# LRmodeTest: Computes an Asymptotic Confidence Interval for the mode of a... In logcondens.mode: Compute MLE of Log-Concave Density on R with Fixed Mode, and Perform Inference for the Mode.

## Description

A likelihood ratio test to test whether 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.

## Usage

 1 2 LRmodeTest(mode, x, xgrid = NULL, w, nn = length(x), alpha, prec=1e-10, print=FALSE) 

## Arguments

 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 log-concave density) or, such data binned. If x is binned, there should be corresponding weights w. Binning is usually handled by passing in a non-NULL value for xgrid. xgrid Governs binning of x and generation of corresponding weights w. See logcondens::preProcess. If w is not NA then xgrid should be NULL. w Numeric vector of length 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. nn 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. alpha Numeric value in [0,1], the level for the test. prec Numeric value, giving the precision passed to activeSetLogCon and activeSetLogCon.mode. print TRUE or FALSE, depending on whether debugging information should be printed or not, respectively.

## Details

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. 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.

## Value

Returns TRUE or FALSE for not reject or to reject mode, respectively.

## References

Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave 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 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.
 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)