Description Usage Arguments Details Value Note Author(s) References See Also Examples

Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed on all intervals spanned by two observations.

1 | ```
modeHunting(X.raw, lower = -Inf, upper = Inf, crit.vals, min.int = FALSE)
``` |

`X.raw` |
Vector of observations. |

`lower` |
Lower support point of |

`upper` |
Upper support point of |

`crit.vals` |
2-dimensional vector giving the critical values for the desired level. |

`min.int` |
If |

In general, the methods `modeHunting`

, `modeHuntingApprox`

, and
`modeHuntingBlock`

compute for a given level *α \in (0, 1)* and the corresponding
critical value *c_{jk}(α)* two sets of intervals

*\mathcal{D}^\pm(α) = \Bigl\{ \mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}} ) > c_{jk}(α) \Bigr\}*

where *\mathcal{I}_{jk}:=(X_{(j)},X_{(k)})* for *0≤ j < k ≤ n+1, k-j> 1* and *c_{jk}* are
appropriate critical values.

Specifically, the function `modeHunting`

computes *\mathcal{D}^\pm(α)* based on the two
test statistics

*T_n^+({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} \Bigl( |T_{jk}({\bf{X}})| / σ_{jk} - Γ \Bigl(\frac{k-j}{n+2}\Bigr)\Bigr)*

and

*T_n({\bf{X}}, \mathcal{I}) = \max_{(j,k) \in \mathcal{I}} ( |T_{jk}({\bf{X}})| / σ_{jk} ),*

using the set *\mathcal{I} := \mathcal{I}_{all}* of all intervals spanned by two observations
*(X_{(j)}, X_{(k)})*:

*\mathcal{I}_{all} = \Bigl\{(j, \ k ) \ : \ 0 ≤ j < k ≤ n+1, \ k - j > 1\Bigr\}.*

We introduced the local test statistics

*T_{jk}({\bf{X}}) := ∑_{i=j+1}^{k-1} ( 2 X_{(i; j, k)} - 1) 1\{X_{(i; j, k)} \in (0,1)\},*

for local order statistics

*X_{(i; j, k)} := \frac{X_{(i)}-X_{(j)}}{X_{(k)} - X_{(j)}},*

the standard deviation *σ_{jk} := √{(k-j-1)/3}* and the additive correction term
*Γ(δ) := √{2 \log(e / δ)}* for *δ > 0*.

If `min.int = TRUE`

, the set *\mathcal{D}^\pm(α)* is replaced by the set *{\bf{D}}^\pm(α)*
of its *minimal elements*. An interval *J \in \mathcal{D}^\pm(α)* is called *minimal* if
*\mathcal{D}^\pm(α)* 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(α) \setminus \bf{D}^\pm(α)* do not contain relevant information.

`Dp` |
The set |

`Dm` |
The set |

`Dp.noadd` |
The set |

`Dm.noadd` |
The set |

Critical values for `modeHunting`

and some combinations of *n* and *α* are provided in the
data set `cvModeAll`

. Critical values for other values of *n* and *α* can be generated
using `criticalValuesAll`

.

Parts of this function were derived from MatLab code provided on Lutz Duembgen's webpage,

http://www.staff.unibe.ch/duembgen.

Kaspar Rufibach, kaspar.rufibach@gmail.com,

http://www.kasparrufibach.ch

Guenther Walther, gwalther@stanford.edu,

www-stat.stanford.edu/~gwalther

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.

`modeHuntingApprox`

, `modeHuntingBlock`

, and `cvModeAll`

.

1 2 3 | ```
## for examples type
help("mode hunting")
## and check the examples there
``` |

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