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Yander Wiel

N, = a,+X£a,

/=o

where the a s are iid N (O, a²) variates and X e[0,l]. An IMA is non-

stationaiy (X * 0) with variance a² (1 + Xt) increasing linearly in /. The increments

form a flrst order moving average. Special cases of the IMA family arise when X = 0 giving an iid process and when X = 1 giving a random walk. For Xe(0, 1) the IMA is equivalent to a random walk observed with iid measurement error (Box and Jenkins, 1976, Chapter 4).

The left column of plots in Figurę 3 shows 250 observations from simulated IMAs with <r = 1 and X varying from 0 to 1 in successive panels. Each panel uses the same set of as. Notice how the IMAs wander morę for larger values of X. The remainder of Figurę 3 is discussed below.

An estimate X of X can be used to form 1-step ahead forecasts of the IMA. The usual forecast of N_t+i based on (N₀, N_t) is

aU,=wv,+(

where the recursion is started from Ń_0H = 0.    jV,_+1/< is an exponentially

weighted moving average (EWMA) of the observed values of the IMA from

^A    A

N₀ through N_t. If X = X, N_nXj, is a minimum mean sąuare error forecast of . The forecast error in period t is

a,=N,-Ń_lM    (1)

This definition gives a, = a,    a,_, j (for / £ 1) so

a, w a, if X « X . The maximum likelihood estimator X minimizes ■

An important baseline for comparing standard control charts has been how ąuickly they detect a sudden shift in the process level. This is measured by