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Invariant Estimation with Application to Linear INVARIANT ESTIMATION WITH APPLICATION TO LINEAR MODELS by Mary Sue Younger Dissertation submitted to the Graduate Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Statistics APPROVED: Dr • Lawrence S • M~yer , ''G;hairman Department of Statistics Dr'. Ra§morid/ijf. Myers Dr. d Harshbarger, H ad Department of Statistics Dep tment of Statistics Dr. I • (fP"Good Dr. Paul W. Hamelman Department of Statistics Department of Business Administration Dr. David L. Klemmack Department of Sociology August, 1972 Blacksburg, Virginia ACKNOWLEDGEMENTS The author would like to express her appreciation to the following people for their help in the preparation of this dissert~tion: To Dr., Boyd Harshbarger for his encouragement throughout the author's years of graduate study and for his securing the financial aid which made graduate study possible. i To D:r. Lawn:::nci<:' S. Mayer, who directed this research with tireless I i energy and without whose confidence, enthusiasm and guid,nce this dis- sertation would ha.ve never bee.n undertaken. I To Dr. Raymond H. Myers for his invaluable suggestions while co- reading this Ule.nuscript and his continued concern and encouragement throughout the author's years of graduate study. To the Faculty serving on the Committee for their in,terest and critic.ism. To the author's colleagues in the College of Busine~s for their ' I interest and concern for the completion of this dissertat,ion, and for their atd in having copies made. To who did the computer work necessar~ to the examples presented. Finally, to who typed th~s manuscript quickly and accurately, and who never became irritated at' having to make so many corre.ctions and revisions. TABLE OF CONTENTS Page ACKNOWLEDGMENTS •••••• . .. ii CHAPTER I: INTRODUCTION • • . .. 1 CHAPTER II: REVIEW OF GROUP THEORY. 5. CHAPTER III: THE STRUCTURE OF THE ESTIMATION PROBLEM. • • 14 Section 3.1: Definition of an Equivariant Estimation Problem • • • • • • • • . 14 Section 3.2: Additional Structure of the Proposed Problems ••• . .. 16 Section 3.3: An Example of the Estimation Structure Under Consideration • • • • • • • • • • • . 18 CHAPTER IV: INVARIANT ESTIMATION OF ORBITS OF THE PARAMETER SPACE • • • • • • • • • • • • • • • • • • • • • . 26 Section 4.1: A Group Theoretic Function Which Indexes Orbits . ................ 26 Section 4.2: A Maximal Invariant Parametric Function • • 31 Section 4.3: An Inva~iant Sufficient Estimator of the Index of Orbits • • • • • • • • • . 33 CHAPTER V: APPLICATION TO THE LINEAR MODEL: NONSTOCHASTIC INPUT MATRIX • • • • • • • • • • • • • • • • • • . 38. CHAPTER VI: COMPARISON OF THE ESTIMATORS OF THE MAXIMAL INVARIANT PARAMETRIC FUNCTION • • • · • • • • . 51 Section 6.1: Properties of !"' . 51 Section 6.2: Properties of the Usual Social Science Estimator • • • • • • • • • • • • •. • .. • • • • 6 7 . CHAPTER VII: RELATIONSHIP OF PROPOSED ESTIMATOR TO THE MINIMUM RISK EQUIVARIANT AND FIDUCIAL ESTIMATORS . 75 Section 7 .l: Relationship to the Minimum Risk Equivariant Estimator • • • • • • . 75 Section 7 .2: Relationship to the Fiducial Estimator. 78, iii iv Page CH:\PTER VIII: APPLICATION TO THE LINEAR MODEL: STOCHASTIC INPUT YJ.ATRIX. • 88 Section 8ol: Alternative Parametric Formulations . 88 SEction 8,2: Properties of the Estimator in Formula t:i.on Oue. • • • • • . 91 Section 8,3: Properties of the Estimator in Formulation Two • . • • • • 97 CHAPTER IX: ESTIJ\llA.TION IN THE LINEAR MODEL: INVARIANCE UNDER NONSINGULAR TRANSFORMATIONS . 103 CHAPTER X: EXAMPLES OF APPLICATION OF THE PROPOSED PROCEDURES 109 Section 10.l: Application to Sociological Data . • • 109 Sect:i.on 10. 2: Application to Engineering Research Data 121 Section 10.3: Summary. 126 REFERENCES , 127 VITA .. • f, ~ 129 LIST OF FIGURES AND TABLES Figure Page 10.Ll Path Diagram showing influence of family characteristics a.nd age> ma.rriage aspirations (from Klemmack) . 110 Table 10.Ll Values of B, 'VB and B* for the regressions indicated in Figure 10.l.l •... 113 '\_, * 10.1.2 Values of B, B and B for pred~cting x10 (FOCI) and x11 (F'OC2) from the other nine van.ables. • • . 114 10 .1. 3 Correlation matrix for all eleven variables 116 Multiple coefficients of determination •.. 118 10.LS Values of (n-1) 112 } • ~for the regressions of FOCl and :F'OC2 from x1 , .•. ,x9 • · · • • · • • · • • · 119 1/2 'V 10.1.6 Values of (n-1) ! = ~ for the five regressions indicated in Figure 10.1.1. , ••.. 120 10.2.1 Correlation matrix for data from Gorman and Toman • 122 10 2.2 Marginal ~t~ndard deviations of x1 , ... 9 X10 and y for Gor1nan 8.D.G .lon18J.1 data . .. " '1 "' " 0 • 0 .. 