modehunt (version 1.0.8)

modeHuntingApprox: 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)

Value

Dp

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

Dm

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

Dp.noadd

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

Dm.noadd

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

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.

Author

Kaspar Rufibach, ,
http://www.kasparrufibach.ch

Guenther Walther, ,
https://gwalther.su.domains/

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 $T_n^{+}(\mathbf{X},{\mathcal{I}}_{app})$ and $T_n(\mathbf{X},{\mathcal{I}}_{app})$.

If min.int = TRUE, the set ${\mathcal{D}}^{\pm }(\alpha )$ is replaced by the set ${\mathbf{D}}^{\pm }(\alpha )$ 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 ${\mathcal{D}}^{\pm }(\alpha )\setminus {\mathbf{D}}^{\pm }\mathbf{(}\alpha \mathbf{)}$ do not contain relevant information.

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