1. Nonparametric Density Estimation
Learn the kernel method for nonparametric density estimation
Learn the difference between fixed kernel window width and variable kernel window width estimates
Learn some of the approaches to kernel window width estimation
Reference
Least squares cross validation
Reading
Silverman Chapter 1, Chapter 2 ( , 2.9), Chapter 3 ( )

2. Parametric vs Nonparametric
Assume data from a known distribution (e.g. Gamma, Normal) and estimate parameters
Nonparametric
Assume that the data has a probability density function but not of a specific known form
Let data speak for themselves
Exploratory data analysis

3. Which method conveys the information best to you ?
Cumulative Density
Probability Density
Equation
Probability Plot

4. San Juan River Annual Streamflow 1906-1983
Histogram, bin width 150
Flow KAF
Number

5. San Juan River Annual Streamflow 1906-1983
Density
Flow KAF

6. Kernel Density Estimator

7. Kernel Density Estimate (KDE)
Place “kernels” at each data point
Sum up the kernels
Width of kernel determines level of smoothing
Determining how to choose the width of the kernel is an important topic!
Narrow kernel
Medium kernel
Wide kernel

8. KDE window width sensitivity

9. Variable Kernel

10. San Juan River Annual Streamflow 1906-1983, Variable Kernel Density Estimate
Flow KAF

11. Some Common Kernels

12. Density estimate with different kernels with the same bandwidth

13. Least Squares Cross Validation
Density
Mo(h)
Flow KAF
Window width h