Description Usage Arguments Details Value Author(s) References See Also Examples
A likelihood ratio test to test whether mode
is the location of
the mode of a (logconcave) density. Uses
activeSetLogCon
and activeSetLogCon.mode
to compute the logconcave MLE and the logconcave MLE where the mode
is restricted to be mode
, respectively.
1 2 
mode 
Numeric value giving the constrained value of the mode location. 
x 
Points at which to compute the unconstrained and constrained
estimators. Either iid data observations (from a logconcave
density) or, such data binned. If 
xgrid 
Governs binning of 
w 
Numeric vector of length 
nn 
The number of data points initially observed. Numeric of length
1. Usually equal to 
alpha 
Numeric value in 
prec 
Numeric value, giving the precision passed to 
print 

Uses activeSetLogCon
and
activeSetLogCon.mode
to compute the logconcave MLE
\hat{f}_n and the logconcave 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 logconcave density f_0
is twice differentiable
at its true mode m
, and f_0
satisfies
f_0''(m)<0
. 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 logconcave
density).
Using the pivotal nature of TLLR, its limit distribution can be
simulated from any given known logconcave 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.
Returns TRUE
or FALSE
for not reject or to reject
mode
, respectively.
Charles R. Doss, cdoss@stat.washington.edu,
http://www.stat.washington.edu/people/cdoss/
Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for logconcave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at http://arxiv.org/abs/0707.4643.
Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a logconcave 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 LogConcave Density Estimation. Journal of Statistical Software, 39(6), 1–28. http://www.jstatsoft.org/v39/i06
Doss, C. R. (2013). ShapeConstrained Inference for ConcaveTransformed 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 logconcave density. Technical Report, University of Washington, in preparation.
LCLRCImode
uses LRmodeTest
to compute asymptotic confidence sets.
1 2 3 4 5 6 7 8 9  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)

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