R: The Tsybakov Mode Estimator

tsybakov {modeest} | R Documentation

## The Tsybakov Mode Estimator

### Description

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

### Usage

      tsybakov(x,
               bw = NULL,
               a,
               alpha = 0.9,
               kernel = "triangular",
               dmp = 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"`, `"triangular"`, `"uniform"`. See `density.default` for more details on some of these kernels.
`dmp` | 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)`.

### 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` if `alpha` is missing), which is the default value advised by Djeddour et al (2003).

### 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.

### Note

The user should preferentially call `tsybakov` through `mlv(x, method = "tsybakov", ...)`. This returns an object of class `mlv`.

### Author(s)

P. Poncet

### 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 recursive du mode et de la valeur modale d'une densite de probabilite. Technical report 105.

- Djeddour K., Mokkadem A. et Pelletier M. (2003). Application du principe de moyennisation a l'estimation recursive du mode et de la valeur modale d'une densite de probabilite. 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)

[Package modeest version 2.1 Index]