emplikAUC (version 0.5)

quantONE: Smoothed quantile estimation from the given x-sample.

Description

This function computes the smoothed quantile estimate, using the smoothing function smooth3. (or it can be changed easily) Otherwise it is similar to R function quantile(). Compare to quantile(x, prob, type=9).

\(myeps\) is a scalar, must > 0. It is the smoothing window width.

Usage

quantONE(x, prob, myeps=(length(x))^(-0.75))

Value

The function quantONE returns a scalar that is the estimated (prob)-th quantile of X distribution.

Arguments

x

a vector of observations, length m, the sample.

prob

a probabilty.

myeps

The smoothing window width, must >0.

Author

Mai Zhou <[e-mail]>.

Details

This function is called by myEstPaucT. It is listed here because the user may find it useful elsewhere.

Known problems: when the input prob is too close to 0 or 1 (distance less than 0.03), the computation will stop. Also, if the solution to the equation $$ \tau ~| ~~~ \frac{1}{m} \sum_{i=1}^m smooth3vec(X_i, \tau, myeps) = prob $$ is not unique, this function only returns one of the solutions.

References

Zhao, Y., Ding, X. and Zhou (2021). Confidence Intervals of AUC and pAUC by Empirical Likelihood. Tech Report. https://www.ms.uky.edu/~mai/research/eAUC1.pdf

Examples

y <- c(10, 209, 273, 279, 324, 391, 566, 785)
x <- c(21, 38, 39, 51, 77, 185, 240, 289, 524)
#### To estimate the 70-th percentile of x distrubution:
quantONE(x=x, prob=0.7)
#### we should get 239.9474.