Vector-valued Parameters Research Articles

DS-HECK: double-lasso estimation of Heckman selection model

We extend the Heckman (1979) sample selection model by allowing for a large number of controls that are selected using lasso under a sparsity scenario. The standard lasso estimation is known to under-select causing an omitted variable bias in addition to the sample selection bias. We outline the required adjustments needed to restore consistency of lasso-based estimation and inference for vector-valued parameters of interest in such models. The adjustments include double lasso for both the selection equation and main equation and a correction of the variance matrix. We also connect the estimator with results on redundancy of moment conditions. We demonstrate the effect of the adjustments using simulations and we investigate the determinants of female labor market participation and earnings in the US using the new approach. The paper comes with dsheckman, a dedicated Stata command for estimating double-selection Heckman models.

Learning Log-Determinant Divergences for Positive Definite Matrices.

Representations in the form of Symmetric Positive Definite (SPD) matrices have been popularized in a variety of visual learning applications due to their demonstrated ability to capture rich second-order statistics of visual data. There exist several similarity measures for comparing SPD matrices with documented benefits. However, selecting an appropriate measure for a given problem remains a challenge and in most cases, is the result of a trial-and-error process. In this paper, we propose to learn similarity measures in a data-driven manner. To this end, we capitalize on the αβ-log-det divergence, which is a meta-divergence parametrized by scalars α and β, subsuming a wide family of popular information divergences on SPD matrices for distinct and discrete values of these parameters. Our key idea is to cast these parameters in a continuum and learn them from data. We systematically extend this idea to learn vector-valued parameters, thereby increasing the expressiveness of the underlying non-linear measure. We conjoin the divergence learning problem with several standard tasks in machine learning, including supervised discriminative dictionary learning and unsupervised SPD matrix clustering. We present Riemannian gradient descent schemes for optimizing our formulations efficiently, and show the usefulness of our method on eight standard computer vision tasks.

FRACTIONAL DEGREE VORTICES FOR A SPINOR GINZBURG–LANDAU MODEL

Recent papers in the physics literature have introduced spin-coupled (or spinor) Ginzburg–Landau models for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors and in optically confined Bose–Einstein condensates. In this paper, we show that such models give rise to new types of vortices, with fractional degree and nontrivial core structure. We illustrate the various possibilites with some specific examples of Dirichlet problems in the unit disk.

Generating Blending Surfaces with a Pseudo-L�vy Series Solution to Fourth Order Partial Differential Equations

In our previous work, a more general fourth order partial differential equation (PDE) with three vector-valued parameters was introduced. This equation is able to generate a superset of the blending surfaces of those produced by other existing fourth order PDEs found in the literature. Since it is usually more difficult to solve PDEs analytically than numerically, many references are only concerned with numerical solutions, which unfortunately are often inefficient. In this paper, we have developed a fast and accurate resolution method, the pseudo-Levy series method. Due to its analytical nature, the comparison with other existing methods indicates that the developed method can generate blending surfaces almost as quickly and accurately as the closed form resolution method, and has higher computational accuracy and efficiency than existing Fourier series and pseudo-spectral methods as well as other numerical methods. In addition, it can be used to solve complex surface blending problems which cannot be tackled by the closed form resolution method. To demonstrate the potential of this new method we have applied it to various surface blending problems, including the generation of the blending surface between parametric primary surfaces, general second and higher degree surfaces, and surfaces defined by explicit equations.

Des vortex fractionnaires pour un modèle Ginzburg–Landau spineur

Recent papers in the physics literature have introduced spin-coupled (or spinor) Ginzburg–Landau models for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors and in optically confined Bose–Einstein condensates. In this Note we observe that such models can lead to new types of vortices, with fractional degree and non-trivial core structure. To cite this article: S. Alama, L. Bronsard, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

A Second-order Adjustment to the Profile Likelihood in the Case of a Multidimensional Parameter of Interest

Summary Inference in the presence of nuisance parameters is often carried out by using the χ2-approximation to the profile likelihood ratio statistic. However, in small samples, the accuracy of such procedures may be poor, in part because the profile likelihood does not behave as a true likelihood, in particular having a profile score bias and information bias which do not vanish. To account better for nuisance parameters, various researchers have suggested that inference be based on an additively adjusted version of the profile likelihood function. Each of these adjustments to the profile likelihood generally has the effect of reducing the bias of the associated profile score statistic. However, these adjustments are not applicable outside the specific parametric framework for which they were developed. In particular, it is often difficult or even impossible to apply them where the parameter about which inference is desired is multidimensional. In this paper, we propose a new adjustment function which leads to an adjusted profile likelihood having reduced score and information biases and is readily applicable to a general parametric framework, including the case of vector-valued parameters of interest. Examples are given to examine the performance of the new adjusted profile likelihood in small samples, and also to compare its performance with other adjusted profile likelihoods.

The Asymptotic Validity of Sequential Stopping Rules for Stochastic Simulations

We establish general conditions for the asymptotic validity of sequential stopping rules to achieve fixed-volume confidence sets for simulation estimators of vector-valued parameters. The asymptotic validity occurs as the prescribed volume of the confidence set approaches 0. There are two requirements: a functional central limit theorem for the estimation process and strong consistency (with-probability-1 convergence) for the variance or "scaling matrix" estimator. Applications are given for: sample means of i.i.d. random variables and random vectors, nonlinear functions of such sample means, jackknifing, Kiefer-Wolfowitz and Robbins-Monro stochastic approximation and both regenerative and nonregenerative steady-state simulation.

A multivariate version of isotonic regression

SUMMARY A multivariate generalization of isotonic regression is given enabling the study of statistical inference for ordered vector-valued parameters or sets of ordered parameters, including multivariate extensions of Bartholomew's 22 and 4 2. An iterative algorithm for computing multivariate isotonic regression is proposed and studied.

Sensitivities in parameter estimation

The method of equivalent observation [1] is extended to vector-valued constant parameters and to uncorrelated measurement noise sequences. It is then used to determine easily calculated partial derivatives of final covariances with respect to parameter variances. Error matrices and bounds are developed in order to determine the range over which linearity may be assumed.

A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters

Abstract Two examples are given from the literature in which results concerning functions of a single variable are applied incorrectly to multivariate functions. One of the examples concerns the proof of a basic theorem in the theory of maximum likelihood estimation of vector-valued parameters. An alternate statement and proof of the theorem are given.

A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters

A Note on the Uniqueness of Roots of the Likelihood Equations for Vector-Valued Parameters