Exploring functional CLT confidence intervals for a population mean in the domain of attraction of the normal law
Let \((X_n)_{n\in \mathbb N}\) be an independent and identically distributed sequence of random variables that belongs to the domain of attraction of the normal law, i.e., such that there exist constants \(a_n\in \mathbb R\) and \(b_n>0\) such that \(b_n^{-1} (\sum_{i=1}^n X_i - a_n)\) converges in distribution to a standard normal random variable. The classical \((1-\alpha)\)-confidence interval \(I_0\) for the expectation \(\mu\) of \(X_1\) is then given by \(I_0 := [\overline{X}_n - n^{-1} s_n z_{\alpha/2}, \overline{X}_n + n^{-1} s_n z_{\alpha/2}]\), where \(\overline{X}_n\) is the sample mean of \(X_1,\ldots, X_n\), \(s_n^2\) the sample variance and \(z_{\alpha/2}\) the \(1-\alpha/2\)-quantile of the standard normal distribution. Based on functional central limit theorems, the authors consider five other confidence intervals \(I_1,\ldots, I_5\) for \(\mu\), e.g., \[ I_2 := \left[ \frac{2}{n-1} \left( \frac{1}{n} \sum_{k=1}^{n-1} \sum_{i=1}^k X_i - z_{\alpha/2} s_n \sqrt{n/3} \right), \frac{2}{n-1} \left( \frac{1}{n} \sum_{k=1}^{n-1} \sum_{i=1}^k X_i + z_{\alpha/2} s_n \sqrt{n/3}\right) \right], \] and compare them empirically to \(I_0\); unlike for \(I_0\) and \(I_2\), the lengths of the other confidence intervals may also be random. It turns out that in all considered cases, \(I_1, \ldots, I_4\) have a longer expected length than \(I_0\) but higher coverage probabilities than \(I_0\), while \(I_5\) has a smaller expected length then \(I_0\) and a smaller coverage probability.
scientific article; zbMATH DE number 6806062