A Bayesian Approach for Bandwidth Selection in Kernel Density Estimation

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A Bayesian Approach for Bandwidth Selection in
Kernel Density Estimation

• C.N.Kuruwita
• Department of Mathematical Sciences
• Clemson University

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What is Density Estimation ?

• The process of obtaining an estimate for an
  unknown probability density function.
• There are two main approaches.
• Parametric.
• Nonparametric.

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Parametric Approach.

• Assumes the data come from a known
• parametric family with density f0(. ?).
• Then estimate the unknown parameter ?
• using a suitable parameter estimation
• method.
• e.g Maximum Likelihood.

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Problems in Parametric Approach

• Restricting the estimator to a certain
  parametric family
• makes important features in the data
  undetected.

Parametric estimate with a lognormal density
Nonparametric estimate
Reference Kernel Smoothing , Wand Jones (1995)
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Nonparametric Approach

• Let the data speak for themselves.

• Problems
• Properties of these estimators are hard to
• determine.
• Computer Intensive.

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Common Nonparametric Density Estimation Methods.

• Kernel density estimators.
• Nearest neighbor method.
• Maximum penalized likelihood estimators.
• Orthogonal series estimators.

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Kernel Density Estimator.

• Definition
• K(.) is a symmetric pdf. (usually)
• h is called the bandwidth or smoothing
  parameter.

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The Problem

• Spill over effect.

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• Data driven bandwidth selection.

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The Approach

• Use an asymmetric kernel with a positive support
  to avoid the spill over effect.
• Assign a prior density for the smoothing
  parameter h.
• Derive the density estimator on a Bayesian
  framework.

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The Lognormal Kernel Density Estimator

• Definition
• K(.) is a lognormal pdf, with scale parameter
  h.
• sj is the jump size of the Kaplan-Meier estimator
  at each observation .
• h is assigned an inverted gamma prior with
  shape parameter ? and scale parameter ?.

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Resulting Bayesian Bandwidth.

• The Bayesian estimator of the smoothing
  parameter h under squared error loss is given
  as
• where

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Asymptotic Properties

• The lognormal KDE converge to the actual
• pdf as .
• i.e a.s ,
• The Bayesian local bandwidths converge to
• zero as .
• i.e a.s ,

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Simulation Study

• Assess the effect of the lognormal kernel
• Compare with Inverse Gaussian kernel.
• Assess the effectiveness of the Bayesian
  bandwidths.
• Compare with Cross Validated bandwidth.
• Simulated Failure Rate Data from Weibull(?,1)
• Decreasing Failure Rate ( DFR )
• Constant Failure Rate ( CFR )
• Increasing Failure Rate ( IFR )
• Censoring levels. 10, 20, 50

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Assessment Criteria

• Pointwise Estimated MSE Ratio
• Estimated Mean Integrated Squared Error
• where N is the number of simulations.

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Estimated Mean Integrated Squared Errors.
IG- Bayes
LN-CV
LN-Bayes
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Application to Real Data
The Problem Estimation of the probability
density of debond strength of carbon
fibers. Data Due to the complexity of the
experiment there were only 12
observations and 3 of which are
censored. Reference Harwell M. (1995)
Microbond Tests for ribbon fibers . M.S.thesis,
Department of Chemical Engineering , Clemson
University.
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Density Estimates
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Conclusion and Future Work

• The lognormal KDE with the Bayesian bandwidths
  shows lot of potential as density estimator.
• Need to explore the boundary effect at the right
  of the support. i.e. when the support is finite
  as 0,? , with

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Thank You