EMS {MGBT}

Expected values of M and S

Description

Compute expected values of M and S given q_\mathrm{min} and define the quantity

z_r = \Phi^{(1)}(q_\mathrm{min})\mbox{,}

where \Phi^{(1)}(\cdot) is the inverse of the standard normal distribution. As result, q_\mathrm{min} is itself a probability because it is an argument to the qnorm() function. The expected value M is defined as

M = \Psi(z_r, 1)\mbox{,}

where \Psi(a,b) is the gtmoms function. The S requires the conditional moments of the Chi-square (CondMomsChi2) defined as the two value vector \mbox{}_2S that provides the values \alpha = \mbox{}_2S_1^2 / \mbox{}_2S_2 and \beta = \mbox{}_2S_2 / \mbox{}_2S_1. The S is then defined by

S = \sqrt{\beta}\biggl(\frac{\Gamma(\alpha+0.5)}{\Gamma(\alpha)}\biggr)\mbox{.}

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_\mathrm{min}.

Value

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

Note

TAC sources call on the explicit first index of M as literally “Em[1]” for the returned vector, which seems unnecessary. This is a potential weak point in design because the gtmoms function is naturally vectorized and could potentially produce a vector of M values. For the implementation here, only the first value in qmin is used and a warning otherwise issued. Such coding prevents the return value from EMS accidentally acquiring a length greater than two. For at least samples of size n=2, overranging in a call to lgamma(alpha) happens for alpha=0. A suppressWarnings() is wrapped around the applicable line. The resulting NaN cascades up the chain, which will end up inside peta, but therein SigmaMp is not finite and a p-value of unity is returned.

Author(s)

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

Source

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

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

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()