Page 198 - Special Topic Session (STS) - Volume 3

STS540 Zhi Song et al.
shifted exponential distribution (denoted by SE(, )) represents the skewed
distribution. The IC sample is from SE(0,1), but the test samples are from a
1
SE(, ) distribution having pdf () = − (−) , ∈ (, ∞) with mean= +

and variance= . To examine the effect of shifts in location and scale
2
parameters, we consider the quartile deviation (QD) for each of the three
distributions. 12 combinations of and values are considered, that is, θ =0,
QD/4 and QD/2 along with δ =1, 1+QD/4, 1+QD/2 and 1+QD. For brevity, we
only represent = 100 , = 5 and = 0.1 case for all schemes for
comparison purposes. A similar conclusion holds for other parameter
conditions. Two situations, namely “Ideal case (True case)” and “Practical case
(Estimated case)” are considered for numerical studies. As their names imply,
these cases are briefly explained in Section 3. The ideal case is unobservable
because the process distribution is unknown. The practical case is the
optimization procedure of what we are facing. Here we choose = 10 and
= 0.04, i.e., CUFAP(10)=0.04. Employing constraints on CUFAP(10)=0.04, we
obtain the optimal chart schemes (ℎ, ) and (ℎ, ̂) , respectively for the ideal
̂
ℎ
ℎ
case and the practical case, to achieve a minimum of CUTSP(10). Usually, exact
amount of possible shift is unknown and therefore, the practitioners will prefer
to choose a combination of (ℎ, ̂) that has overall good performance
̂
ℎ
irrespective of the exact size of shift. To this end, we further introduce the
̅
mean of ℎ, recorded as ℎ. We compute this by calculating the mean of ℎ over
̂
̂
̂
shifts (, ) under consideration. We also consider the mean of ℎ, say ℎ as a
̅
benchmark and for comparison. The comparative results for the two cases are
summarized in Table 1. From Table 1, we find that the optimization model
performs very efficiently for all the three distributions for most of the 11 OOC
scenarios in terms of the proximity between ℎ and ℎ. Furthermore, the last row
̂
̅
under each distribution in Table 1 is ℎ ( ̅ ) and (ℎ, ̅) of each scheme. It is
̅
̂
ℎ ̂
ℎ
observed that they are extremely close.

5. Concluding remarks
In this paper, we present an optimal designing strategy for the
nonparametric EWMA schemes with FIR features to facilitate early detection
of shift with a restriction on the early false alarm probability. Then we apply
this design method to the wellknown EL and EC monitoring schemes for
implementation. It is worth mentioning that the distribution-free characteristic
of the plotting statistic of a nonparametric scheme is, in general, not valid
under a process shift. As a consequence, the pure theoretical optimal charting
scheme is practically unobservable. Noting that the underlying process
distribution is often unknown, we propose a data-dependent estimation
procedure based on KDE for evaluation of the optimal design parameters.
Simulation results show that overall performance of the estimation procedure

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