1.49
1.04
1.31
1.15
1.06
1. 54
1. 24
1.87
1.25
1.19
1.67
1.45
2.27
1.38
1..30
1.72
1.82
3.37
1.46
1.40
1.84
2.09
4.03
1.60
1.51
1.01
1.13
1.01
1.03
1.15
1.26
1.18
1.16
1.20
1.25
1.36
1.27
1.24
1.26
1.38
1.46
1.39
1.38
1. 39
1.49
1.59
1.50
1.49
1.52
1.03
1.06
1.11
1.16
1.12
.98
1.02
1.06
1.12
1.09
1.13
1.17
1.19
1.21
1.17
1.23
1.23
1.28
1.31
1.28
1.38
1.43
1.41
1.37
1.35
1.51
1. 55
1.60
1.58
1.57
1.25
1.23
1.14
1.22
1.18
1.09
1.00
1.01
1.28
1. 23
1.16
1.16
1.18
1.40
1.35
1.26
1.26
1.26
1.47
1.43
1.31'
lAO.
1.41
1.601.56
1.49
1.S:!
1.54
.99
.98
1.02
1.04
1.02
1.15
1.14
1.16
1.18
1.15
1.25
1.24
1.24
1.27
1.25
1.40
1.39
1.39
1.40
1.38
1.51
1.52
1.50
1.53
1.49
1.02
1.12
1.08
1.19
1.04
1.15
1.24
1.22
1.25
1.16
1.25
1.34
1.32
1.37
LIZ
1.04
1.10
1.08
1.28
1.12
1.50
1.56
1.56
1. 59
1.50
1.03
1.01
1.07
1.02
1.40
1.01
1.14
.98
1.14
1.45
1.08
2.50
1.09
.94
1.18
1.01·
1.82 .
1.15
.9&,
1.49
1.22
1.13
1.54
1.26
1.26
1.34
1.23
1.67
1.39
1.46
1.45
1.40
1.32
1.38
1.45
1.46
1.38
1.70
1.26
.93
1.27
1.11
.95
1.10
1.05
1.06
1.08
1.01
1.02
LOS
1.24
1.16
1.25
1.20
1. 2(}'~ 1.29
1.38
1.47
L47
1.49
1.59
1.57
:::l
1.4~
~
0
~
~
+0
;II:
0\
-
-;;-.
~
~
.97
:z
LOS
:z
1.03
LOS
LOS
LZS
1.51
1.62
1.58
1.50
1.84
VARIANCES
Table 4.1:
VaTiances of 43 estimatoTS of location b8sed on 1000 f-tmte <:ado replications.
,
<.,
]8
-
o
a:
0
0
w
co
I.J.I
z:
u
~
1
2
3
4
5
C
0
5%
lot
19%
25%
38%
z:
L!l
+N
Z
o
0
0
0\
.-
-
~
~
-
0
0
z:
+:£
Z
zL!l
z:
0
+-
Z
n
<"')
+0
z:
+N
Z
.
+-
Z
0
+L!l
Z
:::>
......
z:
~
.......
u
u
u
z
0
U
+z:
27.2
8.34
4.99
3.24
2.77
1.60
1. 23
1.24
1. 31
1.43
2.83
1.61
121.
475.
1.34
1.26
2.34
2.02
1.89
1.84
1. 90
1.68
1. 22
1. 21
1.29
1.41
4.35
2.16
1.61
1.57
1.72
16.0
9.12
5.32
2.78
2.49
1.10
1.13
1.17
1.25
1.37
1.40
1.22
1.23
1.30
1.43
8.53
2.70
1. 96
1. 74
1.84
1.58
5.45
17.3
1.32
1.23
1.94
11.5
38.1
1.59
1.54
2.56
26.0
85.3
2.68
3.12
1.55
72.9
289.
1.27
1.19
1.59
381.
2.06
**1c
***
***
1.33
1.25
1.76
1.82
3.10
3.63
4.37
2.76
3.21
3.58.
3.46
1.23
1.29
1.21
1.19
1.21
+
Z
.
3.67
1.80
1.64
1.67
1. 79
2.00
8
9
10
(J.f
DI0
DI5
2.08
2.96
5.80
1.86
1.87
11
D20
12A
17A
2lA
22A
1.95
1.87
1.89
1.96
2.01
1.20
1.27
1.18
1.16
1.18
1.59
1.44
1.37
1.36
1;36
3.80
2.48
2.82
3.11
2.97
1.14
1.26
1.17
1.14
1.17
1.23
1.22
1.21
1.24
1.96
1.68
1.72
1.73
1.78
25A
ADA
HG3
2.07
1.98
2.14
1.93
2.02
1.15
1.19
1.28
1.29
1.28
1.39
1.42
1.74
1.66
1.66
3.60
2.67
4.26
2.80
3.00
1.11
1.15
1.16
1.20
1.16
1.18
1.24
1.30
1.30
1.28
1.89
1.74
1.88
1.84
1.87
3.93
3.01
3.30
2.83
2.91
1.18
1.23
1. 31
1.30
1.30
1.53
1.57
1.69
1.66
1.66
HG4
HG5
HG6
l.81A
1. 81*A
1.88
1.84
1.89
1.99
1.97
1.34
1.29
1.21
1.16
1.19
1.69
1.59
1.60
1.37
1.42
2.60
2.78
4.73
2.90
2.59
1.28
1.26
1.18
1.13
1.15
1.37
1.31
1.24
1.21
1.24
1.83
1.79
1.89
1.73
1.69
2.78
3.01
3.68
3.26
2.84
1.35
1.31
1.25
1.19
1.22
1.69
1.65
1.52
1.54
26
27
28
29
30
1.90A
1.95A
2.00A
l.81D
1.90D
Z.OI
1.17
1.16
1.18
1.24
1.23
1.38
1.38
1.42
1.56
1.56
2.86
3.01
.2.77
2.88
3.06
1.12
1.12
1.15
1.17
1.15
1.21
1.20
1.23
1.26
1.24
1. 75
1.74
1. 74
1. 78
1.80
3.30
3.31
2.98
3.16
3.31
1.20
1.20
1.21
1.26
1.24
1.52
2.01
1.94
1.91
1.94
31
1.95D
1.60D
1.7SD
GAS
1.95
1.87
1.89
1.87
1.97
1.21
1.30
1.28
1.34
1.16
1.56
1.58
1. 57
1.60
1.36
3.21
2.77
2.83
2.67
2.99
1.14
1.24
1.21
1.28
1.14
1.24
1.31
1.29
1.34
1.21
1.B2
1. 76
1.77
1.78
1.73
3.44
3.11
3.11
3.19
3.44
1.24
1.31
1.28
1.35
1.19
1.62
1.61
1.65
1.72
1.52
1.97
2.24
1.97
2.01
2.08
1.22
1.28
1.31
1.17
1.57
1.65
1.95
1.76
1.48
1.94
4.04
5.93
2.90
3.33
2.56
1.14
1.12
1.21
1.13
1.55
1.24
1.28
1.32
1.23
1.59
2.01
2.46
2.05
1.96
2.06
4.57
6.76
3.41
3.84
2.83
1.25
1.31
1.32
1.20
1.61
1. 73
1.93
1.81
1.67
2.00
1.95
2.19
1.98
1.17
1.24
1.24
1.39
1.52
1.42
3.13
4.38
3.73
1.16
1.16
1.20
1.23
1.27
1.25
1.91
1. 73
1.66
1.41
3.25
3.00
1.20
1.28
1.26
1. 58
1.62
1.54
6
50%
7
M
12
13
14
15
16
17
18
19
20
21
2Z
23
Z4
25
32
33
34
3S
36
37
38
39
40
41
4'2
43
HGl
HGZ
AMf
H/L
RN
RN50
RL'lA
RU50
RUA
J~H
5T4
1.~
**ll:
. ***
"**
***
1. 70
1.64
1.67
1.58
1.51
1.50
1. 51
1.72
l.~
1.49
1.63
1.62
VARIANCES
Table 4.1. (continued):
.
