Bandwidth Selection Method Research Articles

BSS algorithm by diffusion mixing non-parametric density estimator

Diffusion mixing estimator (DME)-based non-parametric blind source separation (BSS) algorithm is proposed under the framework of non-Bayesian framework and natural gradient optimisation method. In order to improve the performance of signal separation by BSS, the probability distribution of source signals must be described as accurately as possible. Compared to the non-parametric fixed-width kernel density estimator (FKDE) method, the DME with a new data-driven bandwidth selection method can improve the performance of FKDE, which is inspired via a Langevin diffusion process. Moreover, the direct estimation of the score functions can separate the hybrid mixtures of sources that contain both symmetric and asymmetric distribution source signals and do not need to assume the parametric non-linear functions as them. The effectiveness of the proposed algorithm has been confirmed by simulation experiments.

Data-Driven Bandwidth Selection for Nonstationary Semiparametric Models

This article extends the asymptotic results of the traditional least squares cross-validatory (CV) bandwidth selection method to semiparametric regression models with nonstationary data. Two main findings are that (a) the CV-selected bandwidth is stochastic even asymptotically and (b) the selected bandwidth based on the local constant method converges to 0 at a different speed than that based on the local linear method. Both findings are in sharp contrast to existing results when working with weakly dependent or independent data. Monte Carlo simulations confirm our theoretical results and show that the automatic data-driven method works well.

On the nonparametric smooth estimation of the reversed Hazard Rate function

The Reversed Hazard Rate (RHR) function is an important measure as a tool in the analysis of the reliability of both natural and man-made systems. In this paper, we present several new estimators of the RHR function using nonparametric techniques. These estimators are obtained by incorporating different binning techniques with fixed design local polynomial regression. We show that these estimators are asymptotically unbiased and consistent and, to determine the bandwidth, we propose two simple yet efficient plug-in bandwidth selection methods for even and odd order local polynomial estimators. Simulated and real life data are subsequently used to evaluate the performances of these estimators.

Empirical implementation of nonparametric first-price auction models

Nonparametric estimators provide a flexible means of uncovering salient features of auction data. Although these estimators are popular in the literature, many key features necessary for proper implementation have yet to be uncovered. Here we provide several suggestions for nonparametric estimation of first-price auction models. Specifically, we show how to impose monotonicity of the equilibrium bidding strategy; a key property of structural auction models not guaranteed in standard nonparametric estimation. We further develop methods for automatic bandwidth selection. Finally, we discuss how to impose monotonicity in auctions with differing numbers of bidders, reserve prices, and auction-specific characteristics. Finite sample performance is examined using simulated data as well as experimental auction data.

Density Estimation in Several Populations With Uncertain Population Membership

We devise methods to estimate probability density functions of several populations using observations with uncertain population membership, meaning from which population an observation comes is unknown. The probability of an observation being sampled from any given population can be calculated. We develop general estimation procedures and bandwidth selection methods for our setting. We establish large-sample properties and study finite-sample performance using simulation studies. We illustrate our methods with data from a nutrition study.

A bioinformatic implementation of mutual information as a distance measure for identification of clusters of variables

