Cross-validation Bandwidth Selection Research Articles

A new method for structural safety and reliability analysis of offshore wind turbines

With the motivation to overcome the shortcomings of the Rosenblatt inverse-first-order reliability environmental contour method (IFORM), in this study, the use of bivariate kernel density estimation with smoothed cross-validation bandwidth selection method is proposed for generating more accurate environmental contour lines. The environmental contour lines at a chosen offshore site obtained by using the proposed new method were compared with those obtained by using the Rosenblatt IFORM, and the accuracy and effectiveness of the proposed new method have been fully and clearly substantiated. Next, the 50 year extreme structural dynamic responses of a monopile-supported 5 MW offshore wind turbine installed at this chosen offshore site based on the proposed new method and the Rosenblatt IFORM were calculated. Analysing the calculated results, it can be found that the 50 year extreme fore−aft shear force value based on the 50 year extreme sea state obtained using the proposed new method is 78.9% larger than the corresponding value obtained based on the Rosenblatt IFORM. The calculation results in this paper were further systematically analysed and compared, and the necessity and importance of using more realistic environmental contour lines (such as those generated using the proposed new method) have been finally highlighted.

Nonparametric multidimensional fixed effects panel data models

Multidimensional panel datasets are routinely employed to identify marginal effects in empirical research. Fixed effects estimators are typically used to deal with potential correlation between unobserved effects and regressors. Nonparametric estimators for one-way fixed effects models exist, but are cumbersome to employ in practice as they typically require iteration, marginal integration or profile estimation. We develop a nonparametric estimator that works for essentially any dimension fixed effects model, has a closed form solution and can be estimated in a single step. A cross-validation bandwidth selection procedure is proposed and asymptotic properties (for either a fixed or large time dimension) are given. Finite sample properties are shown via simulations, as well as with an empirical application, which further extends our model to the partially linear setting.

Bayesian selector of adaptive bandwidth for multivariate gamma kernel estimator on [0,∞ ) d

ABSTRACT Bayesian bandwidth selections in multivariate associated kernel estimation of probability density functions are known to improve classical methods such as cross-validation techniques in terms of execution time and smoothing quality. The paper focuses on a basic multivariate gamma kernel which is appropriated to estimate densities with support . For this purpose, we consider a Bayesian adaptive estimation of the bandwidths vector under the usual quadratic loss function. The exact expression of the posterior distribution and the vector of bandwidths are obtained. Simulations studies highlight the excellent performance of the proposed approach, comparing to the global cross-validation bandwidth selection, and under integrated squared errors. Two bivariate and trivariate applications to the Old Faithful geyser data and new ones on drinking water pumps in the Sahel, respectively, are made.

A New Kernel Estimator Based on Scaled Inverse Chi-Squared Density Function

In this work, a new kernel estimator based on scaled inverse chi-squared distribution is proposed to estimate densities having nonnegative support. The optimal rates of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE) are obtained. Adaptive Bayesian bandwidth selection method with Lindley approximation is used for heavy tailed distributions. Simulation studies are performed to compare the performance of the average integrated square error (ISE) by using the bandwidths obtained from the global least squares cross-validation bandwidth selection method and the bandwidths obtained from adaptive Bayesian method with Lindley approximation. Finally, real data sets are presented to illustrate the findings.

Reptiles on the wrong track? Moving beyond traditional estimators with dynamic Brownian Bridge Movement Models

