Page 192 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
response (FIR) feature in an EWMA scheme. Lucas and Crosier [2] first
proposed the FIR feature for CUSUM schemes. They established that this
additional feature helps in detecting early changes more quickly by assigning
some non-zero constant to the starting values of the plotting statistic of the
CUSUM scheme. Lucas and Saccucci [1] introduced a similar FIR feature for the
EWMA schemes. In recent years, many researchers made significant
contributions in literature related to the FIR feature in various SPM schemes.
Most of these papers were excellently written but mainly focused on the out-
of-control (OOC) performance of the schemes with FIR features. There is very
little consideration of the consequences of adding the FIR feature on the early
false alarm probabilities. Knoth [3] exclusively outlined that the FIR feature
certainly helps in achieving higher sensitivity to early changes, but it also
increases the probability of early false alarms. Too many early false alarms may
have an impact on production cost and also lead to a state of confusion about
the suitability of the charting procedure. On the other hand, setting very low
probability of early false alarms reduces the efficiency of the monitoring
scheme, the almost invariably causes a delay in detecting a true in shift in the
process parameters. To this end, we attempt to develop an efficient approach.
We observe that almost all previous works on one-sided monitoring schemes
with FIR feature consider a starting value halfway between the target value
and the control limit. Unlike previous works of fixing a head start value of 50%
for the schemes with FIR features, we recommend an optimal (dynamic) head
start value that optimizes the chart performance. Our proposed approach
restricts probability of an early false alarm to a prefixed value at any situation,
while optimizes the early detection of true signal in presence of the FIR feature.
The rest of the paper is organized as follows. In the next Section, we
introduces four different nonparametric EWMA schemes under consideration.
Section 3 discusses the methodology of optimization of EWMA scheme with
FIR features. We study all these schemes with optimal parameters in Section
4. Several remarks conclude this paper in Section 5.

2. Presentation of the competing distribution-free EWMA schemes
We assume that a reference sample X = (X1,X2, · · ·,Xm) collected from an IC
process with a continuous cumulative distribution function (cdf) F (x). We also
assume that Xi is are independently and identically distributed (i.i.d) random
variable each having cdf F. Let Yj = (Yj1,Yj2, · · · ,Yjn), j = 1, 2, · · · be the jth Phase-
II (test) sample of size n mutually independent of the reference sample, from
a cdf G(x) = F ( −θ ), θ ∈ ℜ, δ > 0, where the constants θ and δ represent the
δ
unknown location and scale parameters, respectively. We further assume that
Yji s are i.i.d for every i (1 ≤ ≤ and j, (j > 1). The process is considered to be

181 |I S I W S C 2 0 1 9