The size of data sets produced in genetic experiments is steadily increasing. Very often there are many more variables than observations, leading to the so-called ``large $p$, small $n$" problem. For such data, clustering and distance based procedures are useful tools for identifying groups of variables associated with outcomes of interest. We develop a novel approach using mutual information as a measure of distance (here dependency) between probability distributions that is valid for comparisons between pairs of variables that are both continuous, both discrete, or one of each. This gives an overall information matrix to be used as a distance matrix in clustering procedures and to define a so-called weighted network of associations between variables. We present computational aspects of implementing our procedures in R. References T. M. Cover and J. A. Thomas. Elements of information theory. Wiley, 2006. doi:10.1002/0471200611. Z. Dawy, B. Goebel, J. Hagenauer, C. Andreoli, T. Meitinger, and J. C. Mueller. Gene mapping and marker clustering using Shannon's mutual information. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 3(1):47--56, 2006. doi:10.1109/TCBB.2006.9. J. J. Faith, B. Hayete, J. T. Thaden, I. Mogno, J. Wierzbowski, G. Cottarel, S. Kasif, J. J. Collins, and T. S. Gardner. Large-scale mapping and validation of Escherichia coli transcriptional regulation from a compendium of expression profiles. PLoS Biol, 5(1):e8, 2007. doi:10.1371/journal.pbio.0050008. T. F. Fuller, A. Ghazalpour, J. E. Aten, T. A. Drake, A. J. Lusis, and S. Horvath. Weighted gene coexpression network analysis strategies applied to mouse weight. Mammalian Genome, 18(6):463--472, 2007. doi:10.1007/s00335-007-9043-3. A. Ghazalpour, S. Doss, B. Zhang, S. Wang, C. Plaisier, R. Castellanos, A. Brozell, E. E. Schadt, T. A. Drake, A. J. Lusis, et al. Integrating genetic and network analysis to characterize genes related to mouse weight. PLoS Genet, 2(8):e130, 2006. doi:10.1371/journal.pgen.0020130. Hung T. Nguyen. On modeling of linguistic information using random sets. Information Sciences, 34(3):265 -- 274, 1984. doi:10.1016/0020-0255(84)90052-5. P. Qiu, A. J. Gentles, and S. K. Plevritis. Fast calculation of pairwise mutual information for gene regulatory network reconstruction. Computer Methods and Programs in Biomedicine, 94(2):177--180, 2009. doi:10.1016/0020-0255(84)90052-5. S. J. Sheather and M. C. Jones. A reliable data-based bandwidth selection method for kernel density estimation. Journal of the Royal Statistical Society. Series B (Methodological), pages 683--690, 1991. M. P. Wand and M. C. Jones. Kernel smoothing. Chapman and Hall/CRC, 1995. B. Zhang and S. Horvath. A general framework for weighted gene co-expression network analysis. Statistical Applications in Genetics and Molecular Biology, 4(1):1128, 2005. doi:10.2202/1544-6115.1128.

Data-Driven Bandwidth Selection for Nonparametric Nonstationary Regressions

We provide a solution to the open problem of bandwidth selection for the nonparametric estimation of potentially non-stationary regressions, a setting in which the popular method of cross-validation has not been justified theoretically. Our procedure is based on minimizing moment conditions involving nonparametric residuals and applies to β-recurrent Markov chains, stationary processes being a special case, as well as nonlinear functions of integrated processes. Local and uniform versions of the criterion are proposed. The selected bandwidths are rate-optimal up to a logarithmic factor, a typical cost of adaptation in other contexts. We further show that the bias induced by (near-) minimax optimality can be removed by virtue of a simple randomized procedure. In a Monte Carlo exercise, we find that our proposed bandwidth selection method, and its subsequent bias correction, fare favorably relative to cross-validation, even in stationary environments.

Data-driven local bandwidth selection for additive models with missing data

This paper deals in the nonparametric estimation of additive models in the presence of missing data in the response variable. Specifically in the case of additive models estimated by the Backfitting algorithm with local polynomial smoothers [1]. Three estimators are presented, one based on the available data and two based on a complete sample from imputation techniques. We also develop a data-driven local bandwidth selector based on a Wild Bootstrap approximation of the mean squared error of the estimators. The performance of the estimators and the local bootstrap bandwidth selection method are explored through simulation experiments.

Fourier series-based direct plug-in bandwidth selectors for kernel density estimation

A class of Fourier series-based direct plug-in bandwidth selectors for kernel density estimation is considered in this paper. The proposed bandwidth estimators have a relative convergence rate n −1/2 whenever the underlying density is smooth enough and the simulation results testify that they present a very good finite sample performance against the most recommended bandwidth selection methods in the literature.

Empirical Likelihood for Regression Discontinuity Design

This paper proposes empirical likelihood based inference methods for causal effects identified from regression discontinuity designs. We consider both the sharp and fuzzy regression discontinuity designs and treat the regression functions as nonparametric. The proposed inference procedures do not require asymptotic variance estimation and the confidence sets have natural shapes, unlike the conventional Wald-type method. These features are illustrated by simulations and an empirical example which evaluates the effect of class size on pupils' scholastic achievements. Bandwidth selection methods, higher-order properties, and extensions to incorporate additional covariates and parametric functional forms are also discussed.

