Stopping on the last success with unknown odds: asymptotic minimax optimality of the plug-in rule

Abstract

We study the last-success problem for sequential Bernoulli trials in the homogeneous setting where $X_1,\ldots,X_n$ are i.i.d. $\mathrm{Bernoulli}(p)$ but the success probability $p\in(0,1)$ is unknown to the decision maker. When $p$ is known, Bruss' sum-the-odds theorem yields an optimal threshold rule with win probability $V_n(p)$; when it is not, the odds driving this threshold must be learned from the very sequence on which one is trying to stop, which turns the problem into a genuinely statistical decision problem over the class of $p$-blind rules---those depending on the data but not on $p$. Writing $W_n^\pi(p)$ for the win probability of such a rule, we show that, for any $p_0\in(0,\tfrac12)$, $$ \lim_{n\to\infty}\sqrt n\,\inf_\pi\sup_{p\in[p_0,1)}\bigl(V_n(p)-W_n^\pi(p)\bigr) = C_\star , $$ with $C_\star:=\tfrac12\sup_{u>0}u\Phi(-u)\approx0.085$ (here, $\Phi$ is the standard normal distribution function), and that this exact constant is attained by the natural plug-in odds rule, which is therefore asymptotically minimax optimal. The result is in fact local: at every transition point $1/k$ of the oracle threshold, the deficit admits an exact local minimax constant proportional to $\gamma_k=\tfrac{1}{\sqrt k}(1-\tfrac1k)^{k-3/2}$, and $C_\star$ is the largest of these, attained at $k=2$. We further quantify the cost of the natural sample-splitting alternative, show that the plug-in rule is asymptotically oracle-optimal in the sparse regime $p=p_n\to0$ with $np_n\to\infty$, and prove that no $p$-blind sequence of rules, even randomized, can converge to the oracle uniformly over $p\in(0,1)$, the obstruction being located in the critical window where $p$ is of order $1/n$.