VMS: Covariance matrix of M and S
In wasquith-usgs/MGBT: Multiple Grubbs-Beck Low-Outlier Test

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

Compute the covariance matrix of M and S given q_\mathrm{min}. Define the vector of four moment expectations

E_{i\in 1,2} = Ψ\bigl(Φ^{(-1)}(q_\mathrm{min}), i\bigr)\mbox{,}

where Ψ(a,b) is the gtmoms function and Φ^{(-1)} is the inverse of the standard normal distribution. Define the scalar quantity Es = EMS(n,r,qmin)[2] as the expectation of S using the EMS function, and define the scalar quantity E_s^2 = E_2 - E_1^2 as the expectation of S^2. Finally, compute the covariance matrix COV of M and S using the V function:

COV_{1,1} = V_{1,1}\mbox{,}

COV_{1,2} = COV_{2,1} = V_{1,2}/2Es\mbox{,}

COV_{2,2} = E_s^2 - (E_s)^2\mbox{.}

1	VMS(n, r, qmin)

`n`	The number of observations;
`r`	The number of truncated observations; and
`qmin`	A nonexceedance probability threshold for X > q_\mathrm{min}.

A 2-by-2 covariance matrix.

Because the gtmoms function is naturally vectorized and TAC sources provide no protection if qmin is a vector (see Note under EMS). For the implementation here, only the first value in qmin is used and a warning issued if it is a vector.

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

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

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

EMS, V, gtmoms

VMS(58,2,.5) # Note that [1,1] is the same as [1,1] for Examples under V().
#            [,1]        [,2]
#[1,] 0.006488933 0.003279548
#[2,] 0.003279548 0.004682506