Nonparametric estimation of varying coefficient error-in-variable models with validation sampling

In this paper, varying coefficient models are investigated with the response and covariate prone to measurement error. Without specifying any error structure equation, four estimators of the coefficient function vector are proposed by using the local linear kernel smoothing technique and also proved to be asymptotically normal. The data-driven bandwidth selection method is discussed. Simulation examples are conducted to evaluate the proposed estimation methods.

On completely data-driven bandwidth selection for single-index models

The class of single-index models (SIMs) has become an important tool for nonparametric regression analysis. As with any other nonparametric regression models, the selection of bandwidth plays an important role in the inferences of the SIMs. However, most results in the literature either take the bandwidths as externally given, or require unpractical assumptions or very restrictive conditions for data-driven bandwidths. We examine the asymptotic properties of a popular bandwidth selection method based on cross-validation that is completely data-driven, under much weaker conditions than those assumed in the literature. And we show that the same bandwidth that is optimal for estimating the index vector, can be used for nearly optimal error variance estimation through the method of varying cross-validation. A simulation study is presented to demonstrate the finite sample performance of the proposed procedures, based on which we recommend a simple 2-step procedure for bandwidth selection, index vector estimation, as well as error variance estimation.

Nonparametric inference of doubly stochastic Poisson process data via the kernel method

Doubly stochastic Poisson processes, also known as the Cox processes, frequently occur in various scientific fields. In this article, motivated primarily by analyzing Cox process data in biophysics, we propose a nonparametric kernel-based inference method. We conduct a detailed study, including an asymptotic analysis, of the proposed method, and provide guidelines for its practical use, introducing a fast and stable regression method for bandwidth selection. We apply our method to real photon arrival data from recent single-molecule biophysical experiments, investigating proteins' conformational dynamics. Our result shows that conformational fluctuation is widely present in protein systems, and that the fluctuation covers a broad range of time scales, highlighting the dynamic and complex nature of proteins' structure.

Improving studies of resource selection by understanding resource use

SUMMARYUnderstanding the resource needs of animals is critical to their management and conservation. Resource utilization functions (RUFs) provide a framework to investigate animal-resource relationships by characterizing variation in the amount of resource use. In this context a ‘resource’ is any aspect of a species' fundamental niche that can be mapped throughout the area of investigation (such as study area or home range). Extensive global positioning system (GPS) data from 17 cougars (Puma concolor) demonstrate the utility and potential challenges of estimating RUFs within the home range for far-ranging species. Ninety-nine per cent utilization distributions (UDs) estimated using bivariate plug in, univariate least-squares cross-validation and reference bandwidth selection methods were compared. Distance to water, per cent clear-cut and regenerating forest, and slope were used to estimate cougar RUFs. UDs derived from GPS data were more refined, and plug-in UDs were least similar to UDs derived from other bandwidths. RUFs were resilient to variation in the smoothing parameter, with all methods yielding coefficients that largely reflected observations of foraging ecology and behaviour. Cougars were individualistic, but use was generally positively associated with the presence of regenerating forest and inversely associated with steep slopes. Advances in technology allow for greater accuracy and resolution of the UD, but software improvements and spatially explicit information on animal behaviour are needed to better understand resource use.

Local Polynomial Modeling of Time-Varying Autoregressive Models With Application to Time–Frequency Analysis of Event-Related EEG

This paper proposes a new local polynomial modeling (LPM) method for identification of time-varying autoregressive (TVAR) models and applies it to time-frequency analysis (TFA) of event-related electroencephalogram (ER-EEG). The LPM method models the TVAR coefficients locally by polynomials and estimates the polynomial coefficients using weighted least-squares with a window having a certain bandwidth. A data-driven variable bandwidth selection method is developed to determine the optimal bandwidth that minimizes the mean squared error. The resultant time-varying power spectral density estimation of the signal is capable of achieving both high time resolution and high frequency resolution in the time-frequency domain, making it a powerful TFA technique for nonstationary biomedical signals like ER-EEG. Experimental results on synthesized signals and real EEG data show that the LPM method can achieve a more accurate and complete time-frequency representation of the signal.

