arXiv 0905.14372009-05-11math.STmath.PRstat.MEstat.TH

简单假设与单侧假设的局部最强序贯检验

Locally most powerful sequential tests of a simple hypothesis vs one-sided alternatives

Andrey Novikov, Petr Novikov

备注：: 30 pages
发表信息：: Journal of Statistical Planning and Inference, v. 140 (2010), no. 3, 750--765
DOI：: 10.1016/j.jspi.2009.09.004

摘要：

设$X_1,X_2,...$是离散时间随机过程，其分布为$P_θ$，$θ\inΘ$，其中$Θ$是实数直线的开子集。我们考虑的问题测试一个简单假设$H_0:$ $θ=θ_0$与复合替代$H_1:$ $θ>θ_0$，其中$θ_0\inΘ$是一些固定点。本文的主要目标是描述这个问题中局部最强序贯检验的结构。对于具有（随机化）停止规则$ψ$和（随机化）决策规则$ϕ$的任何序贯检验$(ψ,ϕ)$，令$α(ψ,ϕ)$为I类错误概率，$\dot β_0(ψ,ϕ)$为幂函数在$θ=θ_0$处的导数，$\mathscr N(ψ)$为检验$(ψ,ϕ)$的平均样本数。然后我们考虑在所有序贯检验类中最大化$\dot β_0(ψ,ϕ)$使得$$ α(ψ,ϕ)\leq α\quad{and}\quad \mathscr N(ψ)\leq \mathscr N, $$的问题，其中$α\in[0,1]$和$\mathscr N\geq 1$是一些限制.假设$\mathscr N(ψ)$是在过程$X_1,X_2...$的某个固定分布（不一定与$P_θ$之一一致）下计算的。刻画了最优序贯检验的结构。

英文摘要：

Let $X_1,X_2,...$ be a discrete-time stochastic process with a distribution $P_θ$, $θ\inΘ$, where $Θ$ is an open subset of the real line. We consider the problem of testing a simple hypothesis $H_0:$ $θ=θ_0$ versus a composite alternative $H_1:$ $θ>θ_0$, where $θ_0\inΘ$ is some fixed point. The main goal of this article is to characterize the structure of locally most powerful sequential tests in this problem. For any sequential test $(ψ,ϕ)$ with a (randomized) stopping rule $ψ$ and a (randomized) decision rule $ϕ$ let $α(ψ,ϕ)$ be the type I error probability, $\dot β_0(ψ,ϕ)$ the derivative, at $θ=θ_0$, of the power function, and $\mathscr N(ψ)$ an average sample number of the test $(ψ,ϕ)$. Then we are concerned with the problem of maximizing $\dot β_0(ψ,ϕ)$ in the class of all sequential tests such that $$ α(ψ,ϕ)\leq α\quad{and}\quad \mathscr N(ψ)\leq \mathscr N, $$ where $α\in[0,1]$ and $\mathscr N\geq 1$ are some restrictions. It is supposed that $\mathscr N(ψ)$ is calculated under some fixed (not necessarily coinciding with one of $P_θ$) distribution of the process $X_1,X_2...$. The structure of optimal sequential tests is characterized.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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