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
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.