Ll')
N
Variances of 43 estimators of location based on 1000 l-bnte Carlo
replications .
19
0::
I.LJ
~
'":
'":
0
0
0
~
co
z:
z::
~
::E
0
M
+0
::>
z:
0
u
1
2
3
4
5
5%
10%
19%
25%
38%
1.13
1.33
1.55
1.83
2.23
1.09
6
7
50%
M
8
(}.f
::;:)
::>
~
~
+~
::>
5
+1.0
::>
~
0
~
z:
~
0
0
~
~
~
z:
0
1.0
+0
z
z
z
CJ\
-0
~
z:
1.0
N
+0
-
01'
0
~
0
z:
~
z
+0
.
~
+ .-
z:
z:
........
LO
oe
1.38
1.69
2.05
1.19
1.31
1.48
1.89
2.25
1.18
1. 25
1.36
1.40
1.54
.99
}.02
1.00
1.09
1.19
1.06
1.07
1.13
1.25
1.41
1.20
1.20
1. 31
1.33
1. 53
1.19
1.12
1.16
1.32
1.30
1.51
1.46
1.40
1.45
1.42
1.00
1.43
1.:59
1.49
1. 75
2.36
.96
.65
2.10
1.63
2.67
1.02
.73
2.07
1.68
1.86
1.06
.89
1.43
1.27
1.52
.98
1.22
1.12
1.09
1.58
1.07
1.23
.92
.96
1. 51
1. 24
2.09
1.34
1.27
1.63
1.31
2.19
1. 27
1.17
1.73
2.13
4.25
1.43
1.45
2.04
2.26
3.85
1.60
1.46
1.17
·1.24
1.18
1.17
1.35
1.50
1.42
1.37
1.37
1.48
1.66
1.43
1.38
1.41
1.27
9
10
DID
DI5
2.40
.94
.63
1. 94
1.56
11
12
D20
12A
17A
2lA
22A
1.26
1.82
1.41
1.26
1.22
1.36
1.82
1.51
1.28
1.28
1. 28
1.96
1.55
1.25
1.22
1.20
1.42
1.28
1.21
1. 21
1.07
1.17
1.07
1.06
1. 07
.94
.94
1."01
.98
1.2Q
1.33
1.27
1.24
1.32
25A
1.15
1. 20
.83
1.33
1.23
1.18
1.;30
.84
1. 35
1.12
1.12
1.46
.90
1.47
1.24
1.20
1.19
1.06
1.36
1.20
1. 04
1.05
1.00
1.04
.98
.95
.97
1.05
1.15
1.18
1.17
1.25
1.26
1.32
1.25
1.19
1.28
1.25
1.24
1.18
1.42
1.46
1.49
1.40
1.41
1.44
1.55
1.81
1.67
1.67
1.67
1.68
1.38
1.20
1.20
J....84
1.89
1.48
1.12
1.12
1.40
1.40
1.36
1.19
1.20
1.14
1.09
1.00
1.06
1.07
1.35
1.25
1.13
.96
.96
1.36
1.33
1.31
1.25
1.Z5
1.35
1. 32
1.16
1.18
1.19
1.30
1.45
1.39
1.44
1.48
1. 73
1.49
1.38
13
14
15
16
17
18
19
20
AIM\
HGl
HG2
HG3
.94
~.13
21
22
23
24
25
HG4
HG6
1.8lA
1. 81*.4.
1.82
1.83
1. SS
1.18
1.15
26
27
28
29
30
1.90A
1. 95A
2.00A
1.81D
1.90D
1.15
1.15
1.26
1.52
1.26
1.19
1.18
1.28
1.57
1.36
1.20
1:12
1.12 . 1.20
1.21
1.25
1.39
1.61
1.20
1.28
1.04
1.04
1.06
1.11
1.07
.95
.96
.94
.97
.94
1.25
1.20
1.24
1.35
1.20
1.22
1.19
1.18
1.22
1.18
1.44
1.43
1.37
1.41
1.40
1.51
1.47
31
32
33
34
35
1. 95D
1.60D
1. 75D
1.26
1.52
1.41
1.97
1.25
1.36
1.57
1.42
1. 98
1.35
1.28
1.61
1.46
2.03
1.27
1.20
1.39
1.26
1.40
1.21
1.07
1.11
1.10
1.02
.93
.94
.97
.92
.99
1.20
1.35
1.29
1.33
1.24
1.17
1.22
1.19
1.35
1.20
1.43
1.41
1.40
1.48
1.41
1.49
1.S5.
