# 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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