I am trying to determine optimal decision boundaries for a sequential probability ratio test. (Bogacz (2006) has something close to what I need, but not quite.) Following Griffith et al. (2021; The statistics of optimal decision making), I treat that as distinguishing between two normal distributions with different means and identical standard deviations, choosing one answer if the upper decision boundary is reached, another answer if the lower decision boundary is reached, and sampling more information in between. Assuming the categories to be distinguished have probabilities p and 1-p, payoff should be:
payoff=(1-p)Φ(x,μ1,σ)-pΦ(x,μ2,σ)-(1-p)(1-Φ(y,μ1,σ))+p(1-Φ(y,μ2,σ))-a(y-x)
where Φ is the cumulative normal distribution, μ1 and μ2 are the means of the underlying normal distributions, σ is their common standard deviation, and x and y are the lower and upper decision boundaries. Then
(1-p)Φ(x,μ1,σ) = payoff for true negatives
pΦ(x,μ2,σ) = payoff for false negatives
(1-p)(1-Φ(y,μ1,σ)) = payoff for false positives
p(1-Φ(y,μ2,σ)) = payoff for true positives
a(y-x) = sampling cost, proportional to the difference between the decision boundaries.
I can approximate the maximum payoff using numerical methods, like this:
Payoff as a function of upper and lower decision boundaries, for p = 0.55
But this seems like something that should have a closed form solution. I could find the derivative if I had only one variable, but I have two, x and y. And the answer I need is the vector (x, y) that gives me both decision boundaries.
(This is for a paper I am working on, so if it gets published and you want your real name in the acknowledgements, send me a message. That's assuming I have understood the SPRT well enough and that the above is a valid approach, which I am not quite sure of.)