1.59
1.62
1.35
1.31
1.00
1.41
1.14
2.39
1.36
1.09
1.62
1.14
2.90
1.24
1.02
1.57
1.13
2.67
1.19
1.04
1.23
1.13
1.86
1.00
1.00
1.05
1.05
1.20
.90
.92
.91
.93
1.16
1.18
1.20
1.35
1.22
1.51
1.11
1.20
1.30
1.17
1.63
1.37
1.49
1.47
1. 36
1. 75
1. S3
1.59
1.72
1.48
2.04
1.25
.91
1. 21
1.28
.89
1.40
1.25
1.02
1.24
1.21
1.08
1.20
1. 06
.99
1.10
.94
1.03
.98
1.24
1.34
1.23
1.18
1.14
1.19
1. 37
1.55
1.47
1.78
HGS
GAS
Af.IT
36
37
38
39
40
HIL
41
42
43
RUA
J0H
R."J
R.'J50
RNA
RU50
5T4
/
1.13
1.~l
1.53
1.41
1.55
l.4S
L38
1.62
PSEUDO-VARIANCES
Table 4.2.:
Pseudo-variances of 43 est~tors of location based on the 2.S%-ile ~ld
1000 Monte Carlo replications.
20
-
i·,
-
a:
w
0
~
W
0
:::>
<:;)
~
1.0
+N
'":
-
-
2:
U
M
~
2:
<:;)
:::>
<:;)
<:;)
<:;)
1.0
+0
<:;)
.....
~
-
<:;)
0
0
0
< :;)
0\
to
U
<:;)
u
0
+1.0
+tn
z;~
27.8
9.14
5.02
3.49
2.66
1.28
1.18
1. 22
1.28
1.33
3.56
1. 95
1.7Z
1.83
1.94
2.16
159.
628.
1. 98
2.28
2.59
680.
1.42
22.0
81.0
1. 27
1.21
65.0
264.
1.90
1.84
2.48
1.59
1.72
1. 77
1.68
4.42
2.54
3.11
3.36
3.06
1.21
1.32
1.19
1.16
1.78
1.85
1.80
1.89
1.80
1. 85
1.77
1.99
1. 94
1.98
3.53
2.57
3.02
2.59
2.77
1.09
1.17
1. 2S
1.33
1.33
1.86
1.93
1.81
1.87
1.81
1. 35
1.28
1.27
1.25
1.34
1.94
2.07
2.32
1.72
1. 59
1. 32
1.29
1.22
1.12
1.24
1.90
1. 8:3}
1. 761.76
1.82 .
1.28
1.28
1.21
1.36
1.37
1.22
1.22
1.25
1.35
1.30
1. 75
1.71
1.83
1.98
2.03
2.62
3.08
3.66
3.11
2.69
3.27
3.27
2.82
3.28
3.38
1.17
1.14
1.18
1.22
1.21
1. 77
1. 7S
1.8Z
1.86
2.56
2.13
2.13
2.23
2.43
1.37
1.36
1.35
1.42
1.22
1.31
1.35
1.30
1.32
1. 24
2.28
1.98
1.98
1.95
1.88
3.51
3.28
3.28
3.31
2.96
1.21
1.22
1.22
1.25
1.12
1. 78
1.94
1.91
1.66
2.12
1.67
1.55
2.07
3.51
5.50
2.53
2.84
2.29
1.28
1.24
1.36
1.18
1.56
1.28
1.28
1.25
1.24
1.60
2.41
3.06
2.21
2.31
2.16
4.64
7.07
3.38
3.79
2.59
1.19
1.27
1.23
1.15
1.42
1.89
2.01
1.96
1. 78
2.27
1.38
1.57
1.48
2.47
2.46
2 .. 59
1.21
1.31
1.22
1.28
1.24
1.25
1.99
2.03
1.92
3.36
3.15
. 2.70
1.14
1.21
1.28
1.9:1.80
1.82
+0
z:
+ .....
z
:z:
2.55
2.28
1.98
1.88
1.89
1.82
1.35
1.31
1.36
1.48
4.62
2.34
1.91
1.59
1.82
16.2
10.5
5.60
2.35
2.09
1.24
1.33
1.41
1.36
1.45
1.59
1. 25
1. 31
1. 31
1.39
7.28
2.97
2.28
1. 92
1. 94
DIO
DIS
2.12
3.15
6.06
1.86
1.92
1.61
6.85
24.7
1.38
1.29
2.08
12.5
41.6
1.63
1.73
2.29
29.0
93.9
2.12
2.63
1. 56
1.85
3.06
1.42
1.44
1.60
9.01
29.5
1.34
1.35
11
12
13
14
15
D20
1ZA
17A
21A
2ZA
1.96
1. 96
1.96
1.92
2.11
1.30
1.38
1.33
1. 21
1.17
25A
ADA
HG1
HG2
HG3
2.09
1.98
2.12
1.93
2.25
1.66
1.48
1.46
1.39
1.44
1.47
1.43
1.86
1.83
1.84
3.43
1.82
2.22
2.47
2.32
16
17
18
19
20
1.31
1.31
1.30
1.27
1.30
1.31
1.26
1.52
1.43
1.48
3.03
2.00
2.98
2.12
2.12
1.26
1.26
1.32
1.31
1. 31
1.30
1.36
1.26
1.25
1.26
1.20
1.23
1.27
1.25
1.26
21
22
23
24
25
HG4
HG5
HG6
1.81A
1. 81 *A
1. 87
1. 88
1. 98
1. 95
1.96
.1.48
1.36
1.31
1.27
1.31
1.82
1.66
1.82
1.50
1.49
2.09
2.33
4.86
2.21
1.83
1.42
1.36
1.41
1.20
1.22
26
27
28
29
30
1.9OA
1.9SA
2.00A
1.8lD
1.90D
2.01
1. 98
'1.94
1.82
1.97
1.31
1.31
1.22
1.38
1.34
1.46
1.41
1.48
1.72
1.72
2.07
2.14
2.00
2.13
2.48
31
32
33
34
35
1.95D
1.60D
1. 75D
GAS
A\fi'
1. 96
1. 82
1.82
2.01
2.01
1.31
1.38
1.36
1.42
1.27
1.72
1.72
1.72
1.69
1.41
36
37
38
39
40
H/L
2.05
2.40
2.17
2.20
2.12
1. 31
1. 38
1.45
1.30·
1.61
1. 99
2.27
1.92
1.29
1.43
1.37
1
5%
10%
19\
25%
38\
50\
2
3
4
5
6
7
8
9
10
/
41
42
43
tol
Qt.!
R.~
RJ%O
RNA
RU50
RUA
J~
5T4
:z:
:::>
.....
Z
+~
+N
:z:
:z:
u
0
z:
+ .....
z:
z:
:z
---
u
***
3.28
3.74
LIZ
1.72
1.7.
1.86
1.~
PSEUDO-VARIANCES
Table 4.2. (conti.nued):
Pseudo-variances of 43 estimators of location based on the 2.St-He
and 1000 ~bnte Carlo replications.
I
_."--~
..
--~~"'-
"T-- -
-- ... _ . __
._ .
~.
