Description Usage Arguments Details Value Author(s) References See Also Examples

Sampling from a given distribution, we estimate via Monte Carlo the limiting distribution of 2-log-likelihood-ratio of the modally-constrained log-concave MLE to the (unconstrained) log-concave MLE.

1 2 |

`rdist` |
A function taking an integer argument |

`mode` |
fixed/known location of mode for constrained estimator. |

`N.MC` |
Number of Monte Carlo simulations to do for the limiting distribution. |

`n.SS` |
Sample Size used for each Monte Carlo. (Each MC simulates |

`xgrid` |
Governs the generation of weights for observations. |

`prec` |
Precision variable |

`seedVal` |
An optional seed value |

`debugging` |
Turns off/on debugging. Any non-character value turns debugging off.
If debugging is a character string, then this string gives the name
of an output file to which |

Computes an estimate of the asymptotic distribution of the likelihood
ratio statistic *2 (\mbox{log} \hat{f}_n - \mbox{log}
\hat{f}_n^0) * under the assumption that the true log-concave density *f_0*
satisfies *f_0''(m)<0* where *m* is the true mode of
*f_0*. The estimate is computed based on a sample of size
`n.SS`

from `rdist`

via `N.MC`

Monte Carlo iterations.

Note: the object `LCTLLRdistn`

was created by output from
this function with `n.SS`

set to 1.2e3 and `N.MC`

set to 1e4.
Thus, `estimateLRdistn`

is _NOT_ needed to simply compute fairly
accurate quantiles of the limit distribution of the likelihood ratio
statistic. `estimateLRdistn`

is more useful for research
purposes. For instance, by passing to `mode`

values that are not
the true mode of `myr`

, the statistic can be studied under the
alternative hypothesis.

A `list(LRs,TLLRs)`

, i.e., "likelihood ratio" and "two log
likelihood ratios". Both are numeric vectors of length `N.MC`

.

Note that theoretically all elements of `LRs`

should be
nonnegative, but in practice some rounding errors can occur when
`n.SS`

is very large.

Charles Doss cdoss@stat.washington.edu,

http://www.stat.washington.edu/people/cdoss/

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 `activeSetLogCon`

and `activeSetLogCon.mode`

,
which compute the unconstrained and constrained MLEs, which form the
likelihood ratio. The object `LCTLLRdistn`

was created by output
from this function.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 | ```
myseed <- 561
{if(require(distr)){
mydistn <- Norm() ##demonstrate use of distr package
myr <- mydistn@r
}
else {
myr <- rnorm
}}
hypothesis.mode <- 0
N.MC <- 100 ## should increase these values for better estimate
n.SS <- 50
LRres <- estimateLRdistn(rdist=myr, mode=hypothesis.mode, N.MC=N.MC, prec=10^-10,
n.SS=n.SS, seedVal=myseed,
debugging=FALSE)
TLLRs <- sort(LRres$TLLRs) ##sort is unnecessary, just for examining data
negIdcs <- TLLRs<=0; ## rounding errors
Nneg <- sum(negIdcs)
print(Nneg)
TLLRs[negIdcs] <- 0
cdf.empirical.f <- ecdf(TLLRs)
xlims <- c(min(TLLRs), max(TLLRs))
xpts <- seq(from=xlims[1], to=xlims[2], by=.001)
plot(xpts, cdf.empirical.f(xpts), type="l",
xlab="TLLRs", ylab="Probability")
#### LCTLLRdistn used 1e4 Monte Carlos with 1.2e3 samples each Monte
####Carlo.
##lines(xpts, LCTLLRdistn@p(xpts), col="blue") ## "object
##'C_R_approxfun' not found" error on winbuilder
``` |

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