Linear Statistical Inference: Proceedings of the International Conference held at Poznań, Poland, June 4–8, 1984

326 Pages·1985·5.269 MB·English

by  H. Caussinus, J. Vaillant (auth.), T. Caliński, W. Klonecki (eds.)

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ctd. on IneIde beck_
Lecture Notes in
Statistics
Edited by D. Brillinger, S. Fienberg, J. Gani,
J. Hartigan, and K. Krickeberg
35
Linear Statistical Inference
Proceedings of the International Conference
held at Poznan, Poland, June 4-8, 1984
Edited by T. Calinski and W. Klonecki
S pri nger-Verlag
Berlin Heidelberg New York Tokyo
Editors
T. Calinski
W. Klonecki
Mathematical Institute of the Polish Academy of Sciences
ul. Kopemika 18, 51-617 Wroclaw, Poland
AMS Subject Classification: 62-06, 62FXX, 62GXX, 62MXX
ISBN-13: 978-0-387-96255-9 e-ISBN-13: 978-1-4615-7353-1
DOl: 10.1007/978-1-4615-7353-1
Library of Congress Cataloging-in-Publication Data. Main entry under title: Linear statistical
inference. (Lecture notea in statistics; 35) 1. Linear models (Statistics)-Congreaaea.1. Calinski, T.
II. KIonecki, W. (Wrtold) III. Series: Lecture notea in statistics (Springer-Verlag);
0A276.L54841985 519.5 85-25096
This work is subject to copyright. All rights are reaerved, whether the whole or part of the material
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payable to ·Verwertungsgeaellschaft Wort", Munich.
@ by Springer-Verlag Berlin Heidelberg 1985
III
FOREWORD
An International Statistical Conference on Linear Inference was held
in Poznan, Poland, on June 4-8, 1984. The conference was organized
under the auspices of the Polish Section of the Bernoulli Society, the
Committee of Mathematical Sciences and the Mathematical Institute of
the ,Polish Academy of Sciences.
The purpose of the meeting was to bring together scientists from vari
ous countries working in the diverse areas of statistical sciences but
showing great interest in the advances of research on linear inference
taken in its broad sense. Thus, the conference programme included ses
sions on Gauss-Markov models, robustness, variance components~ experi
mental design, multiple comparisons, multivariate models, computational
aspects and on some special topics. 38 papers were read within the vari
ous sessions and 5 were presented as posters. At the end of the confer
ence a lively general discussion session was held.
The conference gathered more than ninety participants from 16 countries,
representing both parts of Europe, North America and Asia. Judging from
opinions expressed by many participants, the conference was quite suc
cessful, well contributing to the dissemination of knowledge and the
stimulation of research in different areas linked with statistical li
near inference. If the conference was really a success, it was due to
all its participants who in various ways were devoting their time and
efforts to make the conference fruitful and enjoyable. First who de
serve thanks are all the speakers and authoFs, the chairmen of ses
sions, and the discussants. They made a good job, highly appreciated
by the majority of the conference attendants. A smooth run of the con
ference is to be attributed to the efforts of the local organizing com
mittee skillfully headed by Dr. R. Kala from the Department of Mathe
matical and Statistical Methods of the Poznan Academy of Agriculture
and Dr. R. Zmy~lony from the Wroclaw branch of the Mathematical Insti
tute of the Polish Academy of Sciences.
This volume contains about a half. out of the number of 43 papers pre
sented at the conference, and represents its main features and results.
We would like to express thanks to all authors who decided to undertake
the task of preparing their papers for publishing in the proceedings.
IV
The standard of the papers owes very much to the work of a number of
referees (a list of their names being included at the end of the vol
ume) to whom not only the editors, we think, but also the authors are
very greatful. We apologize, if some of the authors of submitted papers
have possibly found the referee demandb too restrictive.
In the final editorial work we were very much helped by many of our
co-workers in Wroclaw and in Poznan, and also by the administrative
staff of the Mathematical Institute of the Polish Academy of Sciences
in Warsaw. It is impossible to mention the names of all of them, to
whom we owe so much, but we would like to thank in particular Dr. S.
Zontek. Also we would like to thank Ms. A. Go~dzik and Ms. T. Rejnie
wicz for their excellent typing work. Last, but not least, let us ex
press sincere thanks .to the Publisher for their efficient and friendly
co-operation.