-----~~--
\,
DISTRIBUTION
NORI1Al
NORf.1Al
NORI1AL
CAUCHY
CAUCHY
CAUCHY
UNIFORM
UNIFORM
UNIFOR!1
p
.2
.5
.9
.2
.5
.9
.2
.5
.9
w
lJJ
U
U
c(
c(
Z
NUMBER
CODE
V1
«
....
co
1\
5%
10\
19%
2st
5
:)!l\
1
2
3
6
50%
7
8
9
10
M
ClvI
mo
DlS
D20
12A
17A
21A
2ZA
2SA
11
12
13
14
IS
16
17
ADA
eAt;
34
35
36
MIT
39
RNA
41
42
43
--- ---.
H/L
RUA
'.
J01-1
5T4
.....
oc
«
::>
W
U
Z
~
.....
co
.....
oc
«
::>
.02 3.61
-.01 1. 58
.02
3.62
-.01 1.63
.02 3.66
-.01 1.66
- .00 ' 1.74
.02 3.71
.02 3.92
.00 1. 85
-.00
2.01
.02 4.18
.02
3.59
-.01 1. 57
1. 79
.02
3.91
-.01
.00 1. 78
.02
3.82
.00 1.68
.02 3.70
1.62
.02
-.01
3.64
3.81
.00 1.75 " .02
-.01 1.66 .02 3.68
.02
3.64
-.01 1.62
-.01 1.64
.02 3.68
.02 ' 3.62
-.01 1.59
.00 1.64
.02 3.68
3.80
.00 1.80
.01
-.01 1.62 .02
3.64
-.00 1.61
.01
3.64
-.01 1. 59
.02 3.61
-.01 1.62
.02
3.64
-.00 1.63 .02 3.74
-.01
1.66 .02 3.70
W
U
Z
~
.....
co
.03
.01
.02
.02
.02
.02
.03
.04
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
.02
;03
.02
c(
.....
....~IXI
~
61.5
62.0
62.6
63.6
64.3
65.3
61.0
60.1
64.0
63.0
62.2
63.9
62.9
62.4
62.6
61.9
62.4
64.1
62.5
62.2
61.9
62.4
61.1
62.3
• OS
.02
.02
.00
.01
.00
.24
.49
-.01
-.01
.01
.00
.00
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-.00
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~
...~oc
c(
::>
241.
50.0
24.0
11.6
8.94
8.19
3742.
***
10.3
,12.9
15.8
7.33
8.69
9.41
9.08
10.7
7.95
11.1
9.0]
18.8
11.8
9.40
10.3
8.00
l.o:f
U
...co~
.51
.25
.11
-.03
.02
.01
.98
1.98
.01'
.02
.02
.02
.02
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.03
.03
.02
-.03
.02
-.09
.02
.01
-.02
.03
~
....
oc
c(
::>
***
7460.
2115.
]72.
77.6
65.7
***
111"*
92.5
124.
166.
50.9
61.8
68.0
62.5
79.6
55.6
161.
64.3
727'.
84.5
72.0
161.
59.8
W
U
W
U
%
~
.....
co
-35.5
-34.3
-33.1
-31. 3
-29.9
-29.1
-36.2
-41. 2
-30.3
-31. 6
-33.0
-30.5
-32.0
-32.8
-37.8
-33.5
-:B.3
-31.3
-32.8
-34.4
-32.8
-32.8
-38.5
-34.8,
c(
.....
a:
c(
::>
***
*1r*
***
***
"'**
***
"'**
***
***
*,."
**"
***
***
***
***
***
"**
***
***
"**
**"
**ll
***
""It
~
.....
IXI
.00
.00
.00
.00
-.00
-.00
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.01
~
.....
oc
«
::>
1.09
1.87
2.05
2.43
2.81
3.21
1. 50
1.00
2.59
2.13
1.79
2.43
1.99
1.76
1.75
1;62
1. 79
2.48
1. 79
1.75
1.41
1.70
1.321.74
W
U
~
a....
....co
«
::>
-.01
- .01
- .01
-.02
- .02
3.90
4.10
4.28
4.63
4. ~11
5.15
3.68
3.18
4.71
4.38
4.08
4.63
4.26
4.06
4.00
3.89
4.06
3.75
4.04
4.04
3.72
4.06
3.68
-.01
-.01
-.01
- .01
-.01
-.01
-.02
-.01
-.01
- .01
-.01
- .01
-.01
-.01
- .01
-.01
-.01
-.01
-.01
oc
4~OS
W
U
z
O<I:
.....
a:
V'l
«
.....
c::
::>
co
.12
.12
.12
.12
.12
(,11.0
.12
60.i
57.0
57.5
58.0
59.1
.12
56.:'>
.11
S4.9
5~). 4
51;.-1
57.7
59.4
58.2
57.S
S7.9 I
57.3
57.9
59.1
57.7
57.4
56.9
57.8
56.8
57.8 I
.12
.12
.1l
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.12
.11
.12
•
Table 4.3.: Variances for 24 estimators of location based on 1000 Monte Carlorep11cat1ons with first order autore9ress1~e data. Note that *** means
a value larger than 9999.99.
22
5.
Sensitivity Curves.
Tukey (1970) proposes the use of sensitivity curves
to study the finite sample behavior of estimators.
These sensitivity curves
are similar to the influence curves proposed by Parr!pel (1968).
In the
PRS"
stylized sensitivity curves are computed by the following algorithm:
i.
Generate the 19 expected order statistics from a normal sample of
size 19
ii.
To these 19 values add a moving point x and compute the estimator ~
T(x).
iii.
Plot 20T(x)
as a function of x.
Stylized sensitivity curves are produced in the
estirr~tors
and so on.
PRS
for a variety of
including trimmed me&lS, r1-estimators, hampels 9 skipped estimators
We reproduce some additional sensitivity curves.
Again we defer discussion until section 6.
l"~
N
Sensitivity curve for 10% symmetrically trimmed
mean, 10%.
FIGURE 5.1.
-F'
C>
C>
(\)
-F'
C>
<:)
co
<:)
<:)
I
C>
co
C>
C>
I
f\)
-F'
<:)
I
\
:-P -5.00
<:)
C)
I
I
I
I
I
-3.75
-2.50
-1.25
O.
1.25
!
2.Sr-
,
3.75
I
5.00
24
OO'Si
-l.---
00"8-
00"1-
00"1
OO'S
OO'S
~---__l_----_<I~---~I-----___J
U1
C>
<:)
."
......
I
t.0
-.J
U1
G>
c:
;::0
rT1
(Jl
N
(/)
I
1'0
U1
0
(1)
:=l
Vl
ooJ.
rt
ooJ.
