mode is the location of
the mode of a (log-concave) density. Uses
activeSetLogCon and activeSetLogCon.mode
to compute the log-concave MLE and the log-concave MLE where the mode
is restricted to be mode, respectively.
LRmodeTest(mode, x, xgrid = NULL, w, nn = length(x), alpha, prec=1e-10,
print=FALSE)x is binned, there should
be corresponding weights w. Binning is usually handled by
passing in a non-NULL value for xgrid.
x and generation of corresponding weights
w. See logcondens::preProcess.
If w is not NA then xgrid
should be NULL.
length(x) or NA. Weights corresponding to
x. Can be NA (regardless of the value of xgrid)
which indicates the weights are uniform (equal to
1/length(x)). If w is not NA then xgrid
should be NULL. If nn is not equal to length(x)
then w should be given a non-NA value.
If w is not NA, then we assume that x has no
duplicate entries.
The number of data points initially observed. Numeric of length
1. Usually equal to length(x). If some sort of
preProcessing is done in advance, may be not equal to
length(x). To pass in a non-default value for nn
(i.e. something other than length(x)), w must also be
passed in a (numeric vector) value, and xgrid must be
NULL.
[0,1], the level for the test.
activeSetLogCon and
activeSetLogCon.mode.
TRUE or FALSE, depending on whether debugging
information should be printed or not, respectively.
TRUE or FALSE for not reject or to reject
mode, respectively. Uses activeSetLogCon and
activeSetLogCon.mode to compute the log-concave MLE
$\hat{f}_n$ and the log-concave MLE where the mode is restricted
to be mode, $\hat{f}_n^0$. The statistic, Two times the Log
Likelihood Ratio (TLLR) is then defined to be $2( \mbox{log} \hat{f}_n -
\mbox{log} \hat{f}_n^0)$.
Our test is based on the assumption
that the true log-concave density f_0 is twice differentiable
at its true mode m, and f_0 satisfies
f_0''(m)<0< code="">. Under that condition, Doss (2013) conjectures that
the log likelihood ratio statistic is asymptotically pivotal (i.e.,
its limit distribution does not depend on the true log-concave
density).
Using the pivotal nature of TLLR, its limit distribution can be
simulated from any given known log-concave density (e.g., a standard
normal), and the estimated distribution function of this limit is
given by the LCTLLRdistn object. The quantiles of the limit
distribution are used to either reject or not reject the test.
Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log-concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40--68.
Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1--28. http://www.jstatsoft.org/v39/i06
Doss, C. R. (2013). Shape-Constrained Inference for Concave-Transformed Densities and their Modes. PhD thesis, Department of Statistics, University of Washington, in preparation.
Doss, C. R. and Wellner, J. A. (2013). Inference for the mode of a log-concave density. Technical Report, University of Washington, in preparation.
LCLRCImode uses LRmodeTest to compute asymptotic confidence sets.
nn <- 200
myxx <- rnorm(nn) ## no need to sort
## Under null/true hypothesis with or without grid
LRmodeTest(mode=0, x=myxx, xgrid=NULL, alpha=.05)
LRmodeTest(mode=0, x=myxx, xgrid=.05, alpha=.05)
## Under alternative/false hypothesis
LRmodeTest(mode=3, x=myxx, xgrid=NULL, alpha=.05)
Run the code above in your browser using DataLab