T. Calinski
w. Klonecki
v
CONTENTS
1. H. Caussinus SOME GEOMETRIC TOOLS FOR THE GAUSSIAN
J. Vaillant LINEAR MODEL WITH APPLICATIONS TO THE
ANALYSIS OF RESIDUALS . • 1
2. K. Christaf APPROXIMATE DESIGN THEORY FOR A SIMPLE
F. Pukelsheim BLOCK DESIGN WITH RANDOM BLOCK EFFECTS 20
3. L.C.A. Carsten RECTANGULAR LATTICES REVISITED 29
4. C.~l. Dunnett MULTIPLE COMPARISONS BETWEEN SEVERAL
TREATMENTS AND A SPECIFIED 39
TREAT~1ENT
5. H. Drygas MINIMAX-PREDICTION IN LINEAR MODELS 48
6. N. Gaffke SINGULAR INFORMATION MATRICES, DIREC
TIONAL DERIVATIVES AND SUBGRADIENTS IN
OPTIMAL DESIGN THEORY • 61
7. S. Gnat A NOTE ON ADMISSIBILITY OF IMPROVED
J. Kleffe UNBIASED ESTIMATORS IN TWO VARIANCE
COMPONENTS MODELS. 78
8. J. Jureckava LINEAR STATISTICAL INFERENCE BASED ON
L-ESTIMATORS • • • 88
9. S. Kageyama CONNECTED DESIGNS WITH THE MINIMUM
NUMBER OF EXPERH1ENTAL UNITS . • . 99
10. C.G. Khatri SOME REMARKS ON THE SPHERICAL DISTRI-
BUTIONS AND LINEAR MODELS • • 118
11 • J. Kleffe ON COMPUTATION OF THE LOG-LIKELIHOOD
FUNCTIONS UNDER MIXED LINEAR MODELS 135
12. J. Kleffe SOME REMARKS ON IMPROVING UNBIASED
ESTIMATORS BY MULTIPLICATION WITH A
CON STANT • • • • • • • • • • • • • •• 1 50
13. K. Klaczynski ON IMPROVING ESTIMATION IN A RESTRICTED
P. Pardzik GAUSS-MARKOV MODEL • • • • • • • • •• 162
VI
14. M. Krzysko DISTRIBUTION OF THE DISCRIMINANT
FUNCTION • • • • • • • • • . • • • 170
15. L.R. LaMotte ADMISSIBILITY, UNBIASEDNESS AND NON
NEGATIVITY IN THE BALANCED, RANDOM,
ONE-WAY ANOVA MODEL • • • . • 184
16. T. Mathew INFERENCE IN A GENERAL LINEAR MODEL
WITH AN INCORRECT DISPERSION MATRIX 200
17. S. Mejza A SPLIT-PLOT DESIGN WITH WHOLEPLOT
TREATMENTS IN AN INCOMPLETE BLOCK
DESIGN •. 211
18. E. Neuwirth SENSITIVITY OF LINEAR MODELS WITH
RESPECT TO THE COVARIANCE MATRIX • 223
19. K. Nordstrom ON A DECOMPOSTION OF THE SINGULAR
GAUSS-MARKOV MODEL • 231
20. H. Nyquist RIDGE TYPE M-ESTIMATORS • . • • • . 246
21. E. Torgersen MAJORIZATION AND APPROXIMATE MAJO
RIZATION FOR FAMILIES OF MEASURES,
APPLICATIONS TO LOCAL COMPARISON OF
EXPERIMENTS AND THE THEORY OF MAJO-
RIZATION OF VECTORS IN Rn. • • • 259
22. S. Zontek CHARACTERIZATION OF LINEAR ADMIS
SIBLE ESTIMATORS IN THE GAUSS-MAR-
KOV MODEL UNDER NORMALITY. • • •• 311
SOME GEOMETRIC TOOLS FOR THE GAUSSIAN LINEAR MODEL
WITH APPLICATIONS TO THE ANALYSIS OF RESIDUALS
H.Caussinus and J.Vaillant
Laboratoire de Statistique et Probabilites
Universite Paul Sabatier
Toulouse, France
Summary. We consider the gaussian linear model (1)Y = XS+E where
.5f(Y) = Nn (jJ, a2In)' a> 0, jJ E Q (a linear subspace of IRn). This model
is invariant under the group of transformations Y -+ aY + i; (a> 0, i; E Q)
and a maximal invariant is the vector T of normed residuals. Thus, if
(1) is considered as a null hypothesis to be challenged, the restriction
to invariant procedures leads to perform the analysis via T. The matrix
approach is not very convenient to deal with T because the mapping
Y -+ T is not linear. In fact, if model (1) is true, T is uniformly
distributed on the unit sphere S of Ql. • Under some alternatives it
is easy to compute the density of T with respect to the uniform
probability on S, for example when .5f(Y) = Nn (v ,w), v ~ Q •
The formulation which we advocate leads to straightforward results
concerning the optimality of some procedures and enables us to give
a clear account of the assumptions which are used. The problem of k
outliers detection is more specially discussed, including the case where
k is not fixed.
