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均值序贯估计程序的有界遗憾AI 解读

Let $X_1, X_2, \cdots$ be independent observations from a population with mean $\mu$ and variance $\sigma^2$, and suppose that given a sample of size $n$ one wishes to estimate $\mu$ by $\bar{X}_n$, subject to the loss function $L_n = A\sigma^{2\beta-2}(\bar{X}_n - \mu)^2 + n, A > 0, \beta > 0$. If $\sigma$ is known, then the optimal fixed sample size $n_0$ for minimizing the risk $R_n = EL_n$ can be computed, but if $\sigma$ is unknown there is no fixed sample size…

均值估计的一种序贯程序的表现AI 解读

Let $\{X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables with mean $\mu$ and unknown variance $\sigma^2$. We want to estimate $\mu$ by $\bar{X}_n$ with a loss function of $\sigma^{2 \delta - 2}(\bar{X}_n - \mu)^2 + \lambda n$, where $\delta > 0$ and $\lambda \rightarrow 0 +$. For $n_\lambda = o(\lambda^{-1/2})$ and $n_\lambda(\log \lambda)^{-1} \rightarrow -\infty$ as $\lambda \rightarrow 0+$, set $T = \inf\{n \geq n_\lambda: n^{-1} \sum^n_{i=1} (X_i - \bar{X}_n) + b_n \leq \lambda^{1/\delta} a_n\}$…