This thesis is concerned with the optimality of exponential smoothing with arbitrary order, and its sensitivity. The integrated moving average (IMA) process which satisfies the optimality condition of exponential smoothing with minimum forecast error can be given as
$Y(t)=\nabla^nX(t)=\displaystyle\sum^{n}_{i=0}\binom{n}{i}(-\beta)^i(t-i).\forall{i}$
which is a special case of IMA (n,n).
The purpose of this thesis is to examine the optimality of the general IMA processes with the different differencing order and the moving average order. Numerical experiments were done for IMA(m,n) processes with various combinations of m and n, and the corresponding forecast errors were compared.
As a result, the IMA process that the moving order is higher than the differencing order has comparatively smaller forecast error by exponential smoothing. If the difference between the differencing order and the moving order becomes larger, the accuracy of forecast by exponential smoothing declines gradually. The result suggests that exponential smoothing retains its optimality in robust sense, even when the differencing order is not identical with the moving average order. A typical case is when the differencing order is lower than the moving average order by one.
