Exact one-sided simultaneous confidence bands via Uusipaikka’s method

Computation of one-sided simultaneous confidence bands is detailed for a simple linear regression under interval restrictions on the predictor variable, using a method due to Uusipaikka (1983,J. Amer. Statist. Assoc.,78, 638–644). The case of a single interval restriction is emphasized. A WWW-based applet for computing the bands is described.     

On Bootstrapping M-Estimated Residual Processes in Multiple Linear-Regression Models

It is shown, under fairly general conditions, that Efron′s bootstrap procedure captures the limit distribution of weighted empirical processes based on M-estimated residuals in multiple linear regression models. As an application, we construct bootstrap confidence bands for the error distribution function F. The main result can also be used to design distribution-free goodness-of-fit tests for F without any recourse to the split-sample estimation of the regression parameters.     

Weak consistency of the Support Vector Machine Quantile Regression approach when covariates are functions

This paper deals with a nonparametric estimation of conditional quantile regression when the explanatory variable X takes its values in a bounded subspace of a functional space X and the response Y takes its values in a compact of the space Y?R. The functional observations, X₁,…,X_n, are projected onto a finite dimensional subspace having a suitable orthonormal system. The X_i’s will be characterized by their coordinates in this basis. We perform the Support Vector Machine Quantile Regression approach in finite dimension with the selected coefficients. Then we establish weak consistency of this estimator. The various parameters needed for the construction of this estimator are automatically selected by data-splitting and by penalized empirical risk minimization.     

Decomposition of the lattice of pseudovarieties of finite semigroups induced by bands

It has been shown by the authors that the mapping,where B is the pseudovariety of finite bands, is a complete retraction of the lattice L(F) of pseudovarieties of finite semigroups onto the lattice of pseudovarieties of bands. It follows that the classes of the induced congruence on L(F), or on the lattice of subpseudovarieties L(W) for any subpseudovariety W of F, are intervals. In this paper we solve the membership problem for the upper limit of the classes of restricted to L(W) for various , including F itself, and provide bases of pseudoidentities for certain cases. Received April 7, 1999; accepted in final form April 30, 2000.     

The Menger property for infinite ordered sets

It is shown that, if an ordered set P contains at most k pairwise disjoint maximal chains, where k is finite, then every finite family of maximal chains in P has a cutset of size at most k. As a corollary of this, we obtain the following Menger-type result that, if in addition, P contains k pairwise disjoint complete maximal chains, then the whole family, M (P), of maximal chains in P has a cutset of size k. We also give a direct proof of this result. We give an example of an ordered set P in which every maximal chain is complete, P does not contain infinitely many pairwise disjoint maximal chains (but arbitrarily large finite families of pairwise disjoint maximal chains), and yet M (P) does not have a cutset of size <x, where x is any given (infinite) cardinal. This shows that the finiteness of k in the above corollary is essential and disproves a conjecture of Zaguia.     

On Combinatorial Properties of Bands

In this paper we study the D-saturated property of bands defined in terms of their Cayley graphs Cay(S,C), where S is a band and C ? Z(S), the center of S. Also we characterize the Cayley graphs of bands. More generally, for a finite graph Γ =Cay(T, D), where T is a band and D ? Z(T), we give an algorithm for finding all bands S and C ? Z(S) such that Γ =Cay(S, C).     

Positron Emission Tomography and Random Coefficients Regression

We further explore the relation between random coefficients regression (RCR) and computerized tomography. Recently, Beran et al. (1996, Ann. Statist., 24, 2569–2592) explored this connection to derive an estimation method for the non-parametric RCR problem which is closely related to image reconstruction methods in X-ray computerized tomography. In this paper we emphasize the close connection of the RCR problem with positron emission tomography (PET). Specifically, we show that the RCR problem can be viewed as an idealized (continuous) version of a PET experiment, by demonstrating that the nonparametric likelihood of the RCR problem is equivalent to that of a specific PET experiment. Consequently, methods independently developed for either of the two problems can be adapted from one problem to the other. To demonstrate the close relation between the two problems we use the estimation method of Beran, Feuerverger and Hall for image reconstruction in PET.     

Sequential estimation in regression models using analogues of trimmed means

A sequential procedure is proposed for constructing a fixed-size confidence region for the parameters of a linear regression model. The procedure is based on certain regression analogues of trimmed means, as formulated by Welsh (1987,Ann. Statist.,15, 20–36), rather than least squares estimates. For error distributions with continuous, symmetric density and some moment higher than fourth finite, if the design points of the model are bounded, then the procedure is both asymptotically consistent and asymptotically efficient as the size of the region approaches zero.Research supported in part by the National Science Foundation under Grants DMS 85-03321 and 88-02556 and by the Air Force under Grant AFOSR-87-0041.     

