Description Usage Arguments Details Value Note Author(s) References See Also Examples
View source: R/modeHuntingBlock.r
Simultanous confidence statements for the existence and location of local increases and decreases of a density f, computed via the block procedure.
1 2 | modeHuntingBlock(X.raw, lower = -Inf, upper = Inf, d0 = 2,
m0 = 10, fm = 2, crit.vals, min.int = FALSE)
|
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 |
See blocks
for details how \mathcal{I}_{app} is generated and modeHunting
for
a proper introduction to the notation used here.
The function modeHuntingBlock
uses the test statistic T^+_n({\bf X}, \mathcal{B}_r),
where \mathcal{B}_r contains all intervals of Block r, r=1,…,\#blocks.
Critical values for each block individually are received via finding an \tilde α such that
P(B_n({\bf{X}}) > q_{r,\tilde α / (r+tail)^γ} \ for \ at \ least \ one \ r) ≤ α,
where q_{r,α} is the (1-α)–quantile of the distribution of T^+_n({\bf X}, \mathcal{B}_r). We then define the sets \mathcal{D}^\pm(α) as
\mathcal{D}^\pm(α) := \Bigl\{\mathcal{I}_{jk} \ : \ \pm T_{jk}({\bf{X}}) > q_{r,\tilde α / (r+tail)^γ} \, , \ r = 1,… \#blocks\Bigr\}.
Note that γ and tail are automatically determined by crit.vals.
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 \mathcal{D}^+(α) (or \bf{D}^+(α)). |
Dm |
The set \mathcal{D}^-(α) (or \bf{D}^-(α)). |
Critical values for some combinations of n and α are provided in the
data sets cvModeBlock
. Critical values for other
values of n and α can be generated using criticalValuesApprox
.
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.
modeHunting
, modeHuntingApprox
, and cvModeBlock
.
1 2 3 | ## for examples type
help("mode hunting")
## and check the examples there
|
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