MGBT (version 1.0.4)

V: Covariance matrix of M and S-squared

Description

Compute the covariance matrix of $M$ and $S^2$ (S-squared) given $q_{\min }$. Define the vector of four moment expectations $$E_{i\in 1,2,3,4}=\mathrm{\Psi }({\mathrm{\Phi }}^{(-1)}(q_{\min }),i)\text{,}$$ where $\mathrm{\Psi }(a,b)$ is the gtmoms function and ${\mathrm{\Phi }}^{(-1)}$ is the inverse of the standard normal distribution. Using these $E$, define a vector $C_{i\in 1,2,3,4}$ as a system of nonlinear combinations: $$C_1=E_1\text{,}$$ $$C_2=E_2-E_1^2\text{,}$$ $$C_3=E_3-3E_2{E}_1+2E_1^3\text{, and}$$ $$C_4=E_4-4E_3{E}_1+6E_2{E}_1^2-3E_1^4\text{.}$$ Given $k=n-r$ from the arguments of this function, compute the symmetrical covariance matrix $COV$ with variance of $M$ as $$CO{V}_{1,1}=C_2/k\text{,}$$ the covariance between $M$ and $S^2$ as $$CO{V}_{1,2}=CO{V}_{2,1}=\frac{C_3}{\sqrt{k(k-1)}}\text{, and}$$ the variance of $S^2$ as $$CO{V}_{2,2}=\frac{C_4-C_2^2}{k}+\frac{2C_2^2}{k(k-1)}\text{.}$$

Usage

V(n, r, qmin)

Arguments

n

The number of observations;

r

The number of truncated observations; and

qmin

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

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, VMS, gtmoms

Examples

# NOT RUN {
V(58,2,.5)
#            [,1]        [,2]
#[1,] 0.006488933 0.003928333
#[2,] 0.003928333 0.006851120
# }