.. M-ESTIMATORS AND L-ESTIMATORS OF LOCATION: UNIFORM INTEGRABILITY AND ASYMPTOTIC RISK-EFFICIENT SEQUENTIAL VERSIONS by JANA JURECKOVA Department of Probability & Statistics Charles University Prague 8, Czechoslovakia and PRANAB KUMAR SEN Department of Biostatistics University of North Carolina Chapel Hill, NC 27514, U.S.A. Institute of Statistics Mimco Series No. 1280 APRIL 1980
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M-ESTIMATORS AND L-ESTIMATORS OF LOCATION:UNIFORM INTEGRABILITY AND ASYMPTOTICRISK-EFFICIENT SEQUENTIAL VERSIONS
by
JANA JURECKOVADepartment of Probability &Statistics
Charles UniversityPrague 8, Czechoslovakia
and
PRANAB KUMAR SENDepartment of BiostatisticsUniversity of North Carolina
Chapel Hill, NC 27514, U.S.A.
Institute of Statistics Mimco Series No. 1280
APRIL 1980
M-ESTIMATORS AND L-ESTIMATORS OF LOCATION; UNIFORMINTEGRABILITY AND ASYMPTOTIC RISK-EFFICIENT SEQUENTIAL VERSIONS
by
JANA JURECKOVADepartment of Probability &Statistics
Charles UniversityPrague 8, Czechoslovakia
and
PRANAB KUMAR SENlDepartment of BiostatisticsUniversity of North Carolina
Chapel Hill, NC 27514, U.S.A.
ABSTRACT
Sequential M- and L-estimators of location minimizing the risk
asymptotically as the cost of one observation tends to 0 are con-
structed. Their asymptotic risk efficiencies are shown to coincide
with the asymptotic efficiencies of the respective non-sequential
estimators; this enables to construct the asymptotically minimax
sequential M- and L-estimators in the model of contaminacy. The
asymptotic distributions of the stopping times are derived for both
types of estimators. The theorems on uniform integrability and moment
convergence of (non-sequential) M- and L-estimators, developed as the
main tools for the proofs, have an interest of their own.
Key Words &Phrases: M-estimator, L-estimator, asymptotic risk-
efficiency, sequential point estimator, moment convergence.
lResearch of this author was supported by the National Heart, Lungand Blood Institute, Contract NIH-NHLBI-7l-2243 from the NationalInstitutes of Health.
-2-
1. INTRODUCTION
Nonparametric sequential point estimation of location has
received considerable attention during the recent past. Ghosh and
Mukhopadhyay (1979) and Chow and Yu (1980) have considered asymptotically
risk-efficient sequential point estimation of the mean of a population
based on the sequence of sample means and variances, while Sen and
Ghosh (1980) have extended the theory to general U-statistics. Sen
(1980) has considered the problem of estimating the location of
symmetric (but unknown) distribution based on a general class of rank
order (or so called R-) estimators and established the asymptotic risk
efficiency of the proposed sequential procedure. In the classical
non-sequential case, the R-estimators form one of the three main
groups of robust competitors of classical estimation procedures; the
other two major groups are formed by M-estimators and L-estimators
[viz., Huber (1973, 1977)]. The theory of asymptotically risk-
efficient (sequential) point estimation based on a broad class of M-
and L-estimators is developed in the current paper. Unifor,m integra
biZity and moment-convergence properties of these M- and L-estimators
playa fundamental role in this context.
Along with the preliminary notions, the proposed sequential point
estimation procedures are outlined in Section 2. Section 3 is devoted
to the study of uniform integrability and moment convergence of the
M-estimators. Parallel results for the L-estimators are considered in
Section 4. These results are then applied in the proofs of main
theorems of Section 5 concerning the properties of the proposed
sequential procedures. In particular, the Section 5 deals with the
asymptotic risk-efficiency and with the asymptotic normality of the
"
•
-3-
allied stopping times. Similarly as in the case of R-estimators
[Sen (1980)], it is shown that the asymptotic risk efficiencies of
sequential estimators coincide with the asymptotic efficiencies of
their non-sequential versions. This among others enables to extend
the asymptotic minimax properties of M- and L-estimators in the
model of contaminacy to the sequential case.
