# LCLRCImode: Compute Log-Concave Likelihood-Ratio Confidence interval for... In logcondens.mode: Compute MLE of Log-Concave Density on R with Fixed Mode, and Perform Inference 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-α CI is composed of mode values at which the likelihood ratio test does not reject at the α-level.

## Usage

 ```1 2``` ```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 `preProcess`ing 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.

## Details

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

## Value

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

## 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 `LRmodeTest` for the corresponding test.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```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) ```