This function is similar to findUL2 but here the seeking of lower and upper bound are
separate (the other half is findUnew).
The reason for this is: sometime we need to supply the fun
with different nuisance parameter(s) values when seeking Lower or Upper bound.
For example fun returns the -2LLR for a given parameter of interest,
but there are additional nuisance parameter need to be profiled out, and we
need to give a range of the nuisance parameter to be max/min over. This range can be
very different for parameter near Upper bound vs near Lower bound. In the findUL2,
you have to supply a range really wide that (hopefully) works for both Upper and Lower
bound. Here, with separate findLnew and findUnew we can tailor the range
for one end of the confidence interval.
Those nuisance parameter(s) are supplied via the ... input of this function.
Another improvement is that we used the "extendInt" option of the uniroot.
So now we can and did used a smaller default step input, compare to findUL2.
This program uses uniroot( ) to find (only) the lower (Wilks) confidence
limit based on the -2 log likelihood ratio, which the required
input fun is supposed to supply.
Basically, starting from MLE, we search on lower
direction, by step away
from MLE, until we find values that have -2LLR = level.
(the value of -2LLR at MLE is supposed to be zero.)
At curruent implimentation, only handles one dimesional parameter, i.e. only confidence intervals, not confidence regions.
findLnew(step=0.003, initStep=0, fun, MLE, level=3.84146, tol=.Machine$double.eps^0.5,...)A list with the following components:
the lower limit of the confidence interval.
the final step size when search lower limit. An indication of the precision.
The -2LLR value of the final Low value. Should be approximately equal to level. If larger than level,
than the confidence interval limit Low is wrong.
a positive number. The starting step size of the search. Reasonable value should be about 1/5 of the SD of MLE.
a nonnegative number. The first step size of the search. Sometimes, you may want to put a larger innitStep to speed the search.
a function that returns a list. One of the item in the list should be "-2LLR", which is the -2 log (empirical) likelihood ratio.
The first input of fun must be the parameter for which we are seeking the confidence interval. (The MLE or NPMLE of this parameter should be supplied as in the input MLE). The rest of the input to fun are typically the data.
If the first input of fun is set to MLE, then the returned -2LLR should be 0.
The MLE of the parameter. No need to be exact, as long as it is inside the confidence interval.
an optional positive number, controls the confidence level. Default to 3.84146 = chisq(0.95, df=1). Change to 2.70=chisq(0.90, df=1) to get a 90% confidence interval.
Error bound of the final result.
additional arguments, if any, to pass to fun.
Mai Zhou
Basically we repeatedly testing the value of the parameter, until we find those which the -2 log likelihood value is equal to 3.84146 (or other level, if set differently).
Zhou, M. (2016). Empirical Likelihood Method in Survival Analysis. CRC Press.
## example with tied observations. Kaplan-Meier mean=4.0659.
## For more examples see vignettes.
x <- c(1, 1.5, 2, 3, 4, 5, 6, 5, 4, 1, 2, 4.5)
d <- c(1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1)
myfun6 <- function(theta, x, d) {
el.cen.EM2(x, d, fun=function(t){t}, mu=theta)
}
findLnew(step=0.1, fun=myfun6, MLE=4.0659, x=x, d=d)
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