2. THE PROPOSED SEQUENTIAL PROCEDURES
Let {X., i ~ I} be a sequence of independent and identically1
distributed random variables (i.i.d.r.v.) with distribution function
(d.f.) F8
(x) ==F(x -8), xsR == (_00,00), where F (unknown) is symmetric
about 0 and 8 is the unknown location parameter to be estimated. Let
T be a sui table estimator of 8 based on Xl' ... , Xn and assume thatn
2 2 exists for all (2.1 )~e
'J == nE (T - 8) n ~ nO'n n
for some nO (~l) and
2 2'J -+ 'J as n +00, 0 < 'J < 00.
n
We conceive the loss (in estimating 8 by Tn)
2~ (a, c) == a(Tn - 8) + cn,
(2.2)
(2.3)
where a and c are positive constants. Then the risk is
-1 2An(a, c) == E~(a, c) == n a'Jn +cn. (2.4)
We like to minimize (2.4) by a proper choice of n. The optimal choice
of n generally depends on the unknown F, for any fixed c as well
as asymptotically as c 1- O. In this asymptotic case, the optimal
and A ()(a,c)~2'J&nO c
lim q(c)/r(c) ==1. This suggestsdObe a sequence of estimates of 'J
choice of n is no(c), where!<
no(c) ~b'J, b == (a/c) 2
where q(c) ~r(c) denotes that
following procedure: Let {Vn}
and let n' be an initial sample size (~2) and h(>O) be an
(2.5)
the
arbitrary constant. Define
N =min{n ~n':c
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a stopping number N(= N )c" -h }n~b(v +n ) , c>O
n
by
(2.6)•
based on Xl, ... ,XN .c
and consider the sequential point estimator TNc
The risk of estimating 8 by TN is thenc 2
A*(a, c) = aE(TN -8) +cENc (2.7)c
We are primarily interested in showing that
A*(a,c)/A ()(a,c)-d as c+o, (2.8)nO c
which means that the sequential procedure is asymptotically (as c +0)
equally risk-efficient as the optimal non-sequential one, if v were
known.
The convergence (2.8) has been studied by more authors [referred
to in Section 1] in the case that {T }n
is either the sample mean,
U-statistic or some case of R-estimator. In the current paper, we
shall show that (2.8) holds for a broad classes of M-estimators
and L-estimators (i.e., the estimators of maximum-likelihood type and
of linear combination of order statistics type, respectively).
An M-estimator M of 8 is a solution of the equation·n
(2.9)
with respect to t, where ~ is some nondecreasing score function
(so that S (t)n
is \ in t) . More precisely, is defined by
where
M = (M* +M**)/2,n n n
M* = sup{t: S (t) >O} and M** =inf{t: S (t)<O}.n n n n
(2.10)
(2.11 )
Under suitable regularity conditions on ~ and on F [viz., Huber
(1964)], to be specified later on,
1 2L{n~ (Mn - 8) } ~ N(0, v (M)) as n -T 00
where
(2.12)
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V~M) =cr~M/Y (M)' cr~M) = flO1jJ2 (x)dF (x) ,_00
00
Y(M) =y(1jJ, F) = f {-f'(x)/f(x)}1jJ(x)dF(x) (>0)
(2.13)
(2.14)
d d2
and f' (x) =dxf(x) =~ (x) is assumed to exist almost everywhere.dx
In Section 3, we shall show that (2.1) and (2.2) hold for M-estimators
generated
estimate
by a class of bounded 1jJ-functions.
2V(M) as follows. Let
2 -1 n 2s (M) =n L· l1jJ (X. - M ), n ~ 1,n 1= 1 n
In this case, we shall
(2.15)
let <P be the standard normal d.f. and let <P(-TE) =E, O<E<l.
F sF;-sfor which sup f (x) (F (x) [1 - F (x)]) < 00,
x
for some s >1/6. It is easy to verify that the intersection of F~
F*4
is a non-null set.
be the set of distributions defined in (6.4)THEOREM 6.1.
