MGBT (version 1.0.7)

VMS: Covariance matrix of M and S

Description

Compute the covariance matrix of $M$ and $S$ given $q_{\min }$. Define the vector of four moment expectations $$E_{i\in 1,2}=\mathrm{\Psi }({\mathrm{\Phi }}^{(-1)}(q_{\min }),i)\text{,}$$ where $\mathrm{\Psi }(a,b)$ is the gtmoms function and ${\mathrm{\Phi }}^{(-1)}$ 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: $$CO{V}_{1,1}=V_{1,1}\text{,}$$ $$CO{V}_{1,2}=CO{V}_{2,1}=V_{1,2}/2Es\text{,}$$ $$CO{V}_{2,2}=E_s^2-(E_s{)}^2\text{.}$$

Usage

VMS(n, r, qmin)

Arguments

n

The number of observations;

r

The number of truncated observations; and

qmin

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

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
# }