1. 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

2. Problems with the Histogram Method

3. Histogram and KDE
Simulated Example

4. The Naïve Estimator

5. Definition of the Kernel Function

6. Some Commonly Used Kernel Functions

7. Examples of Kernels
4 kernels

8. The Kernel Density Estimator

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

10. Examples
Undersmooth and oversmooth

11. Examples
Uniform kernel used

12. Examples
Optimal kernel used

13. Some Remarks

14. The Convolution Expression

15. Measures of Performance

16. Theoretical Properties: Bias and Variance

17. Theoretical Properties: MSE and the Local Optimal Bandwidth

18. Theoretical Properties: MISE and the Global Optimal Bandwidth

19. Some Remarks

20. The Optimal Kernel

21. 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

22. Example of Kernel Efficiency-1 +1 1

23. Remarks for Kernel Choice