dkern: Aitchison-Aitken kernel
In npmlreg: Nonparametric Maximum Likelihood Estimation for Random Effect Models

Description Usage Arguments Details Author(s) References Examples

Discrete kernel for categorical data with k unordered categories.

1	dkern(x, y, k, lambda)

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

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.

Jochen Einbeck (2006)

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.

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))
  }