# Rosner RST Test Adjusted for Low Outliers The Rosner (1975) method or the essence of the method, given the order statistics $x_{[1:n]} \le x_{[2:n]} \le \cdots \le x_{[(n-1):n]} \le x_{[n:n]}$, is the statistic: [REMOVE_ME]$$ RS_r =\frac{ x_{[r:n]} - \mathrm{mean}\{x_{[(r+1)\rightarrow(n-r):n]}\} }{\sqrt{\mathrm{var}\{x_{[(r+1)\rightarrow(n-r):n]}\}}}\mbox{,} [REMOVE_ME_2]$$ utf8 ## Description The Rosner (1975) method or the essence of the method, given the order statistics $x_{[1:n]} \le x_{[2:n]} \le \cdots \le x_{[(n-1):n]} \le x_{[n:n]}$, is the statistic: $$ RS_r =\frac{ x_{[r:n]} - \mathrm{mean}\{x_{[(r+1)\rightarrow(n-r):n]}\} }{\sqrt{\mathrm{var}\{x_{[(r+1)\rightarrow(n-r):n]}\}}}\mbox{,} $$ ```r RSlo(x, r, n=length(x)) ``` ## Arguments - `x`: The data values and note that base-10 logarithms of these are not computed internally; - `r`: The number of truncated observations; and - `n`: The number of observations. ## Returns The value for $RS_r$. ## Note Regarding `n=length(x)`, it is not clear that TAC intended `n` to be not equal to the sample size. TAC chose to not determine the length of `x` internally to the function but to have it available as an argument. Also `MGBTcohn2011` and `BLlo` were similarly designed. ## Source `LowOutliers_jfe(R).txt` and `LowOutliers_wha(R).txt`---Named `RST` ## References Cohn, T.A., 2013--2016, Personal communication of original R source code: U.S. Geological Survey, Reston, Va. Rosner, Bernard, 1975, On the detection of many outliers: Technometrics, v. 17, no. 2, pp. 221--227. ## Author(s) W.H. Asquith consulting T.A. Cohn sources ## See Also `MGBTcohn2011`, `BLlo` ## Examples ```r # Long CPU time, arguments slightly modified to run faster and TAC had. # TAC has maxr=n/2 (the lower tail) but WHA has changed to maxr=3 for # speed to get the function usable during automated tests. testMGBvsN3 <- function(n=100, maxr=3, nrep=10000) { # TAC named function for(r in 1:maxr) { set.seed(123457) res1 <- replicate(nrep, { x <- sort(rnorm(n)) c(MGBTcohn2011(x,r),BLlo(x,r),RSlo(x,r)) }) r1 <- rank(res1[1,]); r2 <- rank(res1[2,]); r3 <- rank(res1[3,]) v <- quantile(r1,.1); h <- quantile(r2,.1) plot(r1,r2) abline(v=v, col=2); abline(h=h, col=2) message(' BLlo ',r, " ", cor(r1,r2), " ", mean((r1 <= v) & (r2 <= h))) v <- quantile(r1,.1); h <- quantile(r2,.1) plot(r1,r3) abline(v=v, col=2); abline(h=h, col=2) mtext(paste0("Order statistic iteration =",r," of ",maxr)) message('RSTlo ',r, " ", cor(r1,r3), " ", mean((r1 <= v) & (r3 <= h))) } } testMGBvsN3() # ```

RSlo function