Strong consistency of the kernel estimators of conditional density function

Lincheng, Zhao; Liu, Zhijun

doi:10.1007/bf02564838

Cited by 12 publications

(4 citation statements)

References 3 publications

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“…Several nonparametric methods have been proposed for estimating a conditional density: Hyndman et al (1996) and Fan et al (1996) have improved the seminal Nadaraya-Watson-type estimator of Rosenblatt (1969) and Lincheng and Zhijun (1985), as well as De Gooijer and Zerom (2003) who introduced another weighted kernel estimator. For these kernel estimators, different methods have been advocated to tackle the bandwidth selection issue: bootstrap approach (Bashtannyk and Hyndman, 2001) or cross-validation variants, see Fan and Yim (2004); Holmes et al (2010), Ichimura and Fukuda (2010).…”

Section: Motivationsmentioning

confidence: 99%

Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation

Nguyen,

Lacour,

Rivoirard

2021

Preprint

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This paper studies the estimation of the conditional density f (x, •) of Y i given X i = x, from the observation of an i.i.d. sample (X i , Y i ) ∈ R d , i ∈ {1, . . . , n}. We assume that f depends only on r unknown components with typically r d. We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate f . To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty, 2006) to detect the sparsity structure of f . More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Our results also hold for density estimation. The computational complexity of our method is only O(dn log n). A deep numerical study shows nice performances of our approach.

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Section: Motivationsmentioning

confidence: 99%

Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation

Nguyen,

Lacour,

Rivoirard

2021

Preprint

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“…Once β(x) is fitted, we can sample from the estimated density f (x) by importance sampling with g(x) as the proposal distribution. In this paper, the density ratio is estimated by nonparametric methods based on the k-nearest-neighbors kernel density estimator (Lincheng & Zhijun 1985), which approximates the density at a point by a kernel smoother applied to the k nearest neighbors of that point, and the Spectral Series estimators (Izbicki et al 2014), which combines orthogonal series expansion and adaptively chosen bases to construct non-parametric likelihood functions. With the density ratio, we are improving the Gaussian likelihood model using non-parametric methods.…”

Section: Multivariate Likelihood Modelsmentioning

confidence: 99%

Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases

Lin,

Harnois-Déraps,

Eifler

et al. 2019

Preprint

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We study the significance of non-Gaussianity in the likelihood of weak lensing shear twopoint correlation functions, detecting significantly non-zero skewness and kurtosis in onedimensional marginal distributions of shear two-point correlation functions in simulated weak lensing data though the full multivariate distributions are relatively more Gaussian. We examine the implications in the context of future surveys, in particular LSST, with derivations of how the non-Gaussianity scales with survey area. We show that there is no significant bias in one-dimensional posteriors of Ω m and σ 8 due to the non-Gaussian likelihood distributions of shear correlations functions using the mock data (100 deg 2 ). We also present a systematic approach to constructing an approximate multivariate likelihood function by decorrelating the data points using principal component analysis (PCA). When using a subset of the PCA components that account for the majority of the cosmological signal as a data vector, the one-dimensional marginal likelihood distributions of those components exhibit less skewness and kurtosis than the original shear correlation functions. We further demonstrate that the difference in cosmological parameter constraints between the multivariate Gaussian likelihood model and more complex non-Gaussian likelihood models would be even smaller for an LSST-like survey due to the area effect. In addition, the PCA approach automatically serves as a data compression method, enabling the retention of the majority of the cosmological information while reducing the dimensionality of the data vector by a factor of ∼5.

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“…In this paper, we take l = 0, which corresponds to a leave-one-out cross-validated negative log-predictive likelihood. We takef θ to be the hybrid kernel density estimator-k-nearest-neighbor estimator, also known as a kernel nearest-neighbor estimator [39], of the predictive density. The kernel nearest-neighbor estimator of the predictive density of X 0 given X −1 − is given bŷ…”

Section: The Negative Log-predictive Likelihoodmentioning

confidence: 99%

Information-theoretic model selection for optimal prediction of stochastic dynamical systems from data

Darmon

2018

Phys. Rev. E

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In the absence of mechanistic or phenomenological models of real-world systems, data-driven models become necessary. The discovery of various embedding theorems in the 1980s and 1990s motivated a powerful set of tools for analyzing deterministic dynamical systems via delay-coordinate embeddings of observations of their component states. However, in many branches of science, the condition of operational determinism is not satisfied, and stochastic models must be brought to bear. For such stochastic models, the tool set developed for delay-coordinate embedding is no longer appropriate, and a new toolkit must be developed. We present an information-theoretic criterion, the negative log-predictive likelihood, for selecting the embedding dimension for a predictively optimal data-driven model of a stochastic dynamical system. We develop a nonparametric estimator for the negative log-predictive likelihood and compare its performance to a recently proposed criterion based on active information storage. Finally, we show how the output of the model selection procedure can be used to compare candidate predictors for a stochastic system to an information-theoretic lower bound.

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Strong consistency of the kernel estimators of conditional density function

Cited by 12 publications

References 3 publications

Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation

Adaptive greedy algorithm for moderately large dimensions in kernel conditional density estimation

Non-Gaussianity in the Weak Lensing Correlation Function Likelihood -- Implications for Cosmological Parameter Biases

Information-theoretic model selection for optimal prediction of stochastic dynamical systems from data

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