modeest (version 1.06)

tsybakov: Tsybakov Mode Estimator

Description

This mode estimator is based on a gradient-like recursive algorithm. It includes Mizoguchi-Shimura (1976) mode estimator, based on the window training procedure.

Usage

tsybakov(x, 
         bw = NULL, 
         a, 
         alpha = 0.9, 
         kernel = "triangular", 
         djeddour = TRUE, 
         par = shorth(x))

Arguments

x

numeric. Vector of observations.

bw

numeric. Vector of length length(x) giving the sequence of smoothing bandwidths to be used.

a

numeric. Vector of length length(x) used in the gradient algorithm.

alpha

numeric. An alternative way of specifying a. See 'Details'.

kernel

character. The kernel to be used. Available kernels are "biweight", "cosine", "eddy", "epanechnikov", "gaussian", "optcosine", "rectangular", "tri

djeddour

logical. If TRUE, Djeddour et al. version of the estimate is used.

par

numeric. Initial value in the gradient algorithm. Default value is shorth(x).

Value

• A numeric value is returned, the mode estimate.

Warning

The Tsybakov mode estimate as it is presently computed does not work very well. The reasons of this inefficiency are under investigation.

Details

If bw is missing, then bw = (1:length(x))^(-1/7), which is the default value advised by Djeddour et al (2003). If a is missing, then a = (1:length(x))^(-alpha) (with alpha = 0.9 is alpha is missing), which is the default value advised by Djeddour et al (2003).

References

• Mizoguchi R. and Shimura M. (1976). Nonparametric Learning Without a Teacher Based on Mode Estimation.IEEE Transactions on Computers,C25(11):1109-1117.
• Tsybakov A. (1990). Recursive estimation of the mode of a multivariate distribution.Probl. Inf. Transm.,26:31-37.
• Djeddour K., Mokkadem A. et Pelletier M. (2003). Sur l'estimation r�cursive du mode et de la valeur modale d'une densit� de probabilit�.Technical report 105.
• Djeddour K., Mokkadem A. et Pelletier M. (2003). Application du principe de moyennisation � l'estimation r�cursive du mode et de la valeur modale d'une densit� de probabilit�.Technical report 106.

See Also

mlv for general mode estimation

Examples

x <- rbeta(1000, shape1 = 2, shape2 = 5)
## True mode:
betaMode(shape1 = 2, shape2 = 5)
## Estimation:
tsybakov(x, kernel = "triangular")
tsybakov(x, kernel = "gaussian", alpha = 0.99)
M <- mlv(x, method = "tsybakov", kernel = "gaussian", alpha = 0.99)
print(M)
plot(M)