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.

Figure 1. Graphical representation of the process of sequential analysis

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 λ_n the logarithms of λ_n. 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 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 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.