Cimulative Sum (CUSUM) schemes for attributes can be formulated through a Markov chain, and the most important characteristic, ARL, is obtained by solving an inverse matrix whose size is as large as number of possible values the CUSUM takes. The calculation becomes practically impossible when the CUSUM is intended for very small process mean since the corresponding k should be taken a small fraction value and it causes a very large sized computation of an inverse matrix. This study presents an alternative solution without any matrix manipulation based on the authers' result for the sequential probability ratio test. The solution can give an explicit expression of ARL, and its algorithm is also devised. A design table for small valued k to 0.001 is given corresponding to the exisiting table of k larger than 0.25.

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Point estimation of the error variance term in ANOVA for orthogonal arrays is discussed. It is assumed that some error terms are assigned. The procedure which is accompanied with a preliminary test of each effect at some level of significance, and some other procedure of which theoretical features are known are examined. The Monte-Carlo study is taken to estimate the mean square errors of their procedures. It is concluded that the procedure with preliminary test at the level 0.50 has good performance.

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In this paper, we discuss the statistical meaning and the usefulness of the average run length (ARL) through the consideration on the cumulative sum schemes and the runs rule schemes, of which behavior can be represented by Markov models. Observing the geometric approximation to the run length distribution (RLD), we conclude the following. Regarding the type I error, ARL is a useful measure which can give the representation of RLD, because ARL can be considered to be the only parameter of the geometric distribution. Furthermore, observing the conservativeness of the geometric approximation, we can find a different property between the two schemes on the relation between ARL and the probability of the type I error. Regarding the power of test, ARL has no meaning but the mean of RLD from the statistical viewpoint. In a practical representation, however, ARL is useful only under the linear assumption between the cost of out-of control and the delayed time to signal.

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