modehunt (version 1.0.7)

modehunt-package: Multiscale Analysis for Density Functions

Description

Provides five methods and corresponding critical values to perform mode hunting, i.e. to compute multiscale test statistics based on local order statistics and spacings that provide simultaneous confidence statements for the existence and location of local increases and decreases of a density.

Details

ll{ Package: modehunt Type: Package Version: 1.0.7 Date: 2015-07-03 License: GPL (>=2) } In Duembgen and Walther (2008) a multiscale test statistic based on spacings was introduced. This method provides simultaneous confidence statements for the existence and location of local increases and decreases of a density. The procedure guarantees finite--sample significance levels and possesses certain asymptotic optimality and adaptivity properties. However, since the local test statistics are computed on all $O(n^2)$ intervals in the set $$\mathcal{I}_{all} = \Bigl{(j, \ k ) \ : \ 0 \le j < k \le n+1, \ k - j > 1\Bigr},$$ this latter procedure is computationally very expensive. Furthermore, the correction term $\Gamma$ employed by Duembgen and Walther (2008) to prevent the global test statistic to be dominated by the values of the local test statistics on small scales needs in principle to be re--derived for any new local test statistic, a non--trivial task in general. In Rufibach and Walther (2010), two new procedures are proposed: One that within the original framework of Duembgen and Walther (2008) approximates the set $\mathcal{I}_{all}$ by a specific subset of intervals $\mathcal{I}_{app}$ that only contains $O(n \log n )$ intervals. It is shown that considering $\mathcal{I}_{app}$ yields a procedure that is in terms of power asymptotically equivalent to that based on $\mathcal{I}_{all}$, however, computationally much more efficient. Finally, Rufibach and Walther (2010) propose a block procedure. Here, all intervals under consideration are grouped into blocks, where each interval in a block contains approximately the same number of original observations. Critical values are then computed per block. Again, this procedure is basically asymptotically equivalent to the standard approach proposed in Duembgen and Walther (2008), but again computationally much faster. It further offers a (finite--sample) tradeoff between employing or omitting the additive correction $\Gamma$. The initial procedure by Duembgen and Walther (2008) is implemented as the function modeHunting. The help file to the latter function also contains some more description of the mathematical details. criticalValuesAll can be used to compute critical values for this approach and cvModeAll contains a table of critical values (with and without correction term) for some $n$ and $alpha$. The corresonding functions and $p$-values for the approximation are made available as modeHuntingApprox, criticalValuesApprox, and cvModeApprox and for the block method as modeHuntingBlock, criticalValuesBlock, and cvModeBlock.

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.

Examples

## generate random sample
set.seed(1977)
n <- 200; a <- 0; b <- 0.5; s <- 2 / (b - a) 
X.raw <- rlin(n, a, b, s)

## input critical values
alpha <- 0.05
data(cvModeAll); data(cvModeApprox); data(cvModeBlock)
cv.all <- cvModeAll[cvModeAll$alpha == alpha & cvModeAll$n == n, 3:4]
cv.approx <- cvModeApprox[cvModeApprox$alpha == alpha & cvModeApprox$n == n, 3:4]
cv.block <- cvModeBlock[cvModeBlock$alpha == alpha & cvModeBlock$n == n, 3:11]

## standard procedure from Duembgen and Walther (2008)
mod1 <- modeHunting(X.raw, lower = 0, upper = 1, cv.all, min.int = TRUE)

## procedure from Rufibach and Walther (2010) based on I_app
mod2 <- modeHuntingApprox(X.raw, lower = 0, upper = 1, 
                          crit.vals = cv.approx, min.int = TRUE)

## block procedure from Rufibach and Walther (2010)
mod3 <- modeHuntingBlock(X.raw, lower = 0, upper = 1, 
                         crit.vals = cv.block, min.int = TRUE)

## display
mod1; mod2; mod3