Let X be N(0, θ) and, in the notation of this section, let θ^'=4, θ^''=9, αa=0.05, and βa=0.10 .

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Let X be N(0, θ) and, in the notation of this section, let...

Let $X$ be $N(0, \theta)$ and, in the notation of this section, let $\theta^{\prime}=4, \theta^{\prime \prime}=9$, $\alpha_{a}=0.05$, and $\beta_{a}=0.10 .$ Show that the sequential probability ratio test can be based upon the statistic $\sum_{1}^{n} X_{i}^{2} .$ Determine $c_{0}(n)$ and $c_{1}(n)$.

Key Concepts

Sequential Probability Ratio Test (SPRT)

The SPRT is a sequential hypothesis testing procedure in which data is evaluated as it is collected, and a decision to accept one of two competing hypotheses is made as soon as the accumulated evidence is sufficiently strong. The test is based on the likelihood ratio, and predefined error probabilities determine two thresholds. When the likelihood ratio falls outside these bounds, a decision is made immediately, potentially reducing sample size compared to fixed-sample tests.

Likelihood Ratio for Normal Distributions with Unknown Variance

In the context of testing hypotheses for the variance of a normal distribution (with known mean), the likelihood functions under the different variance hypotheses are functions of the observations through their sum of squares. The likelihood ratio comparing two different variance parameters can be expressed solely in terms of the statistic ?X?², which allows the test to be implemented via this single sufficient statistic.

Sufficient Statistic

A sufficient statistic is a function of the data that contains all the information necessary for inference about a parameter. For a sample from a normal distribution with zero mean, the sum of squares, ?X?², is a sufficient statistic for the variance parameter. This property simplifies the form of the likelihood ratio and is central in deriving decision boundaries in sequential testing.

Boundary Determination and Transformation

When constructing the SPRT, the decision boundaries (c?(n) and c?(n)) are derived by transforming the likelihood ratio test into a test based on the sufficient statistic. The accumulated sum of squares is compared against these boundaries, which are determined by the error probabilities (? and ?) and the values of the variance under the null and alternative hypotheses. By equating the log?likelihood ratio to the logarithms of these critical values, one obtains the explicit form of the boundaries as functions of n.

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