SPRT I

Here, we are going to give a brief introduction to SPRT (Sequential Probability Ratio Test).

Basic Idea

As we have seen in this introductory example, that we require the following:

  • A Sequential Sample
  • A Stopping Time Rule

For testing of hypothesis purposes, we may require an extra layer of Testing Rule.


Classical Sampling vs Sequential Sampling

In the classical approach, we need fixed sample size. While here we need variable sample size and a stopping rule, which gives a freedom to do more. In the introductory example, we cannot do by any means a classical fixed sampling methodology, while it can be done by sequential sampling approach.

Example:

\(X_1,X_2, …\) are iid Ber(\(p\)). Then we cannot estimate \(\frac{1}{p}\) by Fixed Classical Sampling Procedures (why?), but we can estimate it by Sequential Sampling Procedures by \(N_0\) as shown in the introductory example.


Wald’s’ SPRT Description := SPRT(\(A_{(\alpha,\beta)},B
_{(\alpha,\beta)} \))

\(X_1,X_2, …\) is the Sequential Sample.

\(H_0\) vs \(H_1\) are the two simple hypotheses.

\(f(X_1, X_2, …, X_m| H_i)\) := \(f_i^{(m)}\), where \(i \in \{0,1\} \).

\(\lambda_m\) := \(\frac{f_1^{(m)} }{f_0^{(m)}}\)

Choose \((A,B)\), such that \( 0 < B < 1 < A < \infty \).

Testing Rule:

At any stage \(m \geq 1\), observations \(X_1,X_2, …, X_m\). Calculate \(\lambda_m\).

If \(\lambda_m \geq A\), reject \(H_0\),
else if \(\lambda_m \leq B\), reject \(H_1\),
else if \( B < \lambda_m < A\), continue sampling, i.e. observe \(X_{m+1}\) and repeat this again.

Stopping Rule:

\(n\) = first \(m\), such that \(\lambda_m \geq A\) or \(\lambda_m \leq B\), provided such a \(m\) exists.

Strength of the Test: \({(\alpha,\beta)}\)

\(\alpha\) = P(reject \(H_0\) | \(H_0\) is true).

\(\beta\) = P(accept \(H_0\) | \(H_1\) is true).

We have to select \(A\) and \(B\) in such a way so that the strength of the test is satisfied.


Now, how to calculate the values of \(A\) and \(B\) such that conditions are satisfied. Also, does the stopping rule converge? These are the questions, we will study in the next post.

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