<
ooJ.
rt
'<
()
C
t->
1'0
U1
~
<
(1)
-to,
0
~
3
(1)
QJ
C>
:=l
3:
I\)
U1
en
C>
<:)
I.fl
N
FIGURE 5.3.
Sensitivity curve for Mestimate, k=1.5, 015.
-+:
a
o
I\)
-+:
o
o
co
o
o
I
o
co
o
o
I
I\)
-P
o
1
-+:
o
o
1
L
I
I
,
,
I
1
,
,
-5.00
-3.75
-2.50
-1. 25
o.
1. 25
2.50
3.75
5.00
\0
N
FIGURE 5.4.
Sensitivity curve for l-step hampel estimate. 25A .
...p
a
a
f\)
...p
a
a
co
a
a
I
a
co
a
a
I
f\)
~
a
I
...p
a
a
\
I
I
I
I
I
I
,
I
-5.00
-3.75
-2.50
-1. 25
o.
1. 25
2.50
3.75
5.00
c--...
N
FIGURE 5.5.
Sensitivity curve for mildly adaptive hampel estimator, ADA.
-+::
w
C>
I\)
-+::
C>
C>
co
C>
C>
I
C>
co
C>
C>
I
I\)
-+::
C>
I
-+::
C>
C>
-5.00
-3.75
-2.50
-1. 25
o.
1. 25
2.50
3.75
5.00
28
S9' 8-'
I
61 'S-
se 1-
I
S'g' 8
SL'l
I
I
(J1
o
a
I
W
--.J
(J1
"Tl
......
G>
C
I
;:0
IV
rr1
U1
.m
a
(J1
tn
(J)
~
.....I
IV
(J1
Vl
.....
ri"
.....
<
.....
ri"
'<
n
0
c:
-S
<
(J)
--!'l
0
-S
ri"
:;,-
(J)
0
>--'.
c:
IV
(J)
(J1
ri"
-S
I
3
(J)
QJ
~
a
I'J
U1
<::)
W
--.J
(J1
.3:
0>
N
FIGURE 5.7.
Sensitivity curve for Hogg adapter using OM, Mand trimmed means, HG1. Notice
that because of the IIregularll spacing of the quantiles, this stylized curve does
not reflect the adaption to OM.
-...,
Cl
o
f\.)
...p
o
o
co
o
o
I
o
co
o
o
I
f\.)
...p
o
1 \
.
o
o
-5.00
I
-3.75
I
-2.50
I
-1.25
I
O.
I
1.25
I
2.50
;
3.75
i
5.00
o
r-r.
FIGURE 5.8.
Sensitivity curve for Hogg-type adaptor based on trimmed means, HG3.
-+'
o
Cl
f\)
-P
a
a
eo
a
a
J
0.
eo
a
a
I
f\)
..p
a
I
-P
o
a
-5.00
-3.75
-2.50
-1.25
o.
1. 25
I
1
I
2.50
3.75
51)0
.....
l'I")
FIGURE 5.10.
Sensitivity curve for a Hogg-type adaptor using hampels, 1.8l*A.
~
a
Cl
f\)
-+'
Cl
<::)
co
<::)
<::)
I
Cl
co
<::)
Cl
I
f\)
...."
Cl
1 --t
.
Cl
Cl
-5.00
I
·-3.75
I
-2.50
:
-1.25
I
O.
I
1.25
I
2.50
I
3.75
I
5.00
N
l"")
FIGURE 5.9.
Sensitivity curve for Hogg-type adaptor using hampels, 1.95A .
.-p
o
=
l\)
--P
o
Cl
c::o
o
o
-<
1
o
c::o
o
o
I
l\)
.-p
o
i
--P
o
o
I
I
I
I
I
r
1
i
-5.00
-3.75
-2.50
-1.25
O.
1.2S
2. ') 0
3.75
><
.s. __
t"'l
t"'l
FIGURE 5.11.
Sensitivity for 1-step rank estimator with uniform scores,
start = adaptor, RUA .
. .p
CJ
C>
f\)
.-1"
C>
=
C:::l
C>
C>
I
C>
C:::l
=
C>
I
f\)
.-1"
C>
I
..p
o
=
I
-5.00
-3.75
-2.50
-i.25
o.
,
.... 2.5
c. =:
~
3. 15
s.o'J
<::T
tJ")
Sensitivity curve for 1-step, rank estimator with uniform scores,
FIGURE 5.12.
start
-P
=
<=>
=
50%, RU50%.
l')
-P
Cl
Cl
co
Cl
Cl
J
Cl
c:::>
Cl
Cl
I
T\)
.f>
Cl
I
;-P
=
Cl
I
-·5.00
I
I
i
-3.75
-2.50
-1.25
I
o.
!
1.25
i l l
2.S8
3.75
5.0fJ
LIi
tr,
FIGURE 5.13.
Sensitivity curve for normal scores rank estimator, RN. The estimator
was computed by a 12 step method of bisection using the median as initial
estimate. Use of other starting estimators will give slightly different
sensitivity curve.
--f"
a
Cl
f\.)
-.p
Cl
=
co
Cl
Cl
J
Cl
co
Cl
Cl
,J
I
r0
-.p
C>
1---1
~
Cl
C>
-5. QO
I
·-3.75
1
--2.50
I
-1.25
I
O.
I
1. 25
I
2.50
I
I
3.75
5.QO
36
6.
Discussion and Conclusions.
Tables 4.1 and 4.2 contain
with fairly comparable variance or pseudo-variance.
n~lY
estimators
In order to w.ake the
better estimators more apparent, we have constnlcted an additional table,
Table 6.1.
Let
Ti~
i = 1, ••• ,1000 represent the 1000 observations of an
estimator of location, and let
be the sample variance. AsstnJin.g the Ti ' 5 are approximately normal, then
1000si
---~2r-- is distributed approximately as a x2 with 999 degrees of freedom.
O"T
Accordingly, the variance of si is
.79920"i as pointed out earlier.
.0019980"* and the variance of 20si is
An estimate of the standard deviation of si
can be fmmd by calculating 1.00199"3 si = .0447s.i.
Table 6.1 was constructed
by calculating the estimated standard deviation for the estimator with the
smallest variance.
The variance of any estir.1ator, whose estimated variance
was no J:lorc than one standard deviation from the minimal variance, was replaced
with a plus sign while the variance of any estimator ~ whose estimated variance
was more than 5 standard deviations from the minimal variance, was replaced
with a minus sign. All others were left blank.
The resultant picture, Table
6.1, leaves a clear impression of the consistently stronger esttnators.
