gtmoms: Moments of Observations Above the Threshold
In wasquith-usgs/MGBT: Multiple Grubbs-Beck Low-Outlier Test

Description Usage Arguments Value Note Author(s) Source References See Also Examples

Moments of observations above the threshold (xsi, x_{si}), which has been standardized to a zero mean and unit standard deviation. Define the standard normal hazard function as

H(x) = φ(x) / (1 - Φ(x))\mbox{,}

where φ(x) is the standard normal density function and Φ(x) is the standard normal distribution (cumulative) function. For a truncation index, r, define the recursion formula, Ψ for gtmoms as

Ψ(x_{si}, r) = (r-1)Ψ(x_{si}, r-2) + x_{si}^{r-1}H(x_{si})\mbox{,}

for which Ψ(x_{si}, 0) = 1 and Ψ(x_{si}, 1) = H(x_{si}).

1	gtmoms(xsi, r)

`xsi`	The lower threshold; and
`r`	The number of truncated observations.

The moments.

AUTHOR TODO—Note that is it not clear in TAC documentation that X_{si} is a scalar or vector quantity, and gtmoms is automatically vectoral in the R idioms if X_{si} is. Also it is not immediately clear X_{si} is or is not one of the order statistics. Based on MGBT operation in USGS-PeakFQ output (USGS, 2014), the threshold is “known” no better in accuracy than one of the sample order statistics, so X_{si} might be written x_{[r:n]}. But this answer could be only restricted to a implementation in software and perhaps not theory. Finally, although the computations involve the standard normal distribution, the standardization form of X_{si} is not yet confirmed during the WHA porting process.

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

LowOutliers_jfe(R).txt, LowOutliers_wha(R).txt, P3_089(R).txt—Named gtmoms

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

U.S. Geological Survey (USGS), 2018, PeakFQ—Flood frequency analysis based on Bulletin 17B and recommendations of the Advisory Committee on Water Information (ACWI) Subcommittee on Hydrology (SOH) Hydrologic Frequency Analysis Work Group (HFAWG), version 7.2: Accessed November 29, 2018, at https://water.usgs.gov/software/PeakFQ/.

CondMomsChi2, gtmoms