Robust Procedures for Some Linear Models with one Observation per Cell Author(s): Kjell Doksum Source: The Annals of Mathematical Statistics, Vol. 38, No. 3 (Jun., 1967), pp. 878-883 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/2239004 Accessed: 21/06/2010 07:20 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ims. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
[email protected].
Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematical Statistics.
http://www.jstor.org
ROBUST PROCEDURES FOR SOME LINEAR MODELS WITH ONE OBSERVATION PER CELL'
BY KJELLDOKSUM University of California, Berkeley, University of Oslo, and Institut de Statistique de l'Universit6 de Paris
1. Introductionand summary.For block designswith one observationper cell, the model often used is the linear model in which the observations Xia (i = 1,
, r;oa = 1, *** ,n) canbewritten
( 1.1)
Xia = V +
= O) ,$a + Yia(E {i= E gua
tz+
wherethe i's are the parametersof interest (treatmenteffect) the ,i's are nuisance parameters(block effect), and the Y's are independentwith commoncontinuous distributionF. The purposeof this note is to discusssome new robusttest statistics (e.g. 2.14 and 2.16) of the null-hypothesisHo: j = 6 = * * * = (,r and to discuss a new robust estimate (3.3) of the contrast 0 = E cit . =
2. Testing for absence of mai effect. The tests of Ho
=
=
will be based on the quantities Uij definedby (2.1)
(')Uij = numberolfpairs (a,fl) and
(Xia
-
Xja + X X
with a
0.
Let X(F) = P( Yll < Y12+ Y13- Y14and Yn, < Y15+ Y16 -Y17) and let ,r} be a set of constants, then the results of Hoeff,r;j = 1, {aij : i = 1, ding (1948) on U-statistics yields. LEMMA 2.1. Suppose j - j = aii/nI, then {n'[Uij - E( Uij)]: i < jl has asymptotically the (r - 1)r variate normal distribution with zero mean and covariance matrix 2 = (Oij,kl) given by (2.2)
Sij,ij
=
-X;
0ij,kI
=
[4-X(F) - 1]
0ij,kI=
ij,kl
=
if
0
if
i,
i = k or j = 1; and if
[1-4X(F)]
j, k, l are distinct;
i = l or j = k.
X(F) has been shown by Lehmann (1964) to satisfy (2.3)
4-