Kernel density estimation via diffusion

We present a new adaptive kernel density estimator based on linear diffusion processes. The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. We present simulation examples in which the proposed approach outperforms existing methods in terms of accuracy and reliability.

Hazard function estimation with cause-of-death data missing at random

Hazard function estimation is an important part of survival analysis. Interest often centers on estimating the hazard function associated with a particular cause of death. We propose three nonparametric kernel estimators for the hazard function, all of which are appropriate when death times are subject to random censorship and censoring indicators can be missing at random. Specifically, we present a regression surrogate estimator, an imputation estimator, and an inverse probability weighted estimator. All three estimators are uniformly strongly consistent and asymptotically normal. We derive asymptotic representations of the mean squared error and the mean integrated squared error for these estimators and we discuss a data-driven bandwidth selection method. A simulation study, conducted to assess finite sample behavior, demonstrates that the proposed hazard estimators perform relatively well. We illustrate our methods with an analysis of some vascular disease data.

A Comparative Study for Bandwidth Selection in Kernel Density Estimation

Nonparametric kernel density estimation method does not make any assumptions regarding the functional form of curves of interest; hence it allows flexible modeling of data. A crucial problem in kernel density estimation method is how to determine the bandwidth (smoothing) parameter. This article examines the most important bandwidth selection methods, in particular, least squares cross-validation, biased crossvalidation, direct plug-in, solve-the-equation rules and contrast methods. Methods are described and expressions are presented. The main practical contribution is a comparative simulation study that aims to isolate the most promising methods. The performance of each method is evaluated on the basis of the mean integrated squared error for small-to-moderate sample size. Simulation results show that the contrast method is the most promising methods based on the simulated families considered.

SMOOTH VARYING-COEFFICIENT ESTIMATION AND INFERENCE FOR QUALITATIVE AND QUANTITATIVE DATA

We propose a semiparametric varying-coefficient estimator that admits both qualitative and quantitative covariates along with a test for correct specification of parametric varying-coefficient models. The proposed estimator is exceedingly flexible and has a wide range of potential applications including hierarchical (mixed) settings, small area estimation, etc. A data-driven cross-validatory bandwidth selection method is proposed that can handle both the qualitative and quantitative covariates and that can also handle the presence of potentially irrelevant covariates, each of which can result in finite-sample efficiency gains relative to the conventional frequency (sample-splitting) estimator that is often found in such settings. Theoretical underpinnings including rates of convergence and asymptotic normality are provided. Monte Carlo simulations are undertaken to assess the proposed estimator’s finite-sample performance relative to the conventional semiparametric frequency estimator and to assess the finite-sample performance of the proposed test for correct parametric specification.

Optimal bandwidth selection for conditional efficiency measures: A data-driven approach

In productivity analysis an important issue is to detect how external (environmental) factors, exogenous to the production process and not under the control of the producer, might influence the production process and the resulting efficiency of the firms. Most of the traditional approaches proposed in the literature have serious drawbacks. An alternative approach is to describe the production process as being conditioned by a given value of the environmental variables (Cazals, Florens and Simar, 2002, Daraio and Simar, 2005). This defines conditional efficiency measures where the production set in the input × output space may depend on the value of the external variables. The statistical properties of nonparametric estimators of these conditional measures are now established (Jeong, Park and Simar, 2008). These involve the estimation of a nonstandard conditional distribution function which requires the specification of a smoothing parameter (a bandwidth). So far, only the asymptotic optimal order of this bandwidth has been established. This is of little interest for the practitioner. In this paper we fill this gap and we propose a data-driven technique for selecting this parameter in practice. The approach, based on a Least Squares Cross Validation procedure (LSCV), provides an optimal bandwidth that minimizes an appropriate (weighted) integrated Squared Error (ISE). The method is carefully described and exemplified with some simulated data with univariate and multivariate environmental factors. An application on real data (performances of Mutual Funds) illustrates how this new optimal method of bandwidth selection works in practice.