logcondens.mode (version 1.0.1)

LCLRCImode: Compute Log-Concave Likelihood-Ratio Confidence interval for the mode.

Description

Compute the confidence interval (CI) for the mode of a log-concave density by "inverting" the likelihood ratio statistic, i.e. the $1-\alpha$ CI is composed of mode values at which the likelihood ratio test does not reject at the $\alpha$-level.

Usage

LCLRCImode(x, xgrid = NULL, w = NA, nn = length(x), alpha = 0.05, prec =
1e-10, CIprec = 1e-04, print = F)

Arguments

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 coverage probability for the confidence interval (i.e., the level for the corresponding test).

prec

Numeric value, giving the precision passed to activeSetLogCon and to activeSetLogCon.mode.

CIprec

Numeric value giving precision for the endpoints of the confidence interval.

print

TRUE or FALSE, depending on whether debugging information should be printed or not, respectively.

Value

Returns a numeric vector of length 2, giving the asymptotic confidence interval for the mode location.

Details

The confidence set is given by the values of the mode that LRmodeTest does not reject. See the details of that function.

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.

See Also

See also LRmodeTest for the corresponding test.

Examples

nn <- 200
myxx <- rnorm(nn) ## no need to sort

LCLRCImode(x=myxx,
           xgrid=NULL,
           w=NA,
           ##nn=nn, 
           alpha=0.05,
           CIprec=1e-04,
           print=FALSE)

LCLRCImode(x=myxx,
           xgrid=.05,
           w=NA,
           ##nn=nn, 
           alpha=0.05,
           CIprec=1e-04,
           print=FALSE)