logcondens.mode (version 1.0.1)

LCTLLRdistn: Limit Distribution of the Likelihood Ratio Statistic

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="" $f_0$.<="" p="">

Usage

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

Source

Obtained via simulation from a Gamma(3,1) distribution with density proportional to $x^2 e^{-x}$ on $(0,\infty)$. 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.

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.

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 Also

See the "distr" package. The LRmodeTest and LCLRCImode functions use LCTLLRdistn.

Examples

LCTLLRdistn@q(.95); ##~1.06 is the 95% quantile