BackgroundAnimal movement expressed through home ranges or space-use can offer insights into spatial and habitat requirements. However, different classes of estimation methods are currently instinctively applied to answer home range, space-use or movement-based research questions regardless of their widely varying outputs, directly impacting conclusions. Recent technological advances in animal tracking (GPS and satellite tags), have enabled new methods to quantify animal space-use and movement pathways, but so far have primarily targeted mammal and avian species.MethodsMost reptile spatial ecology studies only make use of two older home range estimation methods: Minimum Convex Polygons (MCP) and Kernel Density Estimators (KDE), particularly with the Least Squares Cross Validation (LSCV) and reference (href) bandwidth selection algorithms. These methods are frequently applied to answer space-use and movement-based questions. Reptile movement patterns are unique (e.g., low movement frequency, long stop-over periods), prompting investigation into whether newer movement-based methods –such as dynamic Brownian Bridge Movement Models (dBBMMs)– apply to Very High Frequency (VHF) radio-telemetry tracking data. We simulated movement data for three archetypical reptile species: a highly mobile active hunter, an ambush predator with long-distance moves and long-term sheltering periods, and an ambush predator with short-distance moves and short-term sheltering periods. We compared traditionally used estimators, MCP and KDE, with dBBMMs, across eight feasible VHF field sampling regimes for reptiles, varying from one data point every four daylight hours, to once per month.ResultsAlthough originally designed for GPS tracking studies, dBBMMs outperformed MCPs and KDE href across all tracking regimes in accurately revealing movement pathways, with only KDE LSCV performing comparably at some higher frequency sampling regimes. However, the LSCV algorithm failed to converge with these high-frequency regimes due to high site fidelity, and was unstable across sampling regimes, making its use problematic for species exhibiting long-term sheltering behaviours. We found that dBBMMs minimized the effect of individual variation, maintained low error rates balanced between omission (false negative) and commission (false positive), and performed comparatively well even under low frequency sampling regimes (e.g., once a month).ConclusionsWe recommend dBBMMs as a valuable alternative to MCP and KDE methods for reptile VHF telemetry data, for research questions associated with space-use and movement behaviours within the study period: they work under contemporary tracking protocols and provide more stable estimates. We demonstrate for the first time that dBBMMs can be applied confidently to low-resolution tracking data, while improving comparisons across regimes, individuals, and species.

Bayesian selector of adaptive bandwidth for gamma kernel density estimator on simulations and applications

Bayesian bandwidth selections in continuous associated kernel estimation of probability density function are very good alternatives to classical methods like cross-validation techniques. In this paper, we examined the behavior of Bayesian variable bandwidths in gamma kernel estimation, developed theoretically in Wansouwé et al. [Ake: An R Package for Discrete and Continuous Associated Kernel Estimations, The R journal 8 (2016), pp. 259–276], and appropriated to smooth densities of support Simulations studies point out remarkable performance of the proposed approach, comparing to the global cross-validation bandwidth selection, and under integrated squared errors. Two applications related to CO 2 emissions and medical bills of bodily injury claims, respectively, are finally made.

LOCAL COMPOSITE QUANTILE REGRESSION SMOOTHING: A FLEXIBLE DATA STRUCTURE AND CROSS-VALIDATION

In this paper, we study the local composite quantile regression estimator for mixed categorical and continuous data. The local composite quantile estimator is an efficient and safe alternative to the local polynomial method and has been well-studied for continuous covariates. Generalization of the local composite quantile regression estimator to a flexible data structure is appealing to practitioners as empirical studies often encounter categorical data. Furthermore, we study the theoretical properties of the cross-validated bandwidth selection for the local composite quantile estimator. Under mild conditions, we derive the rates of convergence of the cross-validated smoothing parameters to their optimal benchmark values for both categorical and continuous covariates. Monte Carlo experiments show that the proposed estimator may have large efficiency gains compared with the local linear estimator. Furthermore, we illustrate the robustness of the local composite quantile estimator using the Boston housing dataset.

A robust methodology for displaying two-dimensional environmental contours at two offshore sites

This paper proposes a robust methodology for displaying two-dimensional environmental contour lines based on measured ocean wave data at two offshore sites. For implementing the robust environmental contour lines methodology, we propose the use of bivariate kernel density estimation with smoothed cross-validation bandwidth selection. The environmental contours obtained by using the proposed robust methodology have been compared with those obtained by using the Gumbel copula transformation method and another recently published method, and the effectiveness and superiority of our proposed robust methodology have been clearly substantiated. The research results in this paper demonstrate that our proposed robust methodology can be utilized as an effective tool for predicting the long-term extreme dynamic responses of ocean engineering structures.

Local Composite Quantile Regression Smoothing: Flexible Data Structure and Cross-Validation

In this paper, we study the local composite quantile regression estimator for mixed categorical and continuous data. The local composite quantile estimator is an efficient and safe alternative to the local polynomial method and has been well-studied for continuous covariates. Generalization of the local composite quantile regression estimator to a flexible data structure is appealing to practitioners as empirical studies often encounter categorical data. Furthermore, we study the theoretical properties of the cross-validated bandwidth selection for the local composite quantile estimator. Under mild conditions, we derive the rates of convergence of the cross-validated smoothing parameters to their optimal benchmark values for both categorical and continuous covariates. Monte Carlo experiments show that the proposed estimator may have large efficiency gains compared to the local linear estimator. Furthermore, we illustrate the robustness of the local composite quantile estimator using the Boston housing dataset.

