spc (version 0.7.1)

xDgrsr.arl: Compute ARLs of Shiryaev-Roberts schemes under drift

Description

Computation of the (zero-state and other) Average Run Length (ARL) under drift for Shiryaev-Roberts schemes monitoring normal mean.

Usage

xDgrsr.arl(k, g, delta, zr = 0, hs = NULL, sided = "one", m = NULL,
mode = "Gan", q = 1, r = 30, with0 = FALSE)

Value

Returns a single value which resembles the ARL.

Arguments

k

reference value of the Shiryaev-Roberts scheme.

g

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

delta

true drift parameter.

zr

reflection border for the one-sided chart.

hs

so-called headstart (enables fast initial response).

sided

distinguishes between one- and two-sided Shiryaev-Roberts schemes by choosing "one" and "two", respectively. Currentlly, the two-sided scheme is not implemented.

m

parameter used if mode="Gan". m is design parameter of Gan's approach. If m=NULL, then m will increased until the resulting ARL does not change anymore.

q

change point position. For \(q=1\) and \(\mu=\mu_0\) and \(\mu=\mu_1\), the usual zero-state ARLs for the in-control and out-of-control case, respectively, are calculated. For \(q>1\) and \(\mu!=0\) conditional delays, that is, \(E_q(L-q+1|L\geq)\), will be determined. Note that mu0=0 is implicitely fixed. Deploy large q to mimic steady-state. It works only for mode="Knoth".

mode

decide whether Gan's or Knoth's approach is used. Use "Gan" and "Knoth", respectively. "Knoth" is not implemented yet.

r

number of quadrature nodes, dimension of the resulting linear equation system is equal to r+1 (one-sided) or r (two-sided).

with0

defines whether the first observation used for the RL calculation follows already 1*delta or still 0*delta. With q additional flexibility is given.

Author

Sven Knoth

Details

Based on Gan (1991) or Knoth (2003), the ARL is calculated for Shiryaev-Roberts schemes under drift. In case of Gan's framework, the usual ARL function with mu=m*delta is determined and recursively via m-1, m-2, ... 1 (or 0) the drift ARL determined. The framework of Knoth allows to calculate ARLs for varying parameters, such as control limits and distributional parameters. For details see the cited papers.

References

F. F. Gan (1991), EWMA control chart under linear drift, J. Stat. Comput. Simulation 38, 181-200.

S. Knoth (2003), EWMA schemes with non-homogeneous transition kernels, Sequential Analysis 22, 241-255.

S. Knoth (2012), More on Control Charting under Drift, in: Frontiers in Statistical Quality Control 10, H.-J. Lenz, W. Schmid and P.-T. Wilrich (Eds.), Physica Verlag, Heidelberg, Germany, 53-68.

C. Zou, Y. Liu and Z. Wang (2009), Comparisons of control schemes for monitoring the means of processes subject to drifts, Metrika 70, 141-163.

See Also

xewma.arl and xewma.ad for zero-state and steady-state ARL computation of EWMA control charts for the classical step change model.

Examples

if (FALSE) {
## Monte Carlo example with 10^8 replicates
#   delta      arl    s.e.
#   0.0001 381.8240   0.0304
#   0.0005 238.4630   0.0148
#   0.001  177.4061   0.0097
#   0.002  125.9055   0.0061
#   0.005   75.7574   0.0031
#   0.01    50.2203   0.0018
#   0.02    32.9458   0.0011
#   0.05    18.9213   0.0005
#   0.1     12.6054   0.0003
#   0.5      5.2157   0.0001
#   1        3.6537   0.0001
#   3        2.0289   0.0000
k <- .5
L0 <- 500
zr <- -7
r <- 50
g <- xgrsr.crit(k, L0, zr=zr, r=r)
DxDgrsr.arl <- Vectorize(xDgrsr.arl, "delta")
deltas <- c(0.0001, 0.0005, 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.5, 1, 3)
arls <- round(DxDgrsr.arl(k, g, deltas, zr=zr, r=r), digits=4)
data.frame(deltas, arls)
}