# LCTLLRdistn: Limit Distribution of the Likelihood Ratio Statistic In logcondens.mode: Compute MLE of Log-Concave Density on R with Fixed Mode, and Perform Inference for the Mode.

## Description

The `LCTLLRdistn` object gives the (estimated) limit distribution of Two times the log likelihood ratio for the location of the mode of a log-concave density f_0, under the assumption that f_0''(m)<0, where m is the mode of f_0.

## Usage

 `1` ```LCTLLRdistn ```

## Format

`LCTLLRdistn` is an object with formal (S4) class 'distr' and subclass 'DiscreteDistribution' [package "distr"] with 12 slots. It is an estimate of a continuous limit distribution by a discrete one.

@support

Gives the (discrete) support, i.e., the simulated values on which the estimate is based.

@img

Formal class 'Reals' [package "distr"] with 2 slots

@dimension

1

@name

"Real Space"

@param

NULL; unused slot.

@r

function (n); simulates `n` values.

@d

function (x, log = FALSE); constant 0 function.

@p

function (q, lower.tail = TRUE, log.p = FALSE); the cumulative distribution function.

@q

function (p, lower.tail = TRUE, log.p = FALSE); the quantile function.

@.withSim

logi FALSE; for internal use

@.withArith

logi FALSE; for internal use

@.logExact

logi FALSE; for internal use

@.lowerExact

logi TRUE; for internal use

@Symmetry

Formal class 'NoSymmetry' [package "distr"] with 2 slots

@type

character "non-symmetric distribution"

@SymmCenter

NULL

## Details

`LCTLLRdistn` is an object of class "distr" and subclass "DiscreteDistribution" from the package `distr`. The main uses are the three functions `q` (the quantile function), `p` (the cumulative distribution function) and `r` (which returns random samples). Note that `d` always returns 0 since the distribution is estimated discretely.

See the `distr` package for more details.

## Source

Obtained via simulation from a Gamma(3,1) distribution with density proportional to x^2 e^{-x} on (0,∞). We simulated the log likelihood ratio statistic 10^4 times, each time with a sample size of 1.2*10^3. The statistic was computed via the `activeSetLogCon` and `activeSetLogCon.mode` functions.

## 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 prepa- ration.

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 the "distr" package. The `LRmodeTest` and `LCLRCImode` functions use LCTLLRdistn.
 `1` ```LCTLLRdistn@q(.95); ##~1.06 is the 95% quantile ```