Let X₁, X₂,... be independent random variables observed sequentially and such that X₁,..., X_θ−1 have a common probability density p₀, while X_θ, X_θ+1,... are all distributed according to p₁ ≠ p₀. It is assumed that p₀ and p₁ are known, but the time change θ ∈ ℤ⁺ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem:$$\Delta(\theta;{\tau^\alpha})\rightarrow\min_{\tau^\alpha}\;\;\text{subject}\;\text{to}\;\;\alpha(\theta;{\tau^\alpha})\leq\alpha\;\text{for}\;\text{any}\;\theta\geq1,$$where α(θ; τ) = P_θ{τ < θ} is the false alarm probability and Δ(θ; τ) = E_θ(τ − θ)₊ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays.

中文翻译：

---

用于检测分布变化的多重假设检验方法

让X ₁，X ₂，...是独立随机变量顺序地观察到，并且使得X ₁，...，X _{θ -1}有一个共同的概率密度p ₀，而X _θ，X _{θ 1}，...都根据p ₁ ≠ p _0分布。假定p ₀和p ₁是已知的，但随时间的变化θ＆Element;ℤ ⁺是未知的，并且目标是构建一个停止时间τ尽快检测变化点θ。解决该问题的标准方法基本上依赖于一些有关θ的先验信息。例如，在贝叶斯方法中，假设θ是具有已知概率分布的随机变量。在与假设检验有关的方法中，该先验信息隐藏在所谓的平均游程长度中。本文的主要目标是构建没有关于θ的先验信息的停止时间。更正式地说，我们提出一种解决以下最小化问题的方法：$$ \ Delta（\ theta; {\ tau ^ \ alpha}）\ rightarrow \ min _ {\ tau ^ \ alpha} \; \; \ text {subject} \; \ text {to} \; \; \ alpha（ \峰; {\ tau蛋白^ \阿尔法}）\当量\阿尔法\; \文本{为} \; \文本{任何} \; \ THETA \ geq1，$$其中α（θ;τ）= P _θ { τ <θ }是虚警概率和Δ（θ ; τ）= E _θ（τ - θ）₊是平均检测延迟计算对于给定的停止时间τ。与基于顺序最大似然检验的标准CUSUM算法相比，我们的方法与多种假设检验方法有关，尤其允许构造具有几乎贝叶斯检测延迟的通用停止时间。