# Sequential Analysis

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## sequential analysis

[si′kwen·chəl ə′nal·ə·səs]*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## 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 *H*_{1} and *H*_{2} according to the results of independent observations. Hypothesis *H*_{1} states that the random variable *X* has a probability distribution with density *f*_{1} (*x*); hypothesis *H*_{2} states that *X* has density *f*_{2} (*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 λ_{1} = *f*_{2} (*x*_{1})/*f*_{1} (*x*_{1}) is computed, where *x*_{1} is the result of the first observation. If λ_{1} < *A*, then *H*_{1} is accepted. If λ_{1} > *B*, then *H*_{2} is accepted. If *A* ≤ λ_{1} ≤ *B*, then the process is continued: a second observation is made; the quantity λ_{2} = *f*_{2}(*x*_{1})*f*_{2}(*x*_{2})/*f*_{1}(*x*_{1})*f*_{1}(*x*_{2}), where *x*_{2} 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 *H*_{1} or the selection of *H*_{2}. 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 α_{1} and α_{2}, respectively. An error of the first type is the rejection of hypothesis *H*_{1} when it is true, and an error of the second type is the acceptance of *H*_{1} when *H*_{2} is true.

In practice, it is more convenient to consider instead of λ* _{n}* the logarithms of λ

*. For example, let hypothesis*

_{n}*H*

_{1}be that

*X*has a normal distribution

with *a*= 0 and *σ* = 1; let hypothesis *H*_{2} that *X* has a normal distribution with *a*= 0.6 and *σ =* 1; and let α_{1} = 0.01 and α_{2} = 0.03. The corresponding calculations show that in this case *A* = 1/33, *B* = 97, and

Therefore, the inequalities λ_{n} < 1/33 and λ_{n} > 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 *H*_{2} is accepted. If the broken line leaves the region through the lower boundary, then *H*_{1} is accepted. In this example, the method of sequential analysis requires on the average not more than 25 observations to decide between *H*_{1} and *H*_{2}. 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