dkern function

Aitchison-Aitken kernel

Discrete kernel for categorical data with k unordered categories.

dkern(x, y, k, lambda)

Arguments

• x: categorical data vector
• y: postive integer defining a fixed category
• k: positive integer giving the number of categories
• lambda: smoothing parameter

Details

This kernel was introduced in Aitchison & Aitken (1976); see also Titterington (1980).

The setting lambda =1/k corresponds to the extreme case 'maximal smoothing', while lambda = 1 means no smoothing'. Statistically sensible settings are only 1/k$<=$ lambda $<=$1`.

References

Aitchison, J. and Aitken, C.G.G. (1976). Multivariate binary discrimination by kernel method. Biometrika 63, 413-420.

Titterington, D. M. (1980). A comparative study of kernel-based density estimates for categorical data. Technometrics, 22, 259-268.

Author(s)

Jochen Einbeck (2006)

Examples

k<-6; 
dkern(1:k,4,k,0.99)   
# Kernel centered at the 4th component with a very small amount of smoothing.

## The function is currently defined as
function(x,y,k,lambda){
ifelse(y==x, lambda, (1-lambda)/(k-1))
  }