V {MGBT}

Covariance matrix of M and S-squared

Description

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} = \Psi\bigl(\Phi^{(-1)}(q_\mathrm{min}), i\bigr)\mbox{,}

where \Psi(a,b) is the gtmoms function and \Phi^{(-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}{\sqrt{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{.}

Usage

V(n, r, qmin)

Arguments

n

The number of observations;

r

The number of truncated observations; and

qmin

A nonexceedance probability threshold for X > q_\mathrm{min}.

Value

A 2-by-2 covariance matrix.

Note

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.

Author(s)

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

Source

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

References

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

See Also

EMS, VMS, gtmoms

Examples

V(58,2,.5)
#            [,1]        [,2]
#[1,] 0.006488933 0.003928333
#[2,] 0.003928333 0.006851120