Description Usage Format Details Source References See Also Examples

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*.

1 |

`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

`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.

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.

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
``` |

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