BLlo {MGBT}

Barnett and Lewis Test Adjusted for Low Outliers

Description

The Barnett and Lewis (1995, p. 224; T_{\mathrm{N}3}) so-labeled “N3 method” with TAC adjustment to look for low outliers. 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

BL_r = T_{\mathrm{N}3} = \frac{ \sum_{i=1}^r x_{[i:n]} - r \times \mathrm{mean}\{x_{[1:n]}\} } {\sqrt{\mathrm{var}\{x_{[1:n]}\}}}\mbox{,}

for the mean and variance of the observations. Barnett and Lewis (1995, p. 218) brand this statistic as a test of the “k \ge 2 upper outliers” but for the MGBT package “lower” applies in TAC reformulation. Barnett and Lewis (1995, p. 218) show an example of a modification for two low outliers as (2\overline{x} - x_{[2:n]} - x_{[1:n]})/s for the mean \mu and standard deviation s. TAC reformulation thus differs by a sign. The BL_r is a sum of internally studentized deviations from the mean:

SP(t) \le {n \choose k} P\biggl(\bm{t}(n-2) > \biggr[\frac{n(n-2)t^2}{r(n-r)(n-1)-nt^2}\biggl]^{1/2}\biggr)\mbox{,}

where \bm{t}(df) is the t-distribution for df degrees of freedom, and this is an inequality when

t \ge \sqrt{r^2(n-1)(n-r-1)/(nr+n)}\mbox{,}

where SP(t) is the probability that T_{\mathrm{N}3} > t when the inequality holds. For reference, Barnett and Lewis (1995, p. 491) example tables of critical values for n=10 for k \in 2,3,4 at 5-percent significant level are 3.18, 3.82, and 4.17, respectively. One of these is evaluated in the Examples.

Usage

BLlo(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.

Value

The value for BL_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 RSlo were designed similarly.

Author(s)

W.H. Asquith consulting T.A. Cohn sources

Source

LowOutliers_jfe(R).txt and LowOutliers_wha(R).txt—Named BL_N3

References

Barnett, Vic, and Lewis, Toby, 1995, Outliers in statistical data: Chichester, John Wiley and Sons, ISBN~0–471–93094–6.

Cohn, T.A., 2013–2016, Personal communication of original R source code: U.S. Geological Survey, Reston, Va.

See Also

MGBTcohn2011, RSlo

Examples

# See Examples under RSlo()

 # WHA experiments with BL_r()
n <- 10; r <- 3; nsim <- 10000; alpha <- 0.05; Tcrit <- 3.82
BLs <- Ho <- RHS <- SPt <- rep(NA, nsim)
EQ <- sqrt(r^2*(n-1)*(n-r-1)/(n*r+n))
for(i in 1:nsim) { # some simulation results shown below
   BLs[i] <- abs(BLlo(rnorm(n), r)) # abs() correcting TAC sign convention
   t  <- sqrt( (n*(n-2)*BLs[i]^2) / (r*(n-r)*(n-1)-n*BLs[i]^2) )
   RHS[i] <- choose(n,r)*pt(t, n-2, lower.tail=FALSE)
   ifelse(t >= EQ, SPt[i] <- RHS[i], SPt[i] <- 1) # set SP(t) to unity?
   Ho[i]  <- BLs[i] > Tcrit
}
results <- c(quantile(BLs, prob=1-alpha), sum(Ho /nsim), sum(SPt < alpha)/nsim)
names(results) <- c("Critical_value", "Ho_rejected", "Coverage_SP(t)")
print(results) # minor differences are because of random number seeding
# Critical_value    Ho_rejected Coverage_SP(t)
#      3.817236       0.048200       0.050100