标题： 基于$t$的置信区间的长度 显示英文标题

标题： On the lengths of $t$-based confidence intervals

摘要： 给定从$N(\theta,\sigma^2)$中获得的$n=mk$ $iid$ 个样本，其中$\theta$和$\sigma^2$未知，我们有两种方法可以构建基于$t$的置信区间来估计$\theta$。 传统的做法是将这些$n$样本视为$n$组，并计算区间。 第二种，较少使用的做法是将它们分为$m$组，每组包含$k$个元素。 对于这种方法，我们计算每组的均值，这些$k$均值可以被视为来自$N(\theta,\sigma^2/k)$的$iid$样本。 我们可以使用这些$k$值来构建基于$t$的置信区间。 直觉告诉我们，在相同的置信水平$1-\alpha$下，第一种方法应该优于第二种方法。 然而，如果我们以置信区间的期望长度来定义“更好”，那么第二种方法更优，因为从第一种方法获得的置信区间的期望长度比从第二种方法获得的要短。 我们的工作从理论上证明了这种直觉。 我们还指出，当每组中的元素相关时，第一种方法变得无效，而第二种方法可以给我们正确的结果。 我们通过解析表达式来说明这一点。 显示英文摘要

摘要： Given $n=mk$ $iid$ samples from $N(\theta,\sigma^2)$ with $\theta$ and $\sigma^2$ unknown, we have two ways to construct $t$-based confidence intervals for $\theta$. The traditional method is to treat these $n$ samples as $n$ groups and calculate the intervals. The second, and less frequently used, method is to divide them into $m$ groups with each group containing $k$ elements. For this method, we calculate the mean of each group, and these $k$ mean values can be treated as $iid$ samples from $N(\theta,\sigma^2/k)$. We can use these $k$ values to construct $t$-based confidence intervals. Intuition tells us that, at the same confidence level $1-\alpha$, the first method should be better than the second one. Yet if we define "better" in terms of the expected length of the confidence interval, then the second method is better because the expected length of the confidence interval obtained from the first method is shorter than the one obtained from the second method. Our work proves this intuition theoretically. We also specify that when the elements in each group are correlated, the first method becomes an invalid method, while the second method can give us correct results. We illustrate this with analytical expressions.