Table A.8: The type-I and type-II error rates for size 20
Level 6 pI pII
index MSS T2 T2(mod) MSS T2 T2(mod) 1 0.0000 0.2818 0.0182 0.50 0.10 0.75 2 0.0091 0.3273 0.0091 0.60 0.15 0.80 3 0.0091 0.2727 0.0091 0.65 0.25 0.95 4 0.0000 0.3000 0.0182 0.50 0.00 0.75 5 0.0000 0.3091 0.0182 0.60 0.10 0.60 6 0.0000 0.3000 0.0182 0.60 0.00 0.75 7 0.0000 0.3000 0.0091 0.15 0.00 0.10 8 0.0000 0.2818 0.0091 0.65 0.05 0.75 9 0.0000 0.3000 0.0182 0.65 0.05 0.60
Level 5 pI pII
1 0.0000 0.3000 0.0182 0.50 0.00 0.50 2 0.0000 0.3000 0.0182 0.45 0.00 0.55 3 0.0000 0.3000 0.0182 0.40 0.00 0.20 4 0.0000 0.3000 0.0091 0.65 0.00 0.60 5 0.0000 0.3000 0.0091 0.65 0.05 0.70 6 0.0000 0.3273 0.0182 0.70 0.25 0.85 7 0.0000 0.3000 0.0182 0.40 0.00 0.70 8 0.0000 0.3000 0.0182 0.40 0.00 0.65 9 0.0000 0.3000 0.0091 0.15 0.00 0.50
A.5 The Proportion of the Total Variation Explained by the SMSS Chart of
Simulations in Section 5.3.1
Table A.9: The proportion of the total variation explained by the SMSS chart for OC Model (a) and (b)
Model (a) Model (b)
δ K = 2 K = 3 K = 4 K = 5 K = 2 K = 3 K = 4 K = 5 0.6 0.9504 0.9800 0.9879 0.9936 0.9516 0.9803 0.9881 0.9937 1.2 0.9491 0.9802 0.9880 0.9936 0.9537 0.9811 0.9886 0.9939 1.8 0.9468 0.9804 0.9881 0.9937 0.9566 0.9823 0.9893 0.9943 2.4 0.9439 0.9806 0.9882 0.9938 0.9600 0.9837 0.9902 0.9948 3.0 0.9405 0.9810 0.9885 0.9939 0.9638 0.9852 0.9911 0.9953
Table A.10: The proportion of the total variation explained by the SMSS chart for OC Model (c) and (d)
Model (c) Model (d)
δ K = 2 K = 3 K = 4 K = 5 K = 2 K = 3 K = 4 K = 5 1.4 0.9495 0.9801 0.9880 0.9936 0.9528 0.9808 0.9884 0.9938 1.8 0.9479 0.9803 0.9880 0.9937 0.9552 0.9817 0.9889 0.9941 2.2 0.9457 0.9805 0.9882 0.9937 0.9578 0.9828 0.9896 0.9945 2.6 0.9432 0.9807 0.9883 0.9938 0.9604 0.9839 0.9902 0.9948 3.0 0.9407 0.9809 0.9884 0.9939 0.9634 0.9851 0.9909 0.9952
Table A.11: The proportion of the total variation explained by the SMSS chart for OC Model (e)
δ > 0 δ < 0
δ K = 2 K = 3 K = 4 K = 5 δ K = 2 K = 3 K = 4 K = 5 1.143 0.9509 0.9800 0.9879 0.9936 0.875 0.9509 0.9800 0.9879 0.9936 1.333 0.9510 0.9800 0.9879 0.9936 0.750 0.9510 0.9800 0.9879 0.9936 1.600 0.9509 0.9800 0.9879 0.9936 0.625 0.9509 0.9800 0.9879 0.9936 2.000 0.9508 0.9800 0.9879 0.9936 0.500 0.9508 0.9800 0.9879 0.9936
Appendix B
B.1 ARL Calculation of The Combined EWMA Chart
An approximation of the ARL of the CE chart can be obtained via approximating the properties of the continuous-state two-dimensional Markov chain{(W0,i, W1,i, i = 0, 1,· · · )} by a two-dimensional Markov chain with discrete-state space. By the independency of the T02 and T12 statistics as well as the W0,i and W1,i, the two-dimensional chain can be described by two one-two-dimensional chains, one for each individual EWMA chart. Following Morais and Pacheco (2000) and others, the Markovian ARL approximation is introduced as follows.
First, dividing the in-control interval Cl = (0, Ll) into v− 1 subintervals with equal range, l = 0, 1, where L0 and L1 are defined in equations (3.9) and (3.10), respectively. That is, for each subinterval Ej = (ej, ej+1), where ej = Ll(j−1)/(v−
1), j = 1, . . . , v. Define the absorbing state of each chain as (−∞, 0) ∪ (Ll,∞) for each l. Then an approximation of the probability transition matrix of each individual Markov chain is
Pl(δ) =
Ql(δ) [Iv−1− Qi(δ)]× 1v−1
′
,
where 1v−1and 0v−1are vectors of ones and zeros with dimension v−1, respectively, Iv−1 is the identity matrix with rank v− 1, δ is the vector of mean difference between IC and OC cases, and the matrix Ql(δ) has entries given by
ql,jk = P
Under the normality assumption, the T02 and T12 follow the non-central χ2 distri-bution with degrees of freedom K and n− K and the non-centrality parameters δ′P0Λ−10 P0′δ and δ′P1Λ−11 P1′δ, respectively, where P0, P1, Λ0, and Λ1 are given in Section 3.2. Therefore, the entries of Ql(δ) are of the form
ql,jk = Fd(aj,k+1)− Fd(aj,k),
and Fd is the distribution function of χ2 distribution with degrees of freedom d, and d = K if l = 0, n− K if l = 1.
Let RLα0(δ), RLβ1(δ), and RLα,βCE(δ) denote the run length of the T02, T12, and CE charts, respectively, conditional on δ and the initial values of W0,0 and W1,0, which belong to the transient states Eα and Eβ, respectively. Define pα to be a vector with one at the position α and zeros at the rest, and pβ similarly. Then the survival function of RLα0, RLβ1, and RLα,βCE can be approximated by
FRLα,β
where [s] denotes the integer part of s. Finally, the approximations of the ARLs of the T02, T12, and CE charts are given by
Note that the number of partition of the in-control interval, v− 1, should be odd.
Moreover, one should choose a larger v for the wider range of the in-control interval.
In our simulations and real case studies, v is chosen to be 52 for the T02 part of the CE chart, and 102 for the T12 part.
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