1. Kernel estimators ESSI SYRJÄLÄ

2. Introduction

3. More generally

4. Basic asymptotics

5. Kernel estimator

6. R-code: Choices of bandwidth  library(faraway)  data(trees)  attach(trees)  plot(Height ~ Girth, trees,main="bandwidth=1")  # The default uses a uniform kernel but it’s quite rough so we # change it to normal kernel  lines(ksmooth(Girth,Height,"normal",1),lwd=2,col = "red")  plot(Height ~ Girth, trees,main="bandwidth=3")  lines(ksmooth(Girth,Height,"normal",3),lwd=2,col = "red")  plot(Height ~ Girth, trees,main="bandwidth=7")  lines(ksmooth(Girth,Height,"normal",7),lwd=2,col = "red")

7. Kernel estimates with different bandwidths

8. R-code  install.packages("sm")  library(sm)  #Cross-validated choice of bandwidth  hm<-hcv(Girth,Height,display="lines") #hm=2.291831  #This uses Gaussian kernel  sm.regression(Girth,Height,h=hm,xlab="girth",ylab="height")

9. Cross-validation criterion as a function of a smoothing parameter and kernel estimate with this value of the smoothing parameter

10. Exercise  Use data ais from package alr3. Find the best value for the smoothing parameter (bandwidth) by plotting pictures with different bandwidths and then by cross-validation. Notice that you have to define start value and end value (?hcv).  Then do the same thing just for females (when sex is female).

11. References  Faraway, Julian J. Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models, Chapman& Hall/CRC, 2006.  Wikipedia. Kernel density estimation. Edited 1.4.2015. http://en.wikipedia.org/wiki/Kernel_density_estimation  Wikipedia. Big O notation. Edited 12.3.2015. http://en.wikipedia.org/wiki/Big_O_notation#Usage