and let Tl
Let F*1
be the set of sequential estimators satisfying (6.1)
and (6.2) for F s Fi . Then there
Tel) and a sequential L-estimatorN
c
exist a sequential M-estimator
(2) d F*TN an FOE 1 such thatc
,
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e(TN
' FO) ~e(T~i), FO) = [I(Fo)]~~e(T~i), F)c c c
ho Zds faY' i =1 ,2, V TN £ T1 and F £ Fi . .c
(6.5)
PROOF. It follows from J-Juber (1964) that the asymptotic variances of
M-estimators with the ~-functions satisfying (2.32) - (2.34) have a
saddle-point which corresponds to the density
(1 _E)g(k)eq(x+k), x <-k
fo(x) = (l - E)g(X) , -k ~ x ~ k (6.6)
(l - E)g(k)e-q(X+k), k~x
and the corresponding minimax M-estimator is the maximum-likelihood
estimator corresponding to fO' i.e.,
xER, i.e., (6.7)
where q = -
-q
g I (x)g(x)
q
and k
x < -k
-k < x < k
k<x
is related to E according
(6.8)
(6.9)
It follows from Theorem 5.1 that the sequential M-estimator
corresponding to ~O and to the stopping rule (2.19) is the
of (6.5).
T(l)N(M)
csolution
Moreover, it follows from Jaeckel (1971) that the L-estimator
T(2) corresponding to the weight function.n
(6.10) .
is asymptotically equivalent to
(2) .T (L) with the stopping rule
Nc
alternative solution of (6.5).
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T(l)n •
N (L)c
The sequential L-estimator
given by (2.28) is then an •
•
REFERENCES
ANSCOMBE, F.J. (1952). Large sample theory of sequential estimation.Froc. Camb. PhiZ. Soc. ~~, 600~607 .
CHOW, Y.S. and YU, K.F. (1980). The performance of a sequentialprocedure for the estimation of the mean. (Unpublished),
CSORGO, M. and REVESZ, P. (1978). Strong approximations of thequantile process. Ann. Statist. 2, 882-894.
GARDINER, J.C. and SEN, P.K. (1979). Asymptotic normality of avariance estimator of a linear combination of a function oforder statistics. Zeit. Wahrsch. Verw. Geb. ~Q, 205-221.
GHOSH, M. and MUKHOPADHYAY, N. (1979). Sequential point estimationof the mean when the distribution is unspecified. Comm.Statist. A2' 637-652.
HOEFFDING, W. (1963) .random variables.
Probability inequalities for sums of boundedJour. Amer. Statist. Assoc. ~~, 13-29.
HUBER, P.J. (1964). Robust estimation of a location parameter.Ann. Math. Statist. ~2' 73-101.
HUBER, P. J . (1969). Theorie de l' inference statistique robuste.Seminaire de mathematiques superieures. Universite de Montreal.
•HUBER, P.J. (1973) .
and Monte Carlo .Robust regression: Asymptotics, conjecturesAnn. Statist. 1, 799-821.
IIUBER, P.J. (1977). Robust methods of estimations of regressioncoefficients. Opert. Statist. Ser. Stat. ~, 41-53.
JAECKEL, L.A. (1971). Robust estimates of location: Symmetry andasymmetric contamination. Ann. Math. Statist. ~~, 1020-1034.
JURECKOVA, J. (1969). Asymptotic linearity of a rank statistic inregression parameter. Ann. Math. Statist. ~Q, 1889-1900.
JURECKOVA, J. (1977). Asymptotic relations of M-estimates andR-estimates in linear regression model. Ann. Statist. ~, 464-672.
v ..JURECKOVA, J. (1980). Asymptotic representation of M-estimators of
location. Opert. Statist. Ser. Stat. 11, (in press) .
JURECKOVA, J. and SEN, P.K. (1980a). Invariance principles forsome stochastic processes relating to M-estimators and theirrole in sequential statistical inference. Sankhya~ Series A(under consideration for publication).
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JURECKOVA, J. and SEN, P.K. (1980b). Sequential procedures based onM-estimators with discontinuous score functions, Institute ofStatistics Mimeo Series No. l279. The University of NorthCarolina, Chapel Hill, NC.
SEN, P.K. (1959). On the moments of sample quantiles. CalcuttaStatist. Assoc. Bull. 2, 1-19.
SEN, P.K. (1978). An invariance principle for linear combinationsof order statistics. Zeit. Wahrsch. Verw. Geb. ~£, 327-340.
SEN, P.K. (1980). On nonparametric sequeritial point estimation oflocation based on general rank order statistics. Sankhya~ SeY'. Ai£ (to appear).
SEN, P.K. and GHOSH, M. (1980). Sequential point estimation ofestimable parameters based on U-statistics (under considerationfor publication).
von BAlm, B. (1965). On the convergence of moments in the centrallimit theorem. Ann. Math. Statist. ~£, 808-818.