00254 07cc71dc7c8b6b356c430eda738c59e

Yander Wiel

ARLs for large shifts. As with CUSUM schemes, it is virtually impossible to detect smali or medium sized shifts when X is 0.5 or greater.

4. Likelihood ratio scheme: A monitoring scheme based on likelihood ratio statistics can take advantage of the pattemed change that occurs in the forecast error mean when an IMA undergoes a step shift. To make this scheme manageable we limit the data used in period t to the last n + 1 forecast errors (a,__n.....a,) . For the comparisons in the next section we consider n + 1 = 3, 5

and 9 and found that the sample size madę very little difference in ARL performance except in the iid case (X = 0) for smali sized shifts. Then larger values of n were helpful. Based on {a,a,), the likelihood ratio statistic

to test for a step shift occurring anywhere between period t - n and period t is a monotone function of

U. = wax|z![^|    (2)

'    A=0,...,»l I

where Z^ is the ‘Z-statistic” in the regression of («,_*,o, a_t) on

(l,l-y,...,(l-y)*j, namely

CT TT

is sensitive to a geometrically decaying pattem in the forecast errors that starts in period t - k. Thus, U_t should be sensitive to a step shift in the IMA that begins in one of the most recent n + 1 periods. The likelihood ratio monitoring scheme signals when U_t exceeds an action limit h which is picked to yield a desired ARL when no shift occurs. Monte Carlo simulations were used to approximate ARLs for the likelihood ratio scheme. To find h corresponding to a given ARL₀ we first found values of h that gave ARLs bracketing ARL₀. Then we used the bisection method (for finding a root) to home in on a value of h that gave approximately ARL₀. The window length

n gives one some additional flexibility in choosing a likelihood ratio scheme. However, the comparisons in the next subsection show that the choice of n matters little unless y is near 0 in which case it is helpful to use a large window

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