Abstract:Testing by betting has been a cornerstone of the game-theoretic statistics literature. One bets against the null hypothesis, and the accumulated wealth $W_t$ quantifies the evidence against the null hypothesis after $t$ rounds, and the null can be rejected at level $\alpha$ whenever $W_t \geq 1/\alpha$. A key assumption permeating the literature is that one cannot bet more money than they currently have (the wealth must stay nonnegative). In this work, we examine the consequences of allowing the bettor to borrow money in each round (for example after going bankrupt). Specifically, we ask how the threshold of $1/\alpha$ must be accordingly adjusted to retain the desired level $\alpha$. Our findings are twofold. First, if the new rejection rule is $W_t \geq g(\alpha,L_t)$ where $L_t$ is the total liability at time $t$, then we show that $g(\alpha,0)>1/\alpha$ if $g(\alpha,L_t)<\infty$ for any $L_t > 0$; in words, we must pay for the possibility of borrowing, even if in fact we do not borrow. Second, and in contrast to the first, if one employs a path dependent threshold $h(\alpha,W_0,L_1,\dots,W_{t-1},L_t)$, that is a function of past leverage ratios, then there is in fact no extra price to pay for the possibility of borrowing.
From: Hongjian Wang [view email]
[v1]
Tue, 16 Jul 2024 08:05:04 UTC (19 KB)
[v2]
Thu, 16 Oct 2025 15:07:13 UTC (23 KB)
[v3]
Sun, 3 May 2026 16:51:02 UTC (20 KB)
[v4]
Mon, 11 May 2026 22:52:23 UTC (20 KB)
[v5]
Sat, 11 Jul 2026 01:41:01 UTC (314 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。