# Kernel bandwidth estimators for hypervolumes

### Description

Estimates bandwidth vector from data using multiple approaches.

### Usage

1 | ```
estimate_bandwidth(data,method="silverman")
``` |

### Arguments

`data` |
m x n matrix or data frame, where m is the number of observations and n the number of dimensions. |

`method` |
One of |

### Details

The Silverman (`"silverman"`

) estimator is defined as 1.06 * sd(X) * m^(-1/5) where m is the number of observations and X is the data vector in each dimension. Minimizes mean integrated square error along each axis independently. This is the default option due ONLY to computational simplicity.

The plug-in (`"plug-in"`

) estimator is defined using a diagonal plug-in estimator with a 2-stage pilot estimation and a pre-scaling transformation (in `ks::Hpi.diag`

). The resulting diagonal variances are then transformed to standard deviations and multiplied by two to be consistent for the box kernels used here. Available only in n<7 dimensions. Minimizes sum of asymptotic mean squared error.

The cross-validation (`"cross-validation"`

) estimator is defined using a diagonal smoothed cross validation estimator with a 2-stage pilot estimation and a pre-scaling transformation (in `ks::Hscv.diag`

). The resulting diagonal variances are then transformed to standard deviations and multiplied by two to be consistent for the box kernels used here. Available only in n<7 dimensions. Minimizes sum of asymptotic mean squared error.

Note that all estimators are optimal only for normal kernels, whereas the hypervolume algorithms use box kernels - as the number of data points increases, this difference will become increasingly less important.

Computational run-times for the plug-in and cross-validation estimators may become infeasibly large in n>=4 dimensions.

### Value

Vector of length n with each entry corresponding to the estimated bandwidth along each axis.

### References

Duong, T. (2007) ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R. Journal of Statistical Software 21, (7)

### Examples

1 2 3 4 | ```
data(iris)
print(estimate_bandwidth(iris[,1:2],method="silverman"))
print(estimate_bandwidth(iris[,1:2],method="plug-in"))
print(estimate_bandwidth(iris[,1:2],method="cross-validation"))
``` |