Modern applied sciences (including but not limited to economics, medicine, or engineering) rely heavily on asymptotic theory. It allows applied scientists to make quantitative claims about large populations based on limited samples. Most importantly, the superpower of asymptotic theory is providing a near-perfect (asymptotic) mathematical framework to evaluate and quantify the precision of those claims about the true/unknown populations. Without it, we would mostly be guessing with no reliable foundation to evaluate how good or bad those guesses are.
Here we learn the foundation of asymptotic statistical inference. Note by note.
This note explains how to compute the asymptotic variance of the sample mean step by step. We begin with intuition, then move carefully through the math.
Suppose we observe random variables
$$ X_1, X_2, \dots, X_n $$We assume:
The sample mean is defined as
$$ \bar{X}_n = \frac{1}{n} \sum_{i=1}^n X_i $$The word asymptotic refers to what happens as the sample size $n$ becomes very large. The sample mean converges to the true mean $\mu$, but it still fluctuates slightly. However, if we "zoom in" (in a correct scale), the fluctuation also "converges" to a well-known distribution (Central Limit Theorem). The asymptotic variance characterizes the fluctuation as sample size becomes very large.
We first compute the variance of \(\bar{X}_n\) for a fixed sample size.
$$ \begin{aligned} \mathrm{Var}(\bar{X}_n) &= \mathrm{Var}\left( \frac{1}{n} \sum_{i=1}^n X_i \right) \\ &= \frac{1}{n^2} \, \mathrm{Var}\left( \sum_{i=1}^n X_i \right) \end{aligned} $$Since the \(X_i\) are independent,
$$ \mathrm{Var}\left( \sum_{i=1}^n X_i \right) = \sum_{i=1}^n \mathrm{Var}(X_i) = n\sigma^2 $$Therefore,
$$ \mathrm{Var}(\bar{X}_n) = \frac{\sigma^2}{n} $$
This is the finite-sample variance of the sample mean.
(To be continued.)