arXiv 1812.032142018-12-07stat.CO

基于$t$的置信区间的长度

On the lengths of $t$-based confidence intervals

Yu Zhang, Xiangzhong Fang

摘要：

给定来自$N(θ,σ^2)$的$n=mk$ $iid$样本，其中$θ$和$σ^2$未知，我们有两种方法来构建$θ$的基于$t$的置信区间。传统的方法是将这些$n$样本作为$n$组处理并计算间隔。第二种，也是不太常用的方法是将它们分成$m$组，每组包含$k$元素。对于该方法，我们计算每组的平均值，并且这些$k$平均值可以被视为来自$N(θ,σ^2/k)$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$1-α$下，第一种方法应该优于第二种方法。然而，如果我们根据置信区间的预期长度来定义“更好”，那么第二种方法更好，因为从第一种方法获得的置信区间的预期长度比从第二种方法获得的更短。我们的工作从理论上证明了这种直觉。我们还指定，当每个组中的元素是相关的，第一种方法成为无效的方法，而第二种方法可以给我们正确的结果。我们用解析表达式来说明这一点。

英文摘要：

Given $n=mk$ $iid$ samples from $N(θ,σ^2)$ with $θ$ and $σ^2$ unknown, we have two ways to construct $t$-based confidence intervals for $θ$. 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 $N(θ,σ^2/k)$. We can use these $k$ values to construct $t$-based confidence intervals. Intuition tells us that, at the same confidence level $1-α$, 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.

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