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Statistical Process Monitoring with Integrated Moring Ayerage Noise 259 section considers the problem of testing for a shift in the level of an IMA process. Speciflcally, we derive the exact nuli distribution of the likelihood ratio statistic and give a good approximation for the upper taił of the distribution.

We assume that the IMA parameters X and a are known. For large enough samples the distributions derived below are good approximations when consistent estimates of the parameters are used.

Exact nuli distribution Once again it is convenient to work with 1-step forecast errors a, defined by (1). Recall, from Section 2 that a step shift in the level of an IMA results in a pattemed shift in the mean of its forecast errors. Thus, it is appropriate to consider testing the nuli hypothesis H₀ that forecast

errors a_Q,...a, are iid N(0 ,ct² ) against the altemative H_a that they are independent with a, ~ jV(p.,,a²) where

H, =0 i = 01

= * i =

In H_a both k e{0,...,n} and (-00,00) are unspecified.

A likelihood ratio monitoring scheme was developed in the previous section. Taking the window size n + 1 in that context, equal to the total number of observations t + 1, the statistic U, of (2) is related to the likelihood ratio statistic LR for testing H₀ against H_a by

Uj = -21n(LR)

Recall that U_t is the maximum absolute ‘Z-statistic” in the regressions of (a,__k,a_l__k+l,...,a_l) on (l,l-y,...,(l-y)*) for k = 0In this section we write

U = max\Z_k\

k=0...../' ¹