A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models

Abstract Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ² V, where σ² is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X ₁ : X ₂) is an n×(p+q) known design matrix with rank(X) = r ≤ (p+q), and β = (β′ ₁: β′₂ )′ with β₁ and β₂ being p×1 and q×1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M ₂ X ₁β₁under the model and its best linear unbiased estimators under the reduced linear models of are given, where M ₂ = I -X ₂ X ₂ ⁺ . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M ₂ X ₁β₁ under the model and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model. *This work is supported by the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 10226024), Postdoctoral Foundation of China and Lab. of Math. for Nonlinear Sciences at Fudan University. This research is supported in part by The International Organizing Committee and The Local Organizing Committee at the University of Tampere for this Workshop **The work is supported in part by an NSF grant of China. Results in this paper were presented by the first author at The Eighth International Workshop on Matrices and Statistics: Tampere, Finland, August 1999     

Asymptotics of Huber-Dutter Estimators for Partial Linear Model with Nonstochastic Designs

For partial linear model Y=X~τβ_0 _(g0)(T) εwith unknown β_0∈R~d and an unknown smooth function go, this paper considers the Huber-Dutter estimators of β_0, scale σfor the errors and the function go respectively, in which the smoothing B-spline function is used. Under some regular conditions, it is shown that the Huber-Dutter estimators of β_0 and σare asymptotically normal with convergence rate n~((-1)/2) and the B-spline Huber-Dutter estimator of go achieves the optimal convergence rate in nonparametric regression. A simulation study demonstrates that the Huber-Dutter estimator of β_0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator. An example is presented after the simulation study.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

Weighted least-squares estimators of parametric functions of the regression coefficients under a general linear model

The weighted least-squares estimator of parametric functions K β under a general linear regression model { y, X b, s²S }{\{ {\bf y},\,{\bf X \beta}, \sigma^2{\bf \Sigma} \}} is defined to be K[^(b)]{{\bf K}{\hat{\bf {\beta}}}}, where [^(b)]{\hat{{\bf \beta}}} is a vector that minimizes (y − X β)′V(y − X β) for a given nonnegative definite weight matrix V. In this paper, we study some algebraic and statistical properties of K[^(b)]{{\bf K}\hat{{\bf \beta}}} and the projection matrix associated with the estimator, such as, their ranks, unbiasedness, uniqueness, as well as equalities satisfied by the projection matrices.     

Empirical Bayes Estimation in Regression Model

This paper considers the empirical Bayes （EB） estimation problem for the parameter β of the linear regression model y = Xβ＋ ε with ε- N（0, σ^2I） given β. Based on Pitman closeness （PC） criterion and mean square error matrix （MSEM） criterion, we prove the superiority of the EB estimator over the ordinary least square estimator （OLSE）.     

Consistency and normality of Huber-Dutter estimators for partial linear model

For partial linear model Y = Xτβ0 g0(T) with unknown β0 ∈ Rd and an unknown smooth function g0, this paper considers the Huber-Dutter estimators of β0, scale σ for the errors and the function g0 approximated by the smoothing B-spline functions, respectively. Under some regularity conditions, the Huber-Dutter estimators of β0 and σ are shown to be asymptotically normal with the rate of convergence n-1/2 and the B-spline Huber-Dutter estimator of g0 achieves the optimal rate of convergence in nonparametric regression. A simulation study and two examples demonstrate that the Huber-Dutter estimator of β0 is competitive with its M-estimator without scale parameter and the ordinary least square estimator.     

Moment convergence rates in the law of the logarithm for dependent sequences

Let {X _n ; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S _n = Σ _k=1 ⁿ X _k , M _n = max _k≤n |S _k |, n ≥ 1. Suppose σ ² = EX ₁² + 2Σ _k=2^∞ EX ₁ X _k (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M _n −σɛ√n log n}₊ and E{|S _n | − σɛ√n log n}₊ as ɛ ↘ 0 and E{σɛ√π ² π/8logn − M _n }₊ as ɛ ↗ ∞ are obtained.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

Nonparametric estimation for censored lifetimes suffering from unknown selection bias

In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line (σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ₊. Denoting by σ _S and X _S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F _S of the r.v. X _S suffers from a selection bias in the sense that F _S = ∝ w d F/μ _S, where w and μ _S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X _S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator for the d.f. F _S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for . We establish in addition an exponential bound for .      