In terms of the 4 light-tailed alternatives the estimator of choice
to be the outer mean.
appear~,
From Tables 4.1 and 4.2 we observe that HG1, RN and
also perfolTl creditably but significffi1t1y more poorly than
rn~.
J~H
The pseudo-
varaince also suggests that the mean, 1,1, is not too bad.
We feel that the tmiform is a highly artificial (notional in
FRS)
situatio::
The contamination of a normal by a distribution whose support is a bOlmded inte:;
val seems less so.
While this very small selection of short-tailed
alternative~
is inadequate for sweeping comnittments to certain types of estliruators, it is
very suggestive of what is reasonable.
37
_C7'_
_
C1\
0
0
~
z
....,
c::
w
0
LU
~
0
0
S%
10%
-
-
-
50%
8
9
OM
+
010
DIS
020
lZA
17A
21A
Z2A
2SA
-
17
18
19
20
21
22
23
24
In
ZS%
38%
ADA
HGl
HG2
HG3
HG4
HG5
HG6
l.81A
25
26
1. 81*A
27
1.95A
28
2.00'\
1.810
1.900
29
30
31
:S~
1.90A
1.95D
1.60D
33
1. 75D
34
~
3S
36
37
A\IT
38
RNSO
H/L
RN
39 . RNA
40 RlJ50
41 Rlll\
42
43
+
::>
f.1
12
13
14
15
16
+
0
'"
:=>
7
10
11
~
:z
::>
6
3
4
5
0
~
::>
-
2
+
:z
'0
u
~
1
.
~
J¢H
514
-
-
-
--
-
-
-
--
~
0
-
~
0
~
z
- - - - - -- - - - - +- ... + + -- -- -- - - +
- - - - - +
- - - - - - - +
- - - - - - - - - - - - - - - - - - ......
- - - +
- - - +
- - - - - +...
- - - - - ...+
- - - - - -- - - - - - - +
+
- - - - ...
- - - - - +- - - - - -
:z
0
+
:z
C1\
Ii 13'i
I~
IN
10
I
I~
I
0
+
:z
0
C1\
0
0
~
~
z
0
~
z+
-
+
z
L(l
~
z+
:z
L(l
N
z+
+
+
+
+
+
+
,
+
+
+
+
+
+
+
+
-
.
...
+
+
+
...
+
+
+
+
+
+
+
+
...
+
+
+
+
+
+
+
+
+
+
+
...
+
...
...
+
+
+
+
+
+
+
+
+
+
...
+
+
+
+
+
+
+
+
+
+
+
+
+
...
+
...
+
+
+
...
...
+
...
+
+
- - ...- ...
+
...
+
:z
...0"
0
0
~
Z
0
.
~
z+
z+
-
-
-
-
+
+
+
+
+
+
+
+
+
-
+
+
:z
z+
-
+
+
..
+
+
-
-
..
-
+
::>
-
+
-
...
+
-
+
+
+
+
+
+
+
+
+
-
+
+
+
+
Z
-
..-
+
+
-
-
-
I
+
::::
-
'1
- ,l
-
- -
-
...
+
-
-
I
...
+ I
+
+
+
+
+
...
+
I
+
+
+
-
-
+
+
+
+
+
+
+
+
+
...
...
...
...
...
+
I
I
I
...
...
...
+
+
...
...
+
+
...
+
...
+ I
+
...
+
+
-
...
+
+
+
...
-
+
+
- - ...
+
- ...
- ...- + +
-
...
...
-
+
+
+
...
-
-
-
-
+
+
+
+
...
+
-
-
+
...
+
VARIANCtS
Table 6.1.:
Deviations of the variances of -l3 estimators of location. A plus, +, means the
-variance of the estimator is within one standard errOl" of the minimum variance
while a minus. -. means the variance of the estimator is rore than 5 standard
errors away from that of the smallest variance.
!
I
+
+
+
N
~
z+
:=>
...z
0
0
+
+
I
z
+
+
+
-
+
+
+
+
~
uI -- II --.,
'"
lu I I
+
+
+
+
0
z+
-
...
...
+
...
...
...
N
I
lu...., 19
+
...
...
z
L(l
+
+
+
~
+
+
+
+
+
+
+
+
+
+
+
0
~
+
+
- - - - - +- +-
+
~
~
~
0
~
..
.. +.+ +.. ..+ .-
0
C1\
+
-
0
-000
C1\
00
I"')
c:(
c(
0
c:(
e
..--
""
..--
-.010
-.030
-.009
-.008
.007
.020
.030
-.034
-.142
-.233
-.018
-.025
-.119
-.278
-.025
-.094
-.029
-.069
-.034
-.024
.000
.013
.062
- .017
.007
.065
-.044
-.016
.012
.031
-.008
.007
-.019
-.030
-.017
.000
.007
.020
.025
.000
.015
-.033
-.018
-.008
.006
-.078
.008
.013
N
N(O,l)
N+.01N(0,9)
N+.025N(0,9)
N+.05N(0,9)
N+.10N(0,9)
N+.15N(0,9)
N+.25N(0,9)
N+.05N(0,100)
N+.10N(0,100)
N+.25N(0,100)
N+.03C
N+.10C
N+.50C
C
N+.10N/U
N+.25NjU
c:(
N
c:(
N
N
c:(
Ln
N
-.047
-.050
-.051
-.016
.000
.000
.000
-.017
.015
.013
-.044
-.033
-.023
-.044
-.008
.007
.000
.000
.009
.008
.007
.007
-.029
.009
-.007
-.179
.009
.017
-.083
- .172
.017
-.007
..--
-Ie
..--
c:(
c:(
0
0'1
0
0
1-
N
c:(
c:::
.
..--
.
..--
.
..--
-.029
-.040
-.026
-.031
-.028
-.020
.015
-.026
-.029
.120
-.026
-.033
.000
.095
-.025
-.032
-.019
-.020
-.017
-.016
-.007
.000
.010
.000
.007
.037
-.009
-.008
.006
.015
.008
.000
-.019
-.030
- .034
-.016
-.014
-.013
.020
-.026
-.029
.150
.000
-.010
-.009
-.008
-.007
.007
.000
-.008
.000
.051
.000
-.008
-.006
.003
.000
.000
c:(
C
CO
CO
-.026
-.033
.029
.153
-.017
- .013
.010
-.040
-.017
.000
.000
.013
.035
-.017
-.029
.083
-.026
-.025
.000
.105
-.008
.020
--l
c:::t
::I:
Ln
-.084
-.059
-.017
-.008
.052
.013
.020
.000
.015
.007
.000
-.030
.000
-.016
.007
.007
.020
.050
-.179
-.294
-.047
-.069
-.051
-.062
-.056
-.032
.015
-.067
-.029
-.214
-.018
-.008
.006
-.039
.008
.000
-.018
-.033
-.144
-.322
- .041
-.129
- .069
-.041
.047
.098
- .049
-.013
;::
l-
VARIANCES
Table 6.2.:
Logarithm of the relative efficiency. Efficiency in this table is computed as the ratio of the variance of
1.95A to the variance for various alternatives.
