Plug-in bandwidth for for 1- to 6-dimensional data.

1 2 3 4 5 | ```
Hpi(x, nstage=2, pilot, pre="sphere", Hstart, binned=FALSE, bgridsize,
amise=FALSE, deriv.order=0, verbose=FALSE, optim.fun="nlm")
Hpi.diag(x, nstage=2, pilot, pre="scale", Hstart, binned=FALSE, bgridsize,
amise=FALSE, deriv.order=0, verbose=FALSE, optim.fun="nlm")
hpi(x, nstage=2, binned=TRUE, bgridsize, deriv.order=0)
``` |

`x` |
vector or matrix of data values |

`nstage` |
number of stages in the plug-in bandwidth selector (1 or 2) |

`pilot` |
"amse" = AMSE pilot bandwidths |

`pre` |
"scale" = |

`Hstart` |
initial bandwidth matrix, used in numerical optimisation |

`binned` |
flag for binned kernel estimation. Default is FALSE. |

`bgridsize` |
vector of binning grid sizes |

`amise` |
flag to return the minimal scaled PI value |

`deriv.order` |
derivative order |

`verbose` |
flag to print out progress information. Default is FALSE. |

`optim.fun` |
optimiser function: one of |

`hpi(,deriv.order=0)`

is the univariate plug-in
selector of Wand & Jones (1994), i.e. it is exactly the same as
KernSmooth's `dpik`

. For deriv.order>0, the formula is
taken from Wand & Jones (1995). `Hpi`

is a multivariate
generalisation of this. Use `Hpi`

for unconstrained bandwidth matrices and
`Hpi.diag`

for diagonal bandwidth matrices.

The default pilot is `"samse"`

for d=2,r=0, and
`"dscalar"`

otherwise.
For AMSE pilot bandwidths, see Wand & Jones (1994). For
SAMSE pilot bandwidths, see Duong & Hazelton (2003). The latter is a
modification of the former, in order to remove any possible problems
with non-positive definiteness. Unconstrained and higher order
derivative pilot bandwidths are from Chacon & Duong (2010).

For d=1, 2, 3, 4 and `binned=TRUE`

,
estimates are computed over a binning grid defined
by `bgridsize`

. Otherwise it's computed exactly.
If `Hstart`

is not given then it defaults to `Hns(x)`

.

Plug-in bandwidth.
If `amise=TRUE`

then the minimal scaled PI value is returned too.

Chacon, J.E. & Duong, T. (2010) Multivariate plug-in bandwidth
selection with unconstrained pilot matrices. *Test*, **19**, 375-398.

Duong, T. & Hazelton, M.L. (2003) Plug-in bandwidth matrices for
bivariate kernel density estimation. *Journal of Nonparametric
Statistics*. **15**, 17-30.

Sheather, S.J. & Jones, M.C. (1991) A reliable data-based bandwidth selection
method for kernel density estimation. *Journal of the Royal
Statistical Society Series B*. **53**, 683-690.

Wand, M.P. & Jones, M.C. (1994) Multivariate plugin bandwidth
selection. *Computational Statistics*. **9**, 97-116.

1 2 3 |

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