mig_kdens_bandwidth {mig}

Optimal scale matrix for MIG kernel density estimation

Description

Given an n sample from a multivariate inverse Gaussian distribution on the half-space defined by \{\boldsymbol{x} \in \mathbb{R}^d: \boldsymbol{\beta}^\top\boldsymbol{x}>0\}, the function computes the bandwidth (type="isotropic") or scale matrix that minimizes the asymptotic mean integrated squared error away from the boundary. The latter depend on the true unknown density, which is replaced using as plug-in a MIG distribution evaluated at the maximum likelihood estimator. The integral or the integrated squared error are obtained by Monte Carlo integration with N simulations

Usage

mig_kdens_bandwidth(
  x,
  beta,
  shift,
  method = c("amise", "lcv", "lscv", "rlcv"),
  type = c("isotropic", "full"),
  approx = c("mig", "tnorm"),
  transformation = c("none", "scaling", "spherical"),
  N = 10000L,
  buffer = 0.25,
  pointwise = NULL,
  maxiter = 2000L,
  ...
)

Arguments

x

an n by d matrix of observations

beta

d vector defining the half-space

shift

location vector for translating the half-space. If missing, defaults to zero

method

estimation criterion, either amise for the expression that minimizes the asymptotic integrated squared error, lcv for likelihood (leave-one-out) cross-validation, lscv for least-square cross-validation or rlcv for robust cross validation of Wu (2019)

type

string indicating whether to compute an isotropic model or estimate the optimal scale matrix via optimization

approx

string; distribution to approximate the true density function f(x); either mig for multivariate inverse Gaussian, or tnorm for truncated Gaussian.

transformation

string for optional scaling of the data before computing the bandwidth. Either standardization to unit variance scaling, spherical transformation to unit variance and zero correlation (spherical), or none (default).

N

integer number of simulations to evaluate the integrals of the MISE by Monte Carlo

buffer

double indicating the buffer from the halfspace

pointwise

if NULL, evaluates the mean integrated squared error, otherwise a d vector to evaluate the bandwidth or scale pointwise

maxiter

integer; max number of iterations in the call to optim.

...

additional parameters, currently ignored

Value

a d by d scale matrix

References

Wu, X. (2019). Robust likelihood cross-validation for kernel density estimation. Journal of Business & Economic Statistics, 37(4), 761–770. doi:10.1080/07350015.2018.1424633 Bowman, A.W. (1984). An alternative method of cross-validation for the smoothing of density estimates, Biometrika, 71(2), 353–360. doi:10.1093/biomet/71.2.353 Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics, 9(2), 65–78. http://www.jstor.org/stable/4615859