xDgrsr.arl {spc}

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)

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.

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.

Value

Returns a single value which resembles the ARL.

Author(s)

Sven Knoth

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

## Not run: 
## 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)

## End(Not run)