Sequential Analysis
in mathematical statistics, a method for the statistical testing of hypotheses. In this method, the number of observations required is not fixed in advance but is determined during the course of the test. The proper application of a chosen method of sequential analysis often requires considerably fewer observations for the same degree of validity than do methods where the number of observations is fixed in advance. Since the number of observations in sequential analysis is a random variable, this number is smaller only on the average.
For example, suppose the problem consists in a choice between the hypotheses H1 and H2 according to the results of independent observations. Hypothesis H1 states that the random variable X has a probability distribution with density f1 (x); hypothesis H2 states that X has density f2 (x). The problem is solved in the following manner. Two numbers A and B are chosen such that 0 < A < B. After the first observation, the ratio λ1 = f2 (x1)/f1 (x1) is computed, where x1 is the result of the first observation. If λ1 < A, then H1 is accepted. If λ1 > B, then H2 is accepted. If A ≤ λ1 ≤ B, then the process is continued: a second observation is made; the quantity λ2 = f2(x1)f2(x2)/f1(x1)f1(x2), where x2 is the result of the second observation, is analyzed; and appropriate action is taken. The probability is 1 that the process terminates with either the selection of H1 or the selection of H2. The quantities A and B are determined from the condition that the probabilities of errors of the first and second type have the specified values α1 and α2, respectively. An error of the first type is the rejection of hypothesis H1 when it is true, and an error of the second type is the acceptance of H1 when H2 is true.
In practice, it is more convenient to consider instead of λn the logarithms of λn. For example, let hypothesis H1 be that X has a normal distribution
with a= 0 and σ = 1; let hypothesis H2 that X has a normal distribution with a= 0.6 and σ = 1; and let α1 = 0.01 and α2 = 0.03. The corresponding calculations show that in this case A = 1/33, B = 97, and
Therefore, the inequalities λn < 1/33 and λn > 97 are equivalent to the inequalities
and
respectively. The process of sequential analysis in this case admits of a simple graphic representation (see Figure 1). On the xy-plane there are drawn the two straight lines y= 0.3x —5.83 and y = 0.3x + 7.62 and a broken line with vertices at the points
If the broken line first leaves the region bounded by the straight lines through the upper boundary, then H2 is accepted. If the broken line leaves the region through the lower boundary, then H1 is accepted. In this example, the method of sequential analysis requires on the average not more than 25 observations to decide between H1 and H2. More than 49 observations would be required to decide between the hypotheses on the basis of samples of a fixed size.
REFERENCES
Blackwell, D., and M. A. Girshick. Teoriia igr i statisticheskikh reshenii. Moscow, 1958. (Translated from English.)Wald, A. Posledovatel’nyi analiz. Moscow, 1960. (Translated from English.)
Shiriaev, A. N. Statisticheskii posledovatel’nyi analiz. Moscow, 1969.
IU. V. PROKHOROV
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