Q)
t.f')
I
0
N
0
N(O,l)
N+.01N(0,9)
N+.025N(0,9)
N+.05N(0,9)
N+.10N(0,9)
N+.15N(0,9)
N+.25N(0,9)
N+.05N(0,100)
N+.10N(0,100)
N+.25N(0,100)
N+.03C
N+.10C
N+.50C
C
N+.10NjU
N+.25NjU
-.028
.021
.000
.017
.058
-.007
.010
.000
-.163
- .472
- .016
-.064
-.372
-.301
-.060
-.017
c:(
r--..
r-
-.028
-.051
-.057
.052
.007
.028
.010
.008
-.035
-.037
-.038
-.032
-.006
.050
-.043
-.028
I
I
c:(
c:(
r-
N
N
N
N
- .019
.021
-.033
.008
.043
.063
.031
.031
.014
-.143
.056
-.024
-.034
-.027
.018
-.077
-.028
-.021
-.095
.017
.043
.042
-.064
.008
-.021
-.081
.090
-.032
.018
.066
-.017
-.028
.000
.010
.025
.000
.007
.021
-.054
.000
-.042
-.348
.016
.016
-.079
-.076
.045
-.061
c:(
c:(
LJ")
0
c:(
-.009
-.010
-.041
-.073
-.021
-.053
.000
.039
-.014
.068
.016
-.008
-.034
.241
-.026
-.098
I
c:(
c:(
i<
r-
r-
ce
r-
-.019
.000
-.041
.008
-.007
-.027
.015
.031
-.062
-.032
.064
-.024
-.006
.050
.018
-.006
ce
r-
-.028
.000
-.041
.000
-.034
-.040
.010
.000
-.055
.700
.048
-.094
.073
.195
-.084
-.039
c:(
c:(
0
Q)
0
0
I-
r-
N
c:(
.
.000
.010
-.041
-.025
-.007
-.027
-.015
.000
-.035
.033
.000
.000
-.023
.000
-.026
-.Oll
-.019
.021
-.033
.008
.043
.042
.020
.071
-.048
.068
.056
-.024
-.068
.148
-.034
-.039
:::
.1l2
-.031
-.033
-.008
.014
.085
-.015
.031
.000
-.127
.048
-.016
-.095
.100
.018
-.087
-J
::I:
.029
.065
.017
.070
.043
-.040
-.035
.000
- .163
- .495
.000
- .048
-.343
-.350
-.043
-.077
<::t
ILJ")
-.056
-.021
-.025
.000
-.028
-.097
.031
-.045
- .048
- .191
.048
-.024
-.081
.192
- .1l6
-.039
PSEUDO-VARIANCES
Table 6.3.:
Logarithm of relative efficiency. Efficiency in this table is measured as the ratio of pseudo-variance of
1.95A to the pseudo-variances for various alternatives.
40
Just as dramatic as the a1's good performance in the short-tailed situations is its bad performance in every long-tailed situation.
Table 6.1 also
distinguishes several classes of strong performers in heavy-tailed situations.
As a class, the hampels, numbers 12 through 17 appear strong in spite of the
appearance of several minuses.
The adaptive harnpels, mnnbers 24 through 28
also appear very strong with the added bonus of no minus signs.
Single esti-
mators worthy of mention here include D20, NIT, H/L, RNA and RUA.
The estiP.ators,
~'iA
and RUA, are one-step raJL\ estlinators with an
adaptive initial estimator.
There seems to be evidence that the relatively
good performance in these cases is due to the relatively good initial estimator
rather than the one-step rank estimator itself.
estimators,
For example, the correspondin,?
RNSO% and RUSO%, based on the median as an initial estimator do
not perform particularly well.
Also notice in Figures 5.11 and 5.12, the sen-
sitivity curves portrayed there are essentially tl10se for the initial estimator,
For this reason, we excluded Ri'W, and RUA from the follOWing comparisons.
In the PRS, the concept of deficiency is introduced in order to provide a
comparison of two estimators.
The efficiency of
ffil
estllllator under test rela-
tive to a standard estimator is defined by
efficiency
=
variance of standard
variance
and the deficiency by
deficiency
=1
- efficiency.
Notice tl1at a negative deficiency means the estimator is more efficient than
the standard.
Thus deficiency is centered at
0
with negative meaning the
41
star.61l:"d is less efficient and positive meaning more efficient.
We feel the advantage of the zero reference point is outweighed by the
following anomaly.
In a sense, efficiencies of 2 and !z mean the same thing
(interchanging the roles of the standard estimator with the test estimator).
The corresponding deficiencies of
~
and -1 are not symmetrically located
about 0 and, on the intuitive level, not apparently related.
Rather than compute deficiency, we have computed the natural logarithm of
the likelihood ratio which also has a 0 reference (for efficiency 1) and is
syrmnetrical in the sense that
~n2
= -~11.!z.
The logarithm of the efficiency
also has the advantage that in order to shift standards, only a simple subtraction is necessary.
Tables 6.2 and 6.3 give logarithms of efficiencies for a
variety of the better performing estimators relative to 1. gSA.
the
ch~lge
To illustrate
of standard, the log efficiency of D20 relative to 1.9SA under
Cauchy alternative is -.278 while the log efficiency of 1.8l*A relative to
1.9SA is
.153. Thus the log efficiency of D20 relative to 1.8l*A is just
-.278 - .153
= -.431.
Note that a negative log efficiency means the standard
is more efficient while a positive log efficiency means the standard is less
efficient than the test estimator.
In Tables 6.2 and 6.3, 1.95A
uniformly the best.
adaptive haupels.
\\TaS
used as a standard although it \vas not
However, we feel it adequately represents the class of
For the riioment, let us consider Table 6.2 based on variances
When compared to D20, lillA, H/L and ST4,
that there are more negatives than positives.
1. gSA dominates in the sense
In these cases, the magnitude of
the negative log efficiency is generally much greater than that of positive
log efficiency.
For example, the largest negative log efficiency for D20 is
42
-.278 while the largest positive log efficiency is only .030.
exception to this rule in these cases is the positive
The only clear
.120 for ADA.