A convexity theorem for multiplicative functions

We generalize a well known convexity property of the multiplicative potential function. We prove that, given any convex function g : \mathbbR^m ? [0, ￥]{g : \mathbb{R}^m \rightarrow [{0}, {\infty}]}, the function ${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}${({\rm \bf x},{\rm \bf y})\mapsto g({\rm \bf x})^{1+\alpha}{\bf y}^{-{\bf \beta}}, {\bf y}>{\bf 0}}, is convex if β ≥ 0 and α ≥ β ₁ + ··· + β _n . We also provide further generalization to functions of the form (x,y₁, . . . , y_n)? g(x)^1+af₁(y₁)^-b₁ ···f_n(y_n)^-b_n{({\rm \bf x},{\rm \bf y}_1, . . . , {y_n})\mapsto g({\rm \bf x})^{1+\alpha}f_1({\rm \bf y}_1)^{-\beta_1} \cdot \cdot \cdot f_n({\rm \bf y}_n)^{-\beta_n} } with the f _k concave, positively homogeneous and nonnegative on their domains.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

Large deviations for heavy-tailed random sums of independent random variables with dominatedly varying tails

We prove large deviation results on the partial and random sums Sn = ∑i=1n Xi,n≥1; S(t) = ∑i=1N(t) Xi, t≥0, where {N(t);t≥0} are non-negative integer-valued random variables and {Xn;n≥1} are independent non-negative random variables with distribution, Fn, of Xn, independent of {N(t); t≥0}. Special attention is paid to the distribution of dominated variation.     

On the abelian fundamental group scheme of a family of varieties

Let S be a connected Dedekind scheme and X an S-scheme provided with a section x. We prove that the morphism between fundamental group schemes π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) induced by the canonical morphism from X to its Albanese scheme Alb _X/S (when the latter exists) fits in an exact sequence of group schemes 0 → (NS _X/S ^τ )^⋎ → π ₁(X, x) ^ab → π ₁(Alb _X/S , 0_AlbX/S{0_{{\rm{Al}}{{\rm{b}}_{X/S}}}}) → 0, where the kernel is a finite and flat S-group scheme. Furthermore, we prove that any finite and commutative quotient pointed torsor over the generic fiber X _η of X can be extended to a finite and commutative pointed torsor over X.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear EV Models

This paper studies estimation and serial correlation test of a semiparametric varying-coefficient partially linear EV model of the form Y = X^Tβ ＋Z^Tα（T） ＋ε,ξ = X ＋ η with the identifying condition E[（ε,η^T）^T] =0, Cov[（ε,η^T）^T] = σ^2Ip＋1. The estimators of interested regression parameters /3 , and the model error variance σ2, as well as the nonparametric components α（T）, are constructed. Under some regular conditions, we show that the estimators of the unknown vector β and the unknown parameter σ2 are strongly consistent and asymptotically normal and that the estimator of α（T） achieves the optimal strong convergence rate of the usual nonparametric regression. Based on these estimators and asymptotic properties, we propose the VN,p test statistic and empirical log-likelihood ratio statistic for testing serial correlation in the model. The proposed statistics are shown to have asymptotic normal or chi-square distributions under the null hypothesis of no serial correlation. Some simulation studies are conducted to illustrate the finite sample performance of the proposed tests.     

Classification of Ideals of Homeomorphism C*-Algebras and Quasidiagonality of Their Quotient Algebras

For the homeomorphism C*-algebra A(Σ) arising from a topological dynamical system Σ=(X,σ) where σ is a homeomorphism on an arbitrary compact Hausdorff space X, we first give detailed classification of its closed ideals into four classes. In case when X is a compact metric space, we then determine the conditions when the quotient algebras of A(Σ) become quasidiagonal. The case of A(Σ) itself was treated by M. Pimsner.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...