Membership of A ∨ G for classes of finite weakly abundant semigroups

We consider the question of membership of A ∨ G, where A and G are the pseudovarieties of finite aperiodic semigroups, and finite groups, respectively. We find a straightforward criterion for a semigroup S lying in a class of finite semigroups that are weakly abundant, to be in A ∨ G. The class of weakly abundant semigroups contains the class of regular semigroups, but is much more extensive; we remark that any finite monoid with semilattice of idempotents is weakly abundant. To study such semigroups we develop a number of techniques that may be of interest in their own right.     

On a Two-Stage Procedure Having Second-Order Properties with Applications

Under a fairly general setup, we first modify the Stein-type two-stage methodology in order to incorporate some partial information in the form of a known and positive lower bound for the otherwise unknown nuisance parameter, 0(> 0). This revised methodology is then shown to enjoy various customary second-order properties and expansions for functions of the associated stopping variable, under appropriate conditions. Such general machineries are later applied in different types of estimation as well as selection and ranking problems, giving a sense of a very broad spectrum of possibilities. This constitutes natural extensions of these authors' earlier paper (Mukhopadhyay and Duggan (1997a, Sankhya Ser. A, 59, 435 448)) on the fixed-width confidence interval estimation problem exclusively for the mean of a normal distribution having an unknown variance.     

r-k Class estimation in regression model with concomitant variables

We treat with the r-k class estimation in a regression model, which includes the ordinary least squares estimator, the ordinary ridge regression estimator and the principal component regression estimator as special cases of the r-k class estimator. Many papers compared total mean square error of these estimators. Sarkar (1989, Ann. Inst. Statist. Math., 41, 717–724) asserts that the results of this comparison are still valid in a misspecified linear model. We point out some confusions of Sarkar and show additional conditions under which his assertion holds.     

A graphical tool for selecting the number of slices and the dimension of the model in SIR and SAVE approaches

Sliced inverse regression (SIR) and related methods were introduced in order to reduce the dimensionality of regression problems. In general semiparametric regression framework, these methods determine linear combinations of a set of explanatory variables X related to the response variable Y, without losing information on the conditional distribution of Y given X. They are based on a “slicing step” in the population and sample versions. They are sensitive to the choice of the number H of slices, and this is particularly true for SIR-II and SAVE methods. At the moment there are no theoretical results nor practical techniques which allows the user to choose an appropriate number of slices. In this paper, we propose an approach based on the quality of the estimation of the effective dimension reduction (EDR) space: the square trace correlation between the true EDR space and its estimate can be used as goodness of estimation. We introduce a na?ve bootstrap estimation of the square trace correlation criterion to allow selection of an “optimal” number of slices. Moreover, this criterion can also simultaneously select the corresponding suitable dimension K (number of the linear combination of X). From a practical point of view, the choice of these two parameters H and K is essential. We propose a 3D-graphical tool, implemented in R, which can be useful to select the suitable couple (H, K). An R package named “edrGraphicalTools” has been developed. In this article, we focus on the SIR-I, SIR-II and SAVE methods. Moreover the proposed criterion can be use to determine which method seems to be efficient to recover the EDR space, that is the structure between Y and X. We indicate how the proposed criterion can be used in practice. A simulation study is performed to illustrate the behavior of this approach and the need for selecting properly the number H of slices and the dimension K. A short real-data example is also provided.     

A note on the Brück conjecture

In this paper, we firstly consider the Brück conjecture itself and show that it holds exactly for the entire function c(Ae^cz-a)+a{f(z)=\frac{1}{c}(Ae^{cz}-a)+a} , where A, a, c are nonzero constants. Then we give a necessary and sufficient condition that f(z) and f ′(z) share a finite value a CM for some special cases. Finally, we investigate two analogues of the Brück conjecture including the difference analogue of the Brück conjecture raised by Liu and Yang (Arch. Math. 92, 270–278 (2009)) and the shifted analogue of the Brück conjecture raised by Heittokangas et al. (J. Math. Anal. Appl. 355, 352–363 (2009)). And we give some necessary conditions when f(z) shares a finite value a CM with its difference operators or shifts.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

On the growth of generating sets for direct powers of semigroups

For a semigroup S its d-sequence is d(S)=(d ₁,d ₂,d ₃,…), where d _i is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.     

THE ASYMPTOTICALLY OPTIMAL EMPIRICAL BAYES ESTIMATION IN MULTIPLE LINEAR REGRESSION MODEL

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On the LASSO and its Dual

Abstract

Proposed by Tibshirani, the least absolute shrinkage and selection operator (LASSO) estimates a vector of regression coefficients by minimizing the residual sum of squares subject to a constraint on the l ¹-norm of the coefficient vector. The LASSO estimator typically has one or more zero elements and thus shares characteristics of both shrinkage estimation and variable selection. In this article we treat the LASSO as a convex programming problem and derive its dual. Consideration of the primal and dual problems together leads to important new insights into the characteristics of the LASSO estimator and to an improved method for estimating its covariance matrix. Using these results we also develop an efficient algorithm for computing LASSO estimates which is usable even in cases where the number of regressors exceeds the number of observations. An S-Plus library based on this algorithm is available from StatLib.