We can
point out that relative to 1. 81*A? ADA does not exhibit this behavior.
When compared to the hampels, the adaptive hampe1s do not usually dominate
in terms of sign.
Ilowever~
the worst case behavior still applies.
For
example~
25A has a positive log efficiency in 10 of the 16 cases. However, the maximum
positive log efficiency is only .017 whereas the
ciency is - .179.
srr~llest
negative log effi-
For 22A the maximum positive log efficiency is
the Horst negative case is
.015 while
-.051, and so on.
Table 6.3 is based on pseudo-variances.
Aga:LL the typical pattern is that
the adaptive hampels do not unifonnly dOminate the other estimators ~ but their
worst case behavior is usually considerably better than that of the other estimators.
To
summarize~
with adaptive harirpels, one may lose a slight bit of effi-
ciency in sone cases, but gain a rather large amotmt in others.
In this sense
we say that if hampels are good, adaptive hampels are better.
There is also some question in our minds whether Ql and Q2 are the most
suitable measures of heavy-tailedness. Other more precise measures may yield
higher efficiencies. Also we feel work needs to be done iIl adapting to light
tails.
A more sensitive measure of light-tailedness may improve the efficiencie
there also.
Finally, we mention the possibility of adaptive estimators in a
time series context.
to the
PRS.
In regard to adaptive estirration, we must take exception
Properly fonnulated, blatantly adaptive estimators perfonn at
least as well as non-adaptors for sample sizes less than 40.
43
Sensitivity curves for a variety of estimators were produced in the PRS.
Figures 5.1 through 5.4 reproduce for completeness curves fotmd in the PRS.
Our
figtrres 5.3 and 5.4 contain very slight irregularities in their central
portions which we attribute to the use of the median as a starting estimator
in the one-step procedures.
Figure 5.5 is the sensitivity curve for ADA
which was not given in the PRS.
Figure 5.6 is the outer mean is curve.
that it is Ilinsensitiveli to middle observations.
curve for the Hogg 4 way adaptor HGI.
9
Notice
Figure 5.7 is the sensitivity
The point of interest here is that
there is no iiinsensitivityl1 to middle values.
Thus g in this stylized plot
g
HG1 never has an opporttmity to adapt to ON - the tails of the nomal qucU1tiles being too long to allow this adaptation.
In general one may suspect that
the concept of stylized sensitivity curves is not appropriate to adaptors.
This is true of those adaptors appearing in the PRS also.
With these comments we remark that figures 5.8 through 5.10 represent
other adaptive estimators.
Figures 5.11 and 5.12 are of interest as mentioned
before because they are essentially the sensitivity curves for a hampel and for
the median.
It appears to us as though the convergence of rank estimators is
slow and, l1ence g that one-step rank estimators are unsatisfactory for this
reason.
Figure 5.13 portrays the sensitivity curve for a normal scores rank estimator.
The general raggedness again suggests slO\'1 convergence.
of 12 steps was used.
of this estimator
a~d
Here a maximum
Possiblyg more steps would have improved the performance
the appearance of the curve.
Finally we may address ourselves to Table 4.3 and the time series alternative.
A word on the design of the sar.rp1ing situations is in order.
For the
44
first order autoregressive model
a negative value of
p
guarantees a spectrum domLlated by high frequencies.
In this circumstance observations will occur on both "sides l1 of the center.
For positive values of
particularly for
p
p
low frequencies dominate.
Hence for positive
p~
close to 1 the time series could make long excursions on
one "side11 of the center.
Thus the difficulty in location estiIP.ation in a
series occurs when P is positive and close to one or more
time series is close to nonstationarity.
genera1ly~
tiI::.c~
when the
Gastwirth and Rubin (1975) discuss
robust estimation in precisely this situation.
In
particu1ar~
they derive effi-
ciencies of several esti.m:ltors relativo to the mean M. Unforttmately as Table 4.3
clearly demonstrates M is on the whole unsatisfactory.
It is our assessment
that no estimator presently fashionable is very satisfactory L."1 the presence of
positively correlated data.
A more complicated time series situation might have been to examine a
time series which is contaminated by (possibly) uncorrelated observations.
The
general perfonnance as seen in Table 4.3 did not seem to warrant this investigation~
however.
45
List of References
Andrews, D.F., Bickel, P.J.} Hampel, F.R., Huber, P.J., Rogers, W.H., and
Tukey, J.W. (1972), Robust Estimates of Location: Survey and Advances~
Princeton U. Press, Princeton, N.J.
Bickel, P.J. (1975), One-step Huber estimates in the linear model, J. Am.
Statist. Assoc' 3 70, pp. 428-434.
Box, G.E.P. and IVilller, rLE. (1959), A note on the generation of random nonnal
deviates, Ann. Math. Statist' 3 29 3 pp. 610-611.
Gastwirth, J. (1966), On robust procedures, J. Am. Statist. Assoc'
948.
61 3 pp. 929-
3
Gastwirth, J. and Rubin, H. (1975), The behavior of robust estiInators on dependent data, Ann. Statist.~ 3, pp. 1070-1100,
I-Iampel, F. (1968), Contributions to the Theory of Robust Estimation Ph. D.
Dissertation, Berkeley.
3
Hampel, F. (1974), The influence curve aTld its role in robust estimation"
Am. Statist. Assoc' 3 69 3 pp. 383-393.
J.
Hodges, J.L. and Lehmann, E.L. (1963), Estnnates of location based on raru<
tests, Ann. Math. Statist' 3 34~ pp. 598-611.
Hogg, R.V. (1974), Adaptive robust procedures: a partial review and some suggestions for future applications and theory, J. Am. Statist. Assoc' 3 69~
pp. 909-923.
Huber, P.J. (1964), Robust estimation of a location parameter, Ann. Math.
Statist. 3 353 pp. 73-101.
Jolms, M.V. Jr. (1974), Nonparametric estimators of location,
A8soc.~ 69 3 pp. 453-460.
J.
Am. Statist.
Kraft, C. and Van Eeden, C. (1969), Efficient linearized estimates based on
ranks, Proceedings of the First International Sympoaium on Nonparametric
Statistics 3 Cmabridge U. Press, pp. 267-273.
Paley, R.E.A.C. and Wiener, N. (1934), Fourier Transforms in the Complex Domain:;
American I·1athematica1 Society, Providence, R. I .
Parzen, E. (1962), On the estimation of a probability density function and mode,
Ann. Math. Statist' 33 pp. 1065-1076.
3
3
Schweder, T. (1974), Window estimation of the asymptotic variance of rank
estimators of location, unpublished manuscript.