9,745 research outputs found
Effect of W, LR, and LM Tests on the Performance of Preliminary Test Ridge Regression Estimators
This paper combines the idea of preliminary test and ridge regression methodology, when it is suspected that the regression coefficients may be restricted to a subspace. The preliminary test ridge regression estimators (PTRRE) based on the Wald (W), Likelihood Ratio (LR) and Lagrangian Multiplier (LM) tests are considered. The bias and the mean square errors (MSE) of the proposed estimators are derived under both null and alternative hypotheses. By studying the MSE criterion, the regions of optimality of the estimators are determined. Under the null hypothesis, the PTRRE based on LM test has the smallest risk followed by the estimators based on LR and W tests. However, the PTRRE based on W test performs the best followed by the LR and LM based estimators when the parameter moves away from the subspace of the restrictions. The conditions of superiority of the proposed estimator for both ridge parameter k and departure parameter (triangle symbol) are provided. Some graphical representations have been presented which support the findings of the paper. Some tables for maximum and minimum guaranteed relative efficiency of the proposed estimators have been provided. These tables allow us to determine the optimum level of significance corresponding to the optimum estimators among proposed estimators. Finally, we concluded that the optimum choice of the level of significance becomes the traditional choice by using the W test for all non-negative ridge parameter, k.Dominance; Lagrangian Multiplier; Likelihood Ratio Test; MSE; Non-central Chisquare and F; Ridge Regression; Superiority; Wald Test.
Semiparametric theory
In this paper we give a brief review of semiparametric theory, using as a
running example the common problem of estimating an average causal effect.
Semiparametric models allow at least part of the data-generating process to be
unspecified and unrestricted, and can often yield robust estimators that
nonetheless behave similarly to those based on parametric likelihood
assumptions, e.g., fast rates of convergence to normal limiting distributions.
We discuss the basics of semiparametric theory, focusing on influence
functions.Comment: arXiv admin note: text overlap with arXiv:1510.0474
Algorithms for envelope estimation
Envelopes were recently proposed as methods for reducing estimative variation
in multivariate linear regression. Estimation of an envelope usually involves
optimization over Grassmann manifolds. We propose a fast and widely applicable
one-dimensional (1D) algorithm for estimating an envelope in general. We reveal
an important structural property of envelopes that facilitates our algorithm,
and we prove both Fisher consistency and root-n-consistency of the algorithm.Comment: 30 pages, 2 figures, 2 table
Robust Orthogonal Complement Principal Component Analysis
Recently, the robustification of principal component analysis has attracted
lots of attention from statisticians, engineers and computer scientists. In
this work we study the type of outliers that are not necessarily apparent in
the original observation space but can seriously affect the principal subspace
estimation. Based on a mathematical formulation of such transformed outliers, a
novel robust orthogonal complement principal component analysis (ROC-PCA) is
proposed. The framework combines the popular sparsity-enforcing and low rank
regularization techniques to deal with row-wise outliers as well as
element-wise outliers. A non-asymptotic oracle inequality guarantees the
accuracy and high breakdown performance of ROC-PCA in finite samples. To tackle
the computational challenges, an efficient algorithm is developed on the basis
of Stiefel manifold optimization and iterative thresholding. Furthermore, a
batch variant is proposed to significantly reduce the cost in ultra high
dimensions. The paper also points out a pitfall of a common practice of SVD
reduction in robust PCA. Experiments show the effectiveness and efficiency of
ROC-PCA in both synthetic and real data
Robust variable screening for regression using factor profiling
Sure Independence Screening is a fast procedure for variable selection in
ultra-high dimensional regression analysis. Unfortunately, its performance
greatly deteriorates with increasing dependence among the predictors. To solve
this issue, Factor Profiled Sure Independence Screening (FPSIS) models the
correlation structure of the predictor variables, assuming that it can be
represented by a few latent factors. The correlations can then be profiled out
by projecting the data onto the orthogonal complement of the subspace spanned
by these factors. However, neither of these methods can handle the presence of
outliers in the data. Therefore, we propose a robust screening method which
uses a least trimmed squares method to estimate the latent factors and the
factor profiled variables. Variable screening is then performed on factor
profiled variables by using regression MM-estimators. Different types of
outliers in this model and their roles in variable screening are studied. Both
simulation studies and a real data analysis show that the proposed robust
procedure has good performance on clean data and outperforms the two nonrobust
methods on contaminated data
Space Time MUSIC: Consistent Signal Subspace Estimation for Wide-band Sensor Arrays
Wide-band Direction of Arrival (DOA) estimation with sensor arrays is an
essential task in sonar, radar, acoustics, biomedical and multimedia
applications. Many state of the art wide-band DOA estimators coherently process
frequency binned array outputs by approximate Maximum Likelihood, Weighted
Subspace Fitting or focusing techniques. This paper shows that bin signals
obtained by filter-bank approaches do not obey the finite rank narrow-band
array model, because spectral leakage and the change of the array response with
frequency within the bin create \emph{ghost sources} dependent on the
particular realization of the source process. Therefore, existing DOA
estimators based on binning cannot claim consistency even with the perfect
knowledge of the array response. In this work, a more realistic array model
with a finite length of the sensor impulse responses is assumed, which still
has finite rank under a space-time formulation. It is shown that signal
subspaces at arbitrary frequencies can be consistently recovered under mild
conditions by applying MUSIC-type (ST-MUSIC) estimators to the dominant
eigenvectors of the wide-band space-time sensor cross-correlation matrix. A
novel Maximum Likelihood based ST-MUSIC subspace estimate is developed in order
to recover consistency. The number of sources active at each frequency are
estimated by Information Theoretic Criteria. The sample ST-MUSIC subspaces can
be fed to any subspace fitting DOA estimator at single or multiple frequencies.
Simulations confirm that the new technique clearly outperforms binning
approaches at sufficiently high signal to noise ratio, when model mismatches
exceed the noise floor.Comment: 15 pages, 10 figures. Accepted in a revised form by the IEEE Trans.
on Signal Processing on 12 February 1918. @IEEE201
Semiparametric theory and empirical processes in causal inference
In this paper we review important aspects of semiparametric theory and
empirical processes that arise in causal inference problems. We begin with a
brief introduction to the general problem of causal inference, and go on to
discuss estimation and inference for causal effects under semiparametric
models, which allow parts of the data-generating process to be unrestricted if
they are not of particular interest (i.e., nuisance functions). These models
are very useful in causal problems because the outcome process is often complex
and difficult to model, and there may only be information available about the
treatment process (at best). Semiparametric theory gives a framework for
benchmarking efficiency and constructing estimators in such settings. In the
second part of the paper we discuss empirical process theory, which provides
powerful tools for understanding the asymptotic behavior of semiparametric
estimators that depend on flexible nonparametric estimators of nuisance
functions. These tools are crucial for incorporating machine learning and other
modern methods into causal inference analyses. We conclude by examining related
extensions and future directions for work in semiparametric causal inference
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