Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

On robust estimation of the location parameter

Statistical Analysis With Missing Data (2nd ed.) (Book)

Summary
The purpose of this paper is to propose methodologies for statistical inference of low dimensional parameters with high dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broader context. The theoretical results that are presented provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite dimensional covariance matrices. These sufficient conditions allow the number of variables to exceed the sample size and the presence of many small non-zero coefficients. Our methods and theory apply to interval estimation of a preconceived regression coefficient or contrast as well as simultaneous interval estimation of many regression coefficients. Moreover, the method proposed turns the regression data into an approximate Gaussian sequence of point estimators of individual regression coefficients, which can be used to select variables after proper thresholding. The simulation results that are presented demonstrate the accuracy of the coverage probability of the confidence intervals proposed as well as other desirable properties, strongly supporting the theoretical results.

Confidence intervals for low dimensional parameters in high dimensional linear models

High dimensional statistical problems arise from diverse fields of scientific research and technological development. Variable selection plays a pivotal role in contemporary statistical learning and scientific discoveries. The traditional idea of best subset selection methods, which can be regarded as a specific form of penalized likelihood, is computationally too expensive for many modern statistical applications. Other forms of penalized likelihood methods have been successfully developed over the last decade to cope with high dimensionality. They have been widely applied for simultaneously selecting important variables and estimating their effects in high dimensional statistical inference. In this article, we present a brief account of the recent developments of theory, methods, and implementations for high dimensional variable selection. What limits of the dimensionality such methods can handle, what the role of penalty functions is, and what the statistical properties are rapidly drive the advances of the field. The properties of non-concave penalized likelihood and its roles in high dimensional statistical modeling are emphasized. We also review some recent advances in ultra-high dimensional variable selection, with emphasis on independence screening and two-scale methods.

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A Selective Overview of Variable Selection in High Dimensional Feature Space.

The smoothly clipped absolute deviation (SCAD) estimator, proposed by Fan and Li, has many desirable properties, including continuity, sparsity, and unbiasedness. The SCAD estimator also has the (asymptotically) oracle property when the dimension of covariates is fixed or diverges more slowly than the sample size. In this article we study the SCAD estimator in high-dimensional settings where the dimension of covariates can be much larger than the sample size. First, we develop an efficient optimization algorithm that is fast and always converges to a local minimum. Second, we prove that the SCAD estimator still has the oracle property on high-dimensional problems. We perform numerical studies to compare the SCAD estimator with the LASSO and SIS–SCAD estimators in terms of prediction accuracy and variable selectivity when the true model is sparse. Through the simulation, we show that the variance estimator of Fan and Li still works well for some limited high-dimensional cases where the true nonzero coeffic...

Smoothly Clipped Absolute Deviation on High Dimensions

Spectrum occupancy models are very useful in cognitive radio designs. They can be used to increase spectrum sensing accuracy for more reliable operation, to remove spectrum sensing for higher resource usage efficiency, or to select channels for better opportunistic access, among other applications. In this survey, various spectrum occupancy models from measurement campaigns taken around the world are investigated. These models extract different statistical properties of the spectrum occupancy from the measured data. In addition to these models, spectrum occupancy prediction is also discussed, where autoregressive and/or moving-average models are used to predict the channel status at future time instants. After comparing these different methods and models, several challenges are also summarized based on this survey.

/pdf/a-survey-of-measurement-based-spectrum-occupancy-modeling-5h1rhm2b34.pdf

A Survey of Measurement-Based Spectrum Occupancy Modeling for Cognitive Radios

The concept of empirical mode decomposition (EMD) and the Hilbert spectrum (HS) has been developed rapidly in many disciplines of science and engineering since Huang et al. (1998) invented EMD. The key feature of EMD is to decompose a signal into so-called intrinsic mode function (IMF). Furthermore, the Hilbert spectral analysis of intrinsic mode functions provides frequency information evolving with time and quantifies the amount of variation due to oscillation at different time scales and time locations. In this article, we introduce an R package called EMD (Kim and Oh, 2008) that performs oneand twodimensional EMD and HS.

/pdf/emd-a-package-for-empirical-mode-decomposition-and-hilbert-565xptxg0a.pdf

EMD: A Package for Empirical Mode Decomposition and Hilbert Spectrum

SUMMARY We propose a robust curve and surface estimator based on M-type estimators and penalty based smoothing. This approach also includes an application to wavelet regression. The concept of pseudo data, a transformation of the robust additive model to the one with bounded errors, is used to derive some theoretical properties and also motivate a computational algorithm. The resulting algorithm, termed the ES-algorithm, is computationally fast and provides a simple way of choosing the amount of smoothing. Moreover, it is easily described, straightforwardly implemented and can be extended to other wavelet regression settings such as irregularly spaced data and image denoising. Results from a simulation study and real data examples demonstrate the promising empirical properties of the proposed approach.

The Role of Pseudo Data for Robust Smoothing with Application to Wavelet Regression

Summary. The objective is to estimate the period and the light curve (or periodic function) of a variable star. Previously, several methods have been proposed to estimate the period of a variable star, but they are inaccurate especially when a data set contains outliers. We use a smoothing spline regression to estimate the light curve given a period and then find the period which minimizes the generalized cross-validation (GCV). The GCV method works well, matching an intensive visual examination of a few hundred stars, but the GCV score is still sensitive to outliers. Handling outliers in an automatic way is important when this method is applied in a ‘data mining’ context to a vary large star survey. Therefore, we suggest a robust method which minimizes a robust cross-validation criterion induced by a robust smoothing spline regression. Once the period has been determined, a nonparametric method is used to estimate the light curve. A real example and a simulation study suggest that the robust cross-validation and GCV methods are superior to existing methods.

/pdf/period-analysis-of-variable-stars-by-robust-smoothing-1hnl0j7sel.pdf

Hee-Seok Oh

Papers

Smoothly Clipped Absolute Deviation on High Dimensions

A Survey of Measurement-Based Spectrum Occupancy Modeling for Cognitive Radios

EMD: A Package for Empirical Mode Decomposition and Hilbert Spectrum

The Role of Pseudo Data for Robust Smoothing with Application to Wavelet Regression

Period analysis of variable stars by robust smoothing