Description Usage Arguments Value Note Author(s) Source References See Also Examples
Compute the covariance matrix of M and S^2 (S-squared) given q_\mathrm{min}. Define the vector of four moment expectations
E_{i\in 1,2,3,4} = 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. Using these E, define a vector C_{i\in 1,2,3,4} as a system of nonlinear combinations:
C_1 = E_1\mbox{,}
C_2 = E_2 - E_1^2\mbox{,}
C_3 = E_3 - 3E_2E_1 + 2E_1^3\mbox{, and}
C_4 = E_4 - 4E_3E_1 + 6E_2E_1^2 - 3E_1^4\mbox{.}
Given k = n - r from the arguments of this function, compute the symmetrical covariance matrix COV with variance of M as
COV_{1,1} = C_2/k\mbox{,}
the covariance between M and S^2 as
COV_{1,2} = COV_{2,1} = \frac{C_3}{√{k(k-1)}}\mbox{, and}
the variance of S^2 as
COV_{2,2} = \frac{C_4 - C_2^2}{k} + \frac{2C_2^2}{k(k-1)}\mbox{.}
1 | V(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 V
Cohn, T.A., 2013–2016, Personal communication of original R source code: U.S. Geological Survey, Reston, Va.
1 2 3 4 | V(58,2,.5)
# [,1] [,2]
#[1,] 0.006488933 0.003928333
#[2,] 0.003928333 0.006851120
|
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