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For testing whether a die is fair, we have the log likelihood ratio: $$ \Lambda = \frac{1}{2} \sum_{i} \mathrm{observed}_i \log\left(\frac{\mathrm{observed}_i}{\mathrm{expected}_i}\right). $$

Suppose I wanted to apply Wald's Sequential Probability Ratio Test (SPRT) to this case, to roll the die some number of times and terminate based on the observed $\Lambda$. As far as I have read, the conservative bounds for the SPRT are keep rolling the die if and only if $$ a \le \Lambda \le b, $$ where $$a = \log\left(\frac{\beta}{1-\alpha}\right),\quad b=\log\left(\frac{1-\beta}{\alpha}\right)$$ and $\alpha, \beta$ are the type I and type II rates.

However, when the alternative is not specified, we always have $\Lambda \ge 0$ with equality holding when we see perfectly fair outcomes (all sides seen with equal frequency). Thus we can never observe $\Lambda < a < 0$ for typical type I and II rates. That is, the SPRT will never terminate to accept the null.

Are there known positive values for $a$ that satisfy the type I and type II rates?

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  • $\begingroup$ I think the issue is that SPRT is set up for a specific alternative hypothesis, like "this die has 1 in 3 chance of rolling a six", and the test statistic I am using is for the composite hypothesis, like "one of the sides is more likely than the others." $\endgroup$
    – shabbychef
    Commented May 9 at 0:36
  • $\begingroup$ I think maybe I have myself a Sequential Generalized Likelihood Ratio test here... $\endgroup$
    – shabbychef
    Commented May 9 at 1:06

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