modeHuntingApprox {modehunt}

Multiscale analysis of a density on the approximating set of intervals

Description

Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed on the approximating set of intervals.

Usage

modeHuntingApprox(X.raw, lower = -Inf, upper = Inf, 
    d0 = 2, m0 = 10, fm = 2, crit.vals, min.int = FALSE)

Arguments

X.raw

Vector of observations.

lower

Lower support point of f, if known.

upper

Upper support point of f, if known.

d0

Initial parameter for the grid resolution.

m0

Initial parameter for the number of observations in one block.

fm

Factor by which m is increased from block to block.

crit.vals

2-dimensional vector giving the critical values for the desired level.

min.int

If min.int = TRUE, the set of minimal intervals is output, otherwise all intervals with a test statistic above the critical value are given.

Details

See blocks for details how \mathcal{I}_{app} is generated and modeHunting for a proper introduction to the notation used here. The function modeHuntingApprox computes \mathcal{D}^\pm(\alpha) based on the two test statistics T_n^+({\bf{X}}, \mathcal{I}_{app}) and T_n({\bf{X}}, \mathcal{I}_{app}).

If min.int = TRUE, the set \mathcal{D}^\pm(\alpha) is replaced by the set {\bf{D}}^\pm(\alpha) of its minimal elements. An interval J \in \mathcal{D}^\pm(\alpha) is called minimal if \mathcal{D}^\pm(\alpha) contains no proper subset of J. This minimization post-processing step typically massively reduces the number of intervals. If we are mainly interested in locating the ranges of increases and decreases of f as precisely as possible, the intervals in \mathcal{D}^\pm(\alpha) \setminus \bf{D}^\pm(\alpha) do not contain relevant information.

Value

Dp

The set \mathcal{D}^+(\alpha) (or \bf{D}^+(\alpha)), based on the test statistic with additive correction \Gamma.

Dm

The set \mathcal{D}^-(\alpha) (or \bf{D}^-(\alpha)), based on the test statistic with \Gamma.

Dp.noadd

The set \mathcal{D}^+(\alpha) (or \bf{D}^+(\alpha)), based on the test statistic without \Gamma.

Dm.noadd

The set \mathcal{D}^+(\alpha) (or \bf{D}^-(\alpha)), based on the test statistic without \Gamma.

Note

Critical values for modeHuntingApprox and some combinations of n and \alpha are provided in the data set cvModeApprox. Critical values for other values of n and \alpha can be generated using criticalValuesApprox.

Author(s)

Kaspar Rufibach, [e-mail],
http://www.kasparrufibach.ch

Guenther Walther, [e-mail],
www-stat.stanford.edu/~gwalther

References

Duembgen, L. and Walther, G. (2008). Multiscale Inference about a density. Ann. Statist., 36, 1758–1785.

Rufibach, K. and Walther, G. (2010). A general criterion for multiscale inference. J. Comput. Graph. Statist., 19, 175–190.

See Also

modeHunting, modeHuntingBlock, and cvModeApprox.

Examples

## for examples type
help("mode hunting")
## and check the examples there