MGBT (version 1.0.4)

VMS: Covariance matrix of M and S

Description

Compute the covariance matrix of \(M\) and \(S\) given \(q_\mathrm{min}\). Define the vector of four moment expectations $$E_{i\in 1,2} = \Psi\bigl(\Phi^{(-1)}(q_\mathrm{min}), i\bigr)\mbox{,}$$ where \(\Psi(a,b)\) is the gtmoms function and \(\Phi^{(-1)}\) is the inverse of the standard normal distribution. 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: $$COV_{1,1} = V_{1,1}\mbox{,}$$ $$COV_{1,2} = COV_{2,1} = V_{1,2}/2Es\mbox{,}$$ $$COV_{2,2} = E_s^2 - (E_s)^2\mbox{.}$$

Usage

VMS(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.

References

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

See Also

EMS, V, gtmoms

Examples

# NOT RUN {
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
# }