MGBT (version 1.0.7)

EMS: Expected values of M and S

Description

Compute expected values of $M$ and $S$ given $q_{\min }$ and define the quantity $$z_r={\mathrm{\Phi }}^{(1)}(q_{\min })\text{,}$$ where ${\mathrm{\Phi }}^{(1)}(\cdot )$ is the inverse of the standard normal distribution. As result, $q_{\min }$ is itself a probability because it is an argument to the qnorm() function. The expected value $M$ is defined as $$M=\mathrm{\Psi }(z_r,1)\text{,}$$ where $\mathrm{\Psi }(a,b)$ is the gtmoms function. The $S$ requires the conditional moments of the Chi-square (CondMomsChi2) defined as the two value vector ${}_{2}S$ that provides the values $\alpha ={}_2{S}_1^2/{}_2{S}_2$ and $\beta ={}_2{S}_2/{}_2{S}_1$. The $S$ is then defined by $$S=\sqrt{\beta }(\frac{\mathrm{\Gamma }(\alpha +0.5)}{\mathrm{\Gamma }(\alpha )})\text{.}$$

Usage

EMS(n, r, qmin)

Arguments

n

The number of observations;

r

The number of truncated observations? (confirm); and

qmin

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

Value

The expected values of $M$ and $S$ in the form of an R vector.

References

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

See Also

CondMomsChi2, EMS, VMS, V, gtmoms

Examples

# NOT RUN {
EMS(58,2,.5)
#[1] 0.7978846 0.5989138

#  Monte Carlo experiment to test EMS and VMS functions
"test_EMS" <- function(nrep=1000, n=100, r=0, qr=0.2, ss=1) { # TAC named function
   set.seed(ss)
   Moms <- replicate(n=nrep, {
          x <- qnorm(runif(n-r,min=qr,max=1));
          c(mean(x),var(x))}); xsi <- qnorm(qr);
          list(
    MeanMS_obs = c(mean(Moms[1,]), mean(sqrt(Moms[2,])), mean(Moms[2,])),
    EMS        = c(EMS(n,r,qr), gtmoms(xsi,2) - gtmoms(xsi,1)^2),
    CovMS2_obs = cov(t(Moms)),
    VMS2       = V(n,r,qr),
    VMS_obs    = array(c(var(     Moms[1,]),
                         rep(cov( Moms[1,], sqrt(Moms[2,])),2),
                         var(sqrt(Moms[2,]))),    dim=c(2,2)),
    VMS        = VMS(n,r,qr)  )
}
test_EMS()
# }