On the expansion of the mean integrated squared error of a kernel density estimator

2001 ◽  
Vol 52 (4) ◽  
pp. 441-450 ◽  
Author(s):  
Bert van Es
Keyword(s):  
Kernel Density ◽  
Kernel Density Estimator ◽  
Density Estimator ◽  
Mean Integrated Squared Error ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean

scholarly journals Transmetric Density Estimation

2016 ◽  
Vol 5 (2) ◽  
pp. 35
Author(s):  
Sigve Hovda
Keyword(s):  
Density Estimation ◽  
Kernel Density Estimation ◽  
Kernel Density ◽  
Convergence Order ◽  
Graphical Display ◽  
Mean Integrated Squared Error ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean ◽  
Convergence Orders

<div>A transmetric is a generalization of a metric that is tailored to properties needed in kernel density estimation.  Using transmetrics in kernel density estimation is an intuitive way to make assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display.  This framework is required for discussing the estimators that are suggested by Hovda (2014).</div><div> </div><div>Asymptotic arguments for the bias and the mean integrated squared error is difficult in the general case, but some results are given when the transmetric is of the type defined in Hovda (2014).  An important contribution of this paper is that the convergence order can be as high as $4/5$, regardless of the number of dimensions.</div>

scholarly journals Properties of Transmetric Density Estimation

2016 ◽  
Vol 5 (3) ◽  
pp. 63
Author(s):  
Sigve Hovda
Keyword(s):  
Density Estimation ◽  
Kernel Density ◽  
Density Estimator ◽  
Unified Approach ◽  
Squared Error ◽  
Special Cases ◽  
Integrated Squared Error ◽  
The Common ◽  
The Mean ◽  
Convergence Orders

Transmetric density estimation is a generalization of kernel density estimation that is proposed in Hovda(2014) and Hovda (2016), This framework involves the possibility of making assumptions on the kernel of the distribution to improve convergence orders and to reduce the number of dimensions in the graphical display.  In this paper we show that several state-of-the-art nonparametric, semiparametric and even parametric methods are special cases of this formulation, meaning that there is a unified approach. Moreover, it is shown that parameters can be trained using unbiased cross-validation.  When parameter estimation is included, the mean integrated squared error of the transmetric density estimator is lower than for the common kernel density estimator, when the number of dimensions is larger than two.

NONLINEAR WAVELET ESTIMATION OF CONDITIONAL DENSITY UNDER LEFT-TRUNCATED AND α-MIXING ASSUMPTIONS

2011 ◽  
Vol 09 (06) ◽  
pp. 989-1023 ◽  
Author(s):  
SI-LI NIU ◽  
HAN-YING LIANG
Keyword(s):  
Asymptotic Expression ◽  
Kernel Estimator ◽  
Conditional Density ◽  
Left Truncation ◽  
Mean Integrated Squared Error ◽  
Wavelet Estimator ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean ◽  
Mixing Sequence

In this paper, we construct a nonlinear wavelet estimator of conditional density function for a left truncation model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves.

scholarly journals A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Christophe Chesneau
Keyword(s):  
Density Estimation ◽  
General Setting ◽  
Type Model ◽  
Hard Thresholding ◽  
Mean Integrated Squared Error ◽  
Wavelet Estimator ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean ◽  
Weakly Dependent Data

We consider the estimation of an unknown functionffor weakly dependent data (α-mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

NONLINEAR WAVELET DENSITY ESTIMATION FOR TRUNCATED AND DEPENDENT OBSERVATIONS

2011 ◽  
Vol 09 (04) ◽  
pp. 587-609 ◽  
Author(s):  
JUAN-JUAN CAI ◽  
HAN-YING LIANG
Keyword(s):  
Asymptotic Expression ◽  
Kernel Estimator ◽  
Density Estimator ◽  
Mean Integrated Squared Error ◽  
Wavelet Estimator ◽  
Dependent Observations ◽  
Squared Error ◽  
Integrated Squared Error ◽  
Wavelet Density Estimation ◽  
Mixing Sequence

In this paper, we provide an asymptotic expression for mean integrated squared error (MISE) of nonlinear wavelet density estimator for a truncation model. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimator, the MISE expression of the nonlinear wavelet estimator is not affected by the presence of discontinuities in the curves. Also, we establish asymptotic normality of the nonlinear wavelet estimator.

scholarly journals Bootstrap Selector for the Smoothing Parameter of Beran’s Estimator

2021 ◽  
Vol 7 (1) ◽  
pp. 28
Author(s):  
Rebeca Peláez Suárez ◽  
Ricardo Cao Abad ◽  
Juan M. Vilar Fernández
Keyword(s):  
Confidence Intervals ◽  
Simulation Study ◽  
Smoothing Parameter ◽  
Bootstrap Confidence Intervals ◽  
Bootstrap Approximation ◽  
Mean Integrated Squared Error ◽  
Smoothed Bootstrap ◽  
Squared Error ◽  
Integrated Squared Error ◽  
The Mean

This work proposes a resampling technique to approximate the smoothing parameter of Beran’s estimator. It is based on resampling by the smoothed bootstrap and minimising the bootstrap approximation of the mean integrated squared error to find the bootstrap bandwidth. The behaviour of this method has been tested by simulation on several models. Bootstrap confidence intervals are also addressed in this research and their performance is analysed in the simulation study.

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