awstestprop: Propagation condition for adaptive weights smoothing

Description

The function enables testing of the propagation condition in order to select appropriate values for the parameter lambda in function aws.

Usage

awstestprop(dy, hmax, theta = 1, family = "Gaussian", lkern = "Triangle",
            aws = TRUE, memory = FALSE, shape = 2, homogeneous=TRUE, varadapt=FALSE,
            ladjust = 1, spmin=0.25, seed = 1, minlevel=1e-6, maxz=25, diffz=.5,
            maxni=FALSE, verbose=FALSE)
pawstestprop(dy, hmax, theta = 1, family = "Gaussian", lkern = "Triangle",
             aws = TRUE, patchsize=1, shape = 2,
             ladjust = 1, spmin = 0.25, seed = 1, minlevel = 1e-6,
             maxz = 25, diffz = .5, maxni = FALSE, verbose = FALSE)

Value

A list with components

h: Sequence of bandwidths used
z: seq(0,30,.5), the quantiles exceedence probabilities refer to
prob: the matrix of exceedence probabilities, columns corresponding to h
probna: the matrix of exceedence probabilities for corresponding nonadaptive estimates, columns corresponding to h

Arguments

dy: Dimension of grid used in 1D, 2D or 3D. May also be specified as an array of values. In this case data are generated with parameters dy-mean(dy)+theta and the propagation condition is testet as if theta is the true parameter. This can be used to study properties for a slighty misspecified structural assumption.
hmax: Maximum bandwidth.
theta: Parameter determining the distribution in case of family %in% c("Poisson","Bernoulli")
family: family specifies the probability distribution. Default is family="Gaussian", also implemented are "Bernoulli", "Poisson", "Exponential", "Volatility", "Variance" and "NCchi". family="Volatility" specifies a Gaussian distribution with expectation 0 and unknown variance. family="Volatility" specifies that p*y/theta is distributed as \(\chi^2\) with p=shape degrees of freedom. family="NCchi" uses a noncentral Chi distribution with p=shape degrees of freedom and noncentrality parameter theta.
lkern: character: location kernel, either "Triangle", "Plateau", "Quadratic", "Cubic" or "Gaussian"
aws: logical: if TRUE structural adaptation (AWS) is used.
patchsize: patchsize in case of paws.
memory: logical: if TRUE stagewise aggregation is used as an additional adaptation scheme.
shape: Allows to specify an additional shape parameter for certain family models. Currently only used for family="Variance", that is \(\chi\)-Square distributed observations with shape degrees of freedom.
homogeneous: if homgeneous==FALSE and family==Gaussian then create heterogeneous variances according to a chi-squared distribution with number of degrees of freedom given by sphere
varadapt: if varadapt==TRUE use inverse of variance reduction instead of sum of weights in definition of statistical penalty.
ladjust: Factor to increase the default value of lambda
spmin: Determines the form (size of the plateau) in the adaptation kernel. Not to be changed by the user.
seed: Seed value for random generator.
minlevel: Minimum exceedence probability to use in contour plots.
maxz: Maximum of z-scale in plots.
diffz: Gridlength in z
maxni: If TRUE use \(max_{l<=k}(N_i^{(l)}\) instead of \((N_i^{(k)}\) in the definition of the statistical penalty.
verbose: If TRUE provide additional information.

Author

Joerg Polzehl polzehl@wias-berlin.de

Details

Estimates exceedence probabilities

Results for intermediate steps are provided as contour plots. For a good choice of lambda (ladjust) the contours up to probabilities of 1e-5 should be vertical.

References

S. Becker, P. Mathe, Electron. J. Statist. (2013), 2702-2736, doi:10.1214/13-EJS860