Nonparametric density estimation based on beta prime kernel

In this work, we propose beta prime kernel estimator for estimation of a probability density functions defined with nonnegative support. For the proposed estimator, beta prime probability density function used as a kernel. It is free of boundary bias and nonnegative with a natural varying shape. We obtained the optimal rate of convergence for the mean squared error (MSE) and the mean integrated squared error (MISE). Also, we use adaptive Bayesian bandwidth selection method with Lindley approximation for heavy tailed distributions and compare its performance with the global least squares cross-validation bandwidth selection method. Simulation studies are performed to evaluate the average integrated squared error (ISE) of the proposed kernel estimator against some asymmetric competitors using Monte Carlo simulations. Moreover, real data sets are presented to illustrate the findings.

Nonparametric localized bandwidth selection for Kernel density estimation

ABSTRACTAs conventional cross-validation bandwidth selection methods do not work properly in the situation where the data are serially dependent time series, alternative bandwidth selection methods are necessary. In recent years, Bayesian-based methods for global bandwidth selection have been studied. Our experience shows that a global bandwidth is however less suitable than a localized bandwidth in kernel density estimation based on serially dependent time series data. Nonetheless, a difficult issue is how we can consistently estimate a localized bandwidth. This paper presents a nonparametric localized bandwidth estimator, for which we establish a completely new asymptotic theory. Applications of this new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate and the S&P 500 daily return demonstrate the effectiveness and competitiveness of the proposed localized bandwidth.

Fast bandwidth selection for estimation of the pair correlation function

ABSTRACTKernel estimation is a popular approach to estimation of the pair correlation function which is a fundamental spatial point process characteristic. Least squares cross validation was suggested by Guan [A least-squares cross-validation bandwidth selection approach in pair correlation function estimations. Statist Probab Lett. 2007;77(18):1722–1729] as a data-driven approach to select the kernel bandwidth. The method can, however, be computationally demanding for large point pattern data sets. We suggest a modified least squares cross validation approach that is asymptotically equivalent to the one proposed by Guan but is computationally much faster.

Nonparametric Estimation and Forecasting for Time-Varying Coefficient Realized Volatility Models

This article introduces a new specification for the heterogenous autoregressive (HAR) model for the realized volatility of S&P 500 index returns. In this modeling framework, the coefficients of the HAR are allowed to be time-varying with unspecified functional forms. The local linear method with the cross-validation (CV) bandwidth selection is applied to estimate the time-varying coefficient HAR (TVC-HAR) model, and a bootstrap method is used to construct the point-wise confidence bands for the coefficient functions. Furthermore, the asymptotic distribution of the proposed local linear estimators of the TVC-HAR model is established under some mild conditions. The results of the simulation study show that the local linear estimator with CV bandwidth selection has favorable finite sample properties. The outcomes of the conditional predictive ability test indicate that the proposed nonparametric TVC-HAR model outperforms the parametric HAR and its extension to HAR with jumps and/or GARCH in terms of multi-step out-of-sample forecasting, in particular in the post-2003 crisis and 2007 global financial crisis (GFC) periods, during which financial market volatilities were unduly high.

Gains from joint cross validation bandwidth selection for derivatives of conditional multidimensional densities

This paper studies bandwidth selection for kernel estimation of derivatives of multidimensional conditional densities, a non-parametric realm unexplored in the literature. This paper extends Baird [Cross validation bandwidth selection for derivatives of multidimensional densities. RAND Working Paper series, WR-1060; 2014] in its examination of conditional multivariate densities, derives and presents criteria for arbitrary kernel order and density dimension, shows consistency of the estimators, and investigates a minimization criterion which jointly estimates numerator and denominator bandwidths. I conduct a Monte Carlo simulation study for various orders of kernels in the Gaussian family and compare the new cross validation criterion with those implied by Baird [Cross validation bandwidth selection for derivatives of multidimensional densities. RAND Working Paper series, WR-1060; 2014]. The paper finds that higher order kernels become increasingly important as the dimension of the distribution increases. I find that the cross validation criterion developed in this paper that jointly estimates the derivative of the joint density (numerator) and the marginal density (denominator) does orders of magnitude better than criteria that estimate the bandwidths separately. I further find that using the infinite order Dirichlet kernel tends to have the best results.