1. Introduction and notation
Throughout this paper NE(v,W) denotes the normal (gaussian) distribu
tion on the Euclidean space E with mean v and variance operator W.
The scalar product on E is denoted by <.,.> and the corresponding
KEY WORDS: Gauss-Markov model; coordinate-free approach; analysis of
residuals; outliers; Bayes procedures; distributions on spheres.
2
narm by II ·11 ~ The dual space af E is identified to' E by means af
the scalar praduct in the usual way, So' that W is a symmetric linear
2
aperatar from E intO' E. The distributian NE (O,er IE) (er> 0, IE is
the identity aperatar fram E intO' E) is called spherical
narm~l;
it is preserved by any arthaganal transfarmatian and the prajectians
anta arthaganal subspaces are independently spherical narmal an each
af these spaces.
The unit sphere af E is denated by SE' the unifarm prabability an
SE is UE, the arthaganal prajectar anta a linear subspace Q is
denated by lTQ and Q.l is the subspace af E arthaganal to' Q.
When the cantext is clear subspaces as indices may be drapped ar
replaced by simpler anes: far example Sn instead af S ,etc. •. •
lRn
The prabability law af a randam variable (generally vectar) is denated
by •
~(.)
In the classical linear madel, a sequence af real randam variables
Y1'Y2' ••• 'Yn is given. These are cansidered as the coordinates of a
n
vector Y with respect to a canonical basis e = ( e1, ••• ,en I of lR .
If the matrix of variances-covariances of the Yi'S is er2r (known up
to' the factar er2 and nansingular) the metric on lRn is chosen such
that the n x n matrix of the scalar products <ei ,ej >, i, j = 1, •.• ,n,
-1 2
is r ,which implies that the variance operator of Y is er In.
Now the gaussian linear model is
I.i e: Q, (1)
where Q is a given subspace of lRn, dim(Q) = q, 0< q< n.
This model is invariant under the group of transformations Y+ aY+~
(a>O, ~e:Q) and a maximal invariant is the vector of normed residuals
T = lilT .1 (Y)\\ IT .1 (Y) •
Q Q
In the analysis of residuals model (1) is considered as a null
hypothesis to be challenged. The restriction to invariant procedures
leads to performing the analysis through T. In the framework of model
3
(1) the normed residuals,i.e. the coordinates of T in the basis e,
do not have an easily handled distribution. However the distribution
of the vector T is extremely simple since it is nothing more than
the uniform probability on the unit. sphere of d-: I£ (T) = U .1.
Q
Under various alternatives to model (1) it is tempting to express the
distribution of T with respect to in a coordinate free approach.
Ud-
This is done in Section 2 for the case where I£ (Y) = Nn (v , W), v rt Q.
Section 3 gives applications to residual analysis emphasizing the case
of outlier detection. Other applications are indicated in Section 4.
Also in this section we discuss briefly a kind of robustness arising
from the fact that I£(T) could be U.l without I£(Y) being normal.
Q
Finally Section 5 is devoted to the slightly different problem of
"modified residuals". This section has been included for historical
interest, still stressing the usefulness of the geometrical framework.
For earlier papers using the coordinate - free approach to Gauss-Markov
estimation the reader could refer to Kruskal (1960) and Drygas (1970).
However the geometrical tools developed herein are somewhat different
and turn out to be also closely related to papers on spherical symmetry,
e.g. King (1980).
2. Technical results
This section gives some useful probabilistic results. Lemmas 1 and 2
are well known properties of the normal distribution: see e.g. Dempster
(1969), Chapter 12. Lemma 3 can be found in Degerine (1979) or Watson
(1983), Chapter 2 (2.2.6). Lemma 4 is an obvious general result which
leads to a straightforward derivation of Proposition 1.
Lemma 1: If Z is a random vector with values in the Euclidean space
E, dim (E) = m, then
(i)
1
2(Z) = NE(O,IE) tot (ii) are independent
(iii)
Remark: Z is spherically distributed if (i) and (ii) hold.