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Yander Wiel

ARLs for smali shifts in an iid process (X = 0) reąuires k near 0.5 whereas getting reasonably Iow ARLs for large shifts when X is 0.5 or higher reąuires k to be close to 1.0 or higher. This should be useful for control chart design and it follows the traditional wisdom for CUSUM charts that smali values of k produce better ARLs for smali shifts while large values of k give better ARLs for large shifts.

3. EWMA: An EWMA monitoring scheme is based on an exponentially weighted moving average of the forecast errors

Q, =ya, +(i-yte-i

where Q₀ is initialized at zero. The scheme signals when \Q,\ exceeds an action limit h. The weight y e(0,l] and the action limit h are design

parameters. Choosing y and h for an EWMA scheme is similar to choosing k and h for a CUSUM scheme. For a given y, h can be selected to give a desired ARL₀. The right panel of Figurę 4 shows curves of h versus y for three values of ARL₀. Given ARL₀ and y, the appropriate curve can be used to determine

h. Thus, it remains to choose a reasonable value for y. Setting y = 1 produces the usual Shewhart individuals chart with “/i-sigma” limits.

Figurę 6 is to the EWMA as Figurę 5 is to the CUSUM. Its Iayout and use are essentially identical. Columns are indexed by X and rows are indexed by |x. In each panel the top, middle, and bottom curves correspond to EWMAs with ARL₀ values of 500, 250, and 100 respectively. Each curve shows ARLs for given sized shifts as functions of the EWMA design parameter y where the control limit h is selected (from Figurę 4) to give the appropriate ARL₀.

The curves in Figurę 6 look remarkably similar to those in Figurę 5. Of course they are not identical but apparently a CUSUM monitoring scheme with reference level k is roughly equivalently to an EWMA scheme with y = 2k. The next subscction has further comparisons among the different classes of monitoring schemes.

A reasonable rule of thumb for designing EWMA schemes for monitoring IMA processes is to choose y close to X but not outside the rangę [0.1, 0.5]. (If, however, the process is known to be a random walk (X = 1), then it makes sense to take y = 1 and monitor with a Shewhart chart.) Typically, smali values of y give better ARLs for smali shifts and large values give better