Improving cross-validated bandwidth selection using subsampling-extrapolation techniques

Cross-validation methodologies have been widely used as a means of selecting tuning parameters in nonparametric statistical problems. In this paper we focus on a new method for improving the reliability of cross-validation. We implement this method in the context of the kernel density estimator, where one needs to select the bandwidth parameter so as to minimize L2 risk. This method is a two-stage subsampling-extrapolation bandwidth selection procedure, which is realized by first evaluating the risk at a fictional sample size m(m≤sample size n) and then extrapolating the optimal bandwidth from m to n. This two-stage method can dramatically reduce the variability of the conventional unbiased cross-validation bandwidth selector. This simple first-order extrapolation estimator is equivalent to the rescaled “bagging-CV” bandwidth selector in Hall and Robinson (2009) if one sets the bootstrap size equal to the fictional sample size. However, our simplified expression for the risk estimator enables us to compute the aggregated risk without any bootstrapping. Furthermore, we developed a second-order extrapolation technique as an extension designed to improve the approximation of the true optimal bandwidth. To select the optimal choice of the fictional size m given a sample of size n, we propose a nested cross-validation methodology. Based on simulation study, the proposed new methods show promising performance across a wide selection of distributions. In addition, we also investigated the asymptotic properties of the proposed bandwidth selectors.

Adaptive Bayesian bandwidth selection in asymmetric kernel density estimation for nonnegative heavy-tailed data

In this paper, we consider an interesting problem on adaptive Birnbaum–Saunders-power-exponential (BS-PE) kernel density estimation for nonnegative heavy-tailed (HT) data. Treating the variable bandwidths , of adaptive BS-PE kernel as parameters, we then propose a conjugate prior and estimate the 's by using the popular quadratic and entropy loss functions. Explicit formulas are obtained for the posterior and Bayes estimators. Comparison simulations with global unbiased cross-validation bandwidth selection technique were conducted under four HT distributions. Finally, two applications based on HT real data are presented and analyzed.

Cross Validation Bandwidth Selection for Derivatives of Multidimensional Densities

Little attention has been given to the effect of higher order kernels for bandwidth selection for multidimensional derivatives of densities. This paper investigates the extension of cross validation methods to higher dimensions for the derivative of an unconditional joint density. I present and derive different cross validation criteria for arbitrary kernel order and density dimension, and show consistency of the estimator. Doing a Monte Carlo simulation study for various orders of kernels in the Gaussian family and additionally comparing a weighted integrated square error criterion, I find that higher order kernels become increasingly important as the dimension of the distribution increases. I find that standard cross validation selectors generally outperform the weighted integrated square error cross validation criteria. Using the infinite order Dirichlet kernel tends to have the best results.

Semiparametric Localized Bandwidth Selection in Kernel Density Estimation

Since conventional cross-validation bandwidth selection methods don't work for the case where the data considered are dependent time series, alternative bandwidth selection methods are needed. In recent years, Bayesian based global bandwidth selection methods have been proposed. Our experience shows that the use of a global bandwidth is however less suitable than using a localized bandwidth in kernel density estimation in the case where the data are dependent time series as discussed in an empirical application of this paper. Nonetheless, a difficult issue is how we can consistently estimate a localized bandwidth. In this paper, we propose a semiparametric estimation method, for which we establish an asymptotic theory for the proposed semiparametric estimator. A by-product of this bandwidth estimate is a new sampling-based likelihood approach to hyperparameter estimation. Monte Carlo simulation studies show that the proposed hyperparameter estimation method works very well, and that the proposed bandwidth estimator outperforms its competitors. Applications of the new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate, as well as the S&P 500 daily return under conditional heteroscedasticity, demonstrate the effectiveness and competitiveness of the proposed semiparametric localized bandwidth.

Probit Transformation for Kernel Density Estimation on the Unit Interval

Kernel estimation of a probability density function supported on the unit interval has proved difficult, because of the well-known boundary bias issues a conventional kernel density estimator would necessarily face in this situation. Transforming the variable of interest into a variable whose density has unconstrained support, estimating that density, and obtaining an estimate of the density of the original variable through back-transformation, seems a natural idea to easily get rid of the boundary problems. In practice, however, a simple and efficient implementation of this methodology is far from immediate, and the few attempts found in the literature have been reported not to perform well. In this article, the main reasons for this failure are identified and an easy way to correct them is suggested. It turns out that combining the transformation idea with local likelihood density estimation produces viable density estimators, mostly free from boundary issues. Their asymptotic properties are derived, and a practical cross-validation bandwidth selection rule is devised. Extensive simulations demonstrate the excellent performance of these estimators compared to their main competitors for a wide range of density shapes. In fact, they turn out to be the best choice overall. Finally, they are used to successfully estimate a density of nonstandard shape supported on [0, 1] from a small-size real data sample.

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.