Sequential multiple hypothesis testing in presence of control variables

Suppose that at any stage of a statistical experiment a control variable $X$ that affects the distribution of the observed data $Y$ at this stage can be used. The distribution of $Y$ depends on some unknown parameter $\theta$, and we consider the problem of testing multiple hypotheses $H_1: \theta=\theta_1$, $H_2: \theta=\theta_2, …$, $H_k: \theta=\theta_k$ allowing the data to be controlled by $X$, in the following sequential context. The experiment starts with assigning a value $X_1$ to the control variable and observing $Y_1$ as a response. After some analysis, another value $X_2$ for the control variable is chosen, and $Y_2$ as a response is observed, etc. It is supposed that the experiment eventually stops, and at that moment a final decision in favor of one of the hypotheses $H_1,…$, $H_k$ is to be taken. In this article, our aim is to characterize the structure of optimal sequential testing procedures based on data obtained from an experiment of this type in the case when the observations $Y_1, Y_2,…, Y_n$ are independent, given controls $X_1,X_2,…, X_n$, $n=1,2,…$.

💡 Research Summary

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The paper introduces a novel sequential testing framework in which a controllable experimental variable X can be chosen at each stage to influence the distribution of the observed response Y. The underlying statistical model assumes that, conditional on the sequence of controls (X_{1:n}), the observations (Y_{1},\dots,Y_{n}) are independent and follow densities (f_{\theta}(y\mid x)) indexed by an unknown parameter (\theta). The goal is to test a finite set of simple hypotheses (H_{i}:\theta=\theta_{i}) (i = 1,…,k) while respecting pre‑specified error‑rate constraints (\alpha_{ij}=P{\text{declare }H_{j}\mid H_{i}\text{ true}}).

The authors formulate the design problem as a joint optimization over a control policy (\pi) (which selects the next value of X based on all past data) and a stopping rule (\tau) (the random time at which the experiment terminates and a final decision is made). The performance criterion is the expected sample size (\mathbb{E}_{\theta}