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.

1 |

`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 |

`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 |

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.)

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 |

`Uvalue` |
Ditto. Should be approximately equa to level. |

Mai Zhou

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

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

1 2 3 4 5 6 7 8 | ```
## 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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