Nonparametric Kernel Density Estimation Near the Boundary

Malec, Peter; Schienle, Melanie

doi:10.2139/ssrn.2342165

Cited by 21 publications

(33 citation statements)

References 43 publications

(18 reference statements)

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“…In this section the practical performances of the Local Likelihood-based estimators are compared to those of their competitors. The same 7 test R + -supported densities as in Malec and Schienle (2014) were considered.…”

Section: Simulation Studymentioning

confidence: 99%

“…Although those methods may slightly improve the performance compared to their basic versions, they are more complicated to implement. For instance, Malec and Schienle (2014)…”

Section: Simulation Studymentioning

confidence: 99%

“…Although more types of asymmetric kernels, such as log-normal and Birnbaum-Saunders (Jin and Kawczak, 2003) or Inverse Gaussian and reciprocal Inverse Gaussian (Scaillet, 2004), were investigated in the subsequent literature, those showed little advantage over Chen's Gamma kernel density estimator which arguably remains the main reference for asymmetric kernel estimation on R + . Its properties were further investigated in Bouezmarni and Scaillet (2005), Hagmann and Scaillet (2007), Zhang (2010) and Malec and Schienle (2014), and other ideas related to asymmetric kernel estimation were described in Kuruwita et al (2010), Comte and Genon-Catalot (2012), Mnatsakanov and Sarkisian (2012), Jeon and Kim (2013), Koul and Song (2013), Marchant et al (2013), Igarashi and Kakizawa (2014) and Igarashi (2016). Recently, Hirukawa and Sakudo (2015) described a family of 'generalised Gamma kernels' which includes a variety of similar asymmetric kernels in an attempt to standardise those results.…”

Section: Introductionmentioning

confidence: 99%

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Local-Likelihood Transformation Kernel Density Estimation for Positive Random Variables

Geenens

Wang

2018

Journal of Computational and Graphical Statistics

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The kernel estimator is known not to be adequate for estimating the density of a positive random variable X. The main reason is the well-known boundary bias problems that it suffers from, but also its poor behaviour in the long right tail that such a density typically exhibits. A natural approach to this problem is to first estimate the density of the logarithm of X, and obtaining an estimate of the density of X using standard results on functions of random variables ('back-transformation'). Although intuitive, the basic application of this idea yields very poor results, as was documented earlier in the literature. In this paper, the main reason for this underachievement is identified, and an easy fix is suggested. It is demonstrated that combining the transformation with local likelihood density estimation methods produces very good estimators of R + -supported densities, not only close to the boundary, but also in the right tail. The asymptotic properties of the proposed 'local likelihood transformation kernel density estimators' are derived for a generic transformation, not only for the logarithm, which allows one to consider other transformations as well. One of them, called the 'probex' transformation, is given more focus. Finally, the excellent behaviour of those estimators in practice is evidenced through a comprehensive simulation study and the analysis of several real data sets. A nice consequence of articulating the method around local-likelihood estimation is that the resulting density estimates are typically smooth and visually pleasant, without oversmoothing important features of the underlying density.

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Section: Simulation Studymentioning

confidence: 99%

“…Although those methods may slightly improve the performance compared to their basic versions, they are more complicated to implement. For instance, Malec and Schienle (2014)…”

Section: Simulation Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Local-Likelihood Transformation Kernel Density Estimation for Positive Random Variables

Geenens

Wang

2018

Journal of Computational and Graphical Statistics

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“…. ., x n } from a distribution with unknown density of f(x), the associated kernel density estimator can be expressed as [Malec and Schienle, 2014] f^ðxÞ5 1 nb…”

Section: Copula-based Particle Filtermentioning

confidence: 99%

Development of a copula‐based particle filter (CopPF) approach for hydrologic data assimilation under consideration of parameter interdependence

Fan

Huang

Baetz

et al. 2017

Water Resources Research

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In this study, a copula-based particle filter (CopPF) approach was developed for sequential hydrological data assimilation by considering parameter correlation structures. In CopPF, multivariate copulas are proposed to reflect parameter interdependence before the resampling procedure with new particles then being sampled from the obtained copulas. Such a process can overcome both particle degeneration and sample impoverishment. The applicability of CopPF is illustrated with three case studies using a twoparameter simplified model and two conceptual hydrologic models. The results for the simplified model indicate that model parameters are highly correlated in the data assimilation process, suggesting a demand for full description of their dependence structure. Synthetic experiments on hydrologic data assimilation indicate that CopPF can rejuvenate particle evolution in large spaces and thus achieve good performances with low sample size scenarios. The applicability of CopPF is further illustrated through two real-case studies. It is shown that, compared with traditional particle filter (PF) and particle Markov chain Monte Carlo (PMCMC) approaches, the proposed method can provide more accurate results for both deterministic and probabilistic prediction with a sample size of 100. Furthermore, the sample size would not significantly influence the performance of CopPF. Also, the copula resampling approach dominates parameter evolution in CopPF, with more than 50% of particles sampled by copulas in most sample size scenarios. However, the EnKF can achieve good performances with small ensembles; but related studies have claimed Key Points: A copula-based particle filter approach is developed for hydrological data assimilation Synthetic experiments and real-case studies demonstrate performance of the CopPF method Results suggest that CopPF provides greater opportunities to access new samples which in turn lead to more accurate predictions

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“…KDE is a fundamental data-smoothing problem satisfying the continuity or differentiability property such that data are sampled from a given distribution. Although there are various kernel functions [17], this work refers to a Gaussian kernel function, which is most often used in the literature and is adequate to address non-Gaussian noise [7].…”

Section: Non-gaussian Square-root Unscented Particle Filtermentioning

confidence: 99%

A Hybrid Approach-based Sparse Gaussian Kernel Model for Vehicle State Estimation during the Free and Complete GPS Outages

Havyarimana¹,

Xiao²,

Wang³

2016

ETRI J

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To improve the ability to determine a vehicle's movement information even in a challenging environment, a hybrid approach called non-Gaussian square rootunscented particle filtering (nGSR-UPF) is presented. This approach combines a square root-unscented Kalman filter (SR-UKF) and a particle filter (PF) to determinate the vehicle state where measurement noises are taken as a finite Gaussian kernel mixture and are approximated using a sparse Gaussian kernel density estimation method. During an outage-free GPS period, the updated mean and covariance, computed using SR-UKF, are estimated based on a GPS observation update. During a complete GPS outage, nGSR-UPF operates in prediction mode. Indeed, because the inertial sensors used suffer from a large drift in this case, SR-UKF-based importance density is then responsible for shifting the weighted particles toward the high-likelihood regions to improve the accuracy of the vehicle state. The proposed method is compared with some existing estimation methods and the experiment results prove that nGSR-UPF is the most accurate during both outage-free and complete-outage GPS periods.

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Nonparametric Kernel Density Estimation Near the Boundary

Cited by 21 publications

References 43 publications

Local-Likelihood Transformation Kernel Density Estimation for Positive Random Variables

Local-Likelihood Transformation Kernel Density Estimation for Positive Random Variables

Development of a copula‐based particle filter (CopPF) approach for hydrologic data assimilation under consideration of parameter interdependence

A Hybrid Approach-based Sparse Gaussian Kernel Model for Vehicle State Estimation during the Free and Complete GPS Outages

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