xgrsr.ad {spc}

Compute steady-state ARLs of Shiryaev-Roberts schemes

Description

Computation of the steady-state Average Run Length (ARL) for Shiryaev-Roberts schemes monitoring normal mean.

Usage

xgrsr.ad(k, g, mu1, mu0 = 0, zr = 0, sided = "one", MPT = FALSE, r = 30)

Arguments

k

reference value of the Shiryaev-Roberts scheme.

g

control limit (alarm threshold) of Shiryaev-Roberts scheme.

mu1

out-of-control mean.

mu0

in-control mean.

zr

reflection border to enable the numerical algorithms used here.

sided

distinguishes between one- and two-sided schemes by choosing "one" and"two", respectively. Currently only one-sided schemes are implemented.

MPT

switch between the old implementation (FALSE) and the new one (TRUE) that considers the completed likelihood ratio. MPT contains the initials of G. Moustakides, A. Polunchenko and A. Tartakovsky.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1.

Details

xgrsr.ad determines the steady-state Average Run Length (ARL) by numerically solving the related ARL integral equation by means of the Nystroem method based on Gauss-Legendre quadrature.

Value

Returns a single value which resembles the steady-state ARL.

Author(s)

Sven Knoth

References

S. Knoth (2006), The art of evaluating monitoring schemes – how to measure the performance of control charts? S. Lenz, H. & Wilrich, P. (ed.), Frontiers in Statistical Quality Control 8, Physica Verlag, Heidelberg, Germany, 74-99.

G. Moustakides, A. Polunchenko, A. Tartakovsky (2009), Numerical comparison of CUSUM and Shiryaev-Roberts procedures for detectin changes in distributions, Communications in Statistics: Theory and Methods 38, 3225-3239.

See Also

xewma.arl and xcusum-arl for zero-state ARL computation of EWMA and CUSUM control charts, respectively, and xgrsr.arl for the zero-state ARL.

Examples

## Small study to identify appropriate reflection border to mimic unreflected schemes
k <- .5
g <- log(390)
zrs <- -(0:10)
ZRxgrsr.ad <- Vectorize(xgrsr.ad, "zr")
ads <- ZRxgrsr.ad(k, g, 0, zr=zrs)
data.frame(zrs, ads)

## Table 2 from Knoth (2006)
## original values are
#  mu   arl
#  0    689
#  0.5  30
#  1    8.9
#  1.5  5.1
#  2    3.6
#  2.5  2.8
#  3    2.4
#
k <- .5
g <- log(390)
zr <- -5 # see first example
mus <- (0:6)/2
Mxgrsr.ad <- Vectorize(xgrsr.ad, "mu1")
ads <- round(Mxgrsr.ad(k, g, mus, zr=zr), digits=1)
data.frame(mus, ads)

## Table 4 from Moustakides et al. (2009)
## original values are
# gamma  A        STADD/steady-state ARL
# 50     28.02    4.37
# 100    56.04    5.46
# 500    280.19   8.33
# 1000   560.37   9.64
# 5000   2801.75  12.79
# 10000  5603.7   14.17
Gxgrsr.ad  <- Vectorize("xgrsr.ad", "g")
As <- c(28.02, 56.04, 280.19, 560.37, 2801.75, 5603.7)
gs <- log(As)
theta <- 1
zr <- -6
ads <- round(Gxgrsr.ad(theta/2, gs, theta, zr=zr, r=100), digits=2)
data.frame(As, ads)