Find the Wilks Confidence Interval from the Given (empirical) Likelihood Ratio Function

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Description

This program uses simple search to find the upper and lower (Wilks) confidence limits based on the -2 log likelihood ratio, which the required input fun is supposed to supply.

Basically, starting from MLE, we search on both directions, 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.

For examples of using this function to find confidence interval, see the pdf vignettes file.

Usage

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findUL (step = 0.01, initStep =0, fun, MLE, level = 3.84, ...)

Arguments

step

a positive number. The starting step size of the search. Reasonable value should be about 1/5 of the SD of MLE.

initStep

a nonnegative number. The first step size of the search. Sometimes, you may want to put a larger innitStep to speed the search.

fun

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.

MLE

The MLE of the parameter. No need to be exact, as long as it is inside the confidence interval.

level

an optional positive number, controls the confidence level. Default to 3.84 = chisq(0.95, df=1). Change to 2.70=chisq(0.90, df=1) to get a 90% confidence interval.

...

additional arguments, if any, to pass to fun.

Details

Basically we repeatedly testing the value of the parameter, until we find those which the -2 log likelihood value is equal to 3.84 (or other level, if set differently).

If there is no value exactly equal to 3.84, we stop at the value which result a -2 log likelihood just below 3.84. (as in the discrete case, like quantiles.)

Value

A list with the following components:

Low

the lower limit of the confidence interval.

Up

the upper limit of the confidence interval.

FstepL

the final step size when search lower limit. An indication of the precision.

FstepU

Ditto. An indication of the precision of the upper limit.

Lvalue

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.

Uvalue

Ditto. Should be approximately equa to level.

Author(s)

Mai Zhou

References

Zhou, M. (2016). Empirical Likelihood Method in Survival Analysis. CRC Press.

Zhou, M. (2002). Computing censored empirical likelihood ratio by EM algorithm. JCGS

Examples

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## 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)
}
findUL(step=0.2, fun=myfun6, MLE=4.0659, x=x, d=d)

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