0 " e " • • • 123 'V 10.:Z.3 Values of f3, B, B and B * for the Gor;man and Toman data. 12.4 o.c (11-1) 1/2 '\,, 10 .2 "lf Values .l t -- B for Gorman and Toman data. 125 v CHAPTER I INTRODUCTION The principle of adopting a group of transformations on the sample space :Ln a statistical estimation problem is quite common. Usually the group structure adopted affects the problem in one of two ways: i) The group structure is isomorphic to the parameter space and is assumed to "ca.rry" the esU.mation problem in a natural manner. Attention is re- stricted to cst:tmators which are also "carried11 by the group in a nat- ural manner. ii) The group structure generates orbits on the parameter space and the problem ts to estimate in which orbit a parameter lies. Attention. is restricted to estimators which are unaffected by the group. Estimation problems of type i) e.re called ~_q~ivariant estimation .E.EE.!21.£~ and have been widely studied. For reviews of this work see Zacks (1971), Ferguson (1967). Fraser (1968), Pitman (1939), Brown (1966), and Younger (1969), Equivaria.nt estimation procedures are usu- ally adopted whe.n it is impl)ssible to find a "best" estimator within thf:! class of a.11 estimators; by restricting attention to the class of equivar:i.ant estimators an optimal estimator may appear. Much of the literattn:e deals with the properties of the "best" equivariant esti- ma tor when consi.dere.d as a member of the class of all estimators. Hora and Buehler (1966) show the relationship between the best fiducial and best equivariant: estimators. Estimation problems of type ii) are called inv_ariant estimatio~ E!PJ:?.le~.' These ha 11e been considered by Wijsman (1965), Stein (1959), .,. - (-q-·;) ".- /19 ' . T'' -· c-12tcr ,L.:i, , h.ioo \.L 55), Hall, ~.1Jsrr..an and Ghosh (196::>), and Zacks (1971), Jnvar:i.ant estimation problem~: arise when the orbit of the 1 2 pa.rarneter unde.r a certain group, but not the parameter itself, must be estimated. The group can be thought of as a "nuisance" group, since estimau.on :Ls restricted to be modulo the group. The most notable re- s1il.ts dealing with these problems include the Stein theorem relating invariance and sufL'Lc:i.ency, as presented by Zacks (1971) or Hall, Wij s- man and Ghosh (1965); Wijsman's (1965) results obtaining probability ratios of m.snd.nial invariants, utiliztng invariant measures and differ- ent:.ial manifolds; and th~:. results in multi.variate distribution theory using invarianc:e under the orthogonal group presented by James (1954, 196l:). In thls dj_ssertation we study estimation problems which give rise to both types of group structures. In particular, the parameter space m.ipportr; a nuisance. gToup of transformations and the objective is to estimate the orb.it in which the parameter lies, thus giving rise to an i.nva:r:i.ant est:lmation problem. In addition, a second group can be de- f:i.ned such that the set of ordered p:;<irs from the two groups is given a g:roup structm:e and becomes isomorphic to the parameter space, thus de.fi.ni:ng an cqu:lvariant esti.ma.t:Lon problem. Part of the goal will be to show that if the relationship between the two groups satisfies cer- tain propertiE:~s, then an estimator of orbits in the invariant estima- tion problem can be naturally obtained from the best estimator in the equivariant estimation problem. The motiva.tion for the problem studied here is the problem of est:Lmat:iI:g the pararnet1::rs in the general linear model, the nuisance group being the group of scale changes on the dependent and independent vsri.abli.~s. Thi!_, problem is of interest since. in many areas of research, 3 mc~1t n.otc.bly in the soc.:La1 sciences, an estimate of the. relationship between variables is required which is independent of the units in which the variables are measured" The invariant estimator commonly used is cl:lllecl ;.:; "b~;t:a coefficientr: or "standardize.d regression co- eff:Lcf.e.nt," but the.;c,;o .ccppears to have been no rigorous derivation of th Li meaGure.rne:nt and no investigation of its properties. In Chapter II is a review of the definitions and concepts from group theory necessary to the development of invariant estimators. \.Jli:U<:. rnost of t11e notions presented c:an be found in any elementary text on group theory, some less conventional ideas are introduced in order to fac:LUta.te. the application of group theory to statistical problems, ChElpter III cons!.ders the structure of the estimation problem, defining the pars.meter~ sample~ and estimation spaces of interest and !;he groups of tn.111.sfonnations defined on them. Chapter IV develops the theon:.>.t:Lc baG:i.n for· invariant estimation of the orbit in which Application of th~; theory developed in Chapters III and IV to the probJ.(:.m of invariant est.imat.:i.on in the linear model for the case in which the. indr2pe,:.1'.d.ent ve.rinbles a.n: nonrandom is contained in Chapter V.
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