KERNEL DENSITY ESTIMATION

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Presentation transcript:

KERNEL DENSITY ESTIMATION Lecture 3 KERNEL DENSITY ESTIMATION Problems with the Histogram Method The Naïve Estimator Kernel Functions The Kernel Density Estimator Theoretical Properties The Optimal Kernel

Problems with the Histogram Method

Histogram and KDE Simulated Example

The Naïve Estimator

Definition of the Kernel Function

Some Commonly Used Kernel Functions

Examples of Kernels 4 kernels

The Kernel Density Estimator

Examples x=randn(200,1); >> [xpdf,h]=kde0(x);h=.36572 by normal reference method

Examples Undersmooth and oversmooth

Examples Uniform kernel used

Examples Optimal kernel used

Some Remarks

The Convolution Expression

Measures of Performance

Theoretical Properties: Bias and Variance

Theoretical Properties: MSE and the Local Optimal Bandwidth

Theoretical Properties: MISE and the Global Optimal Bandwidth

Some Remarks

The Optimal Kernel

Eff(K) K(t) 1 0.9295 0.9512 0.9859 0.9939 Uniform Gaussian Triangular Kernel Efficiency Efficiencies of some commonly used kernels 1 0.9295 0.9512 0.9859 0.9939 Uniform Gaussian Triangular Biweight Epanechnikov Eff(K) K(t) Kernel 2 1

Example of Kernel Efficiency -1 +1 1

Remarks for Kernel Choice