Question

Let \( \{ X_n \}_{n \geq 1} \) be a sequence of i.i.d. random variables with common distribution function \( F \), and let \( F_n \) be the empirical distribution function based on \( \{ X_1, X_2, \dots, X_n \} \). Then, for each fixed \( x \in (-\infty, \infty) \), which one of the following options is correct?

• A. \( \sqrt{n}(F_n(x) - F(x)) \xrightarrow{P} 0 { as } n \to \infty \)
• B. \( \frac{n(F_n(x) - F(x))}{\sqrt{F(x)(1 - F(x))}} \xrightarrow{d} Z { as } n \to \infty, { where } Z \sim N(0,1) \)
• C. \( F_n(x) \xrightarrow{a.s.} F(x) { as } n \to \infty \)
• D. \( \lim_{n \to \infty} n {Var}(F_n(x)) = 0 \)

Hint:

The empirical distribution function \( F_n(x) \) converges almost surely to the true distribution function \( F(x) \), and the Central Limit Theorem provides the asymptotic normality of the difference between \( F_n(x) \) and \( F(x) \).

Answer 1

The Correct Option is C

Step 1: Understanding the empirical distribution function.
The empirical distribution function \( F_n(x) \) is the proportion of data points in the sample \( \{ X_1, X_2, \dots, X_n \} \) that are less than or equal to \( x \), i.e.,
\[ F_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}_{\{ X_i \leq x \}}. \]
Since the \( X_i \)'s are i.i.d., \( F_n(x) \) is a consistent estimator for \( F(x) \), meaning \( F_n(x) \xrightarrow{a.s.} F(x) \) as \( n \to \infty \) (option (C)).
Step 2: Analyzing the asymptotic behavior.
By the Central Limit Theorem (CLT), the difference \( \sqrt{n}(F_n(x) - F(x)) \) converges in distribution to a normal distribution with mean 0 and variance \( F(x)(1 - F(x)) \). This corresponds to option (B), but we are asked for the correct statement for fixed \( x \), so this is not the best choice.
Step 3: Verifying other options.
Option (A) is incorrect because \( \sqrt{n}(F_n(x) - F(x)) \) converges in distribution, not in probability.
Option (C) is correct as \( F_n(x) \) converges almost surely to \( F(x) \) by the Glivenko-Cantelli theorem, a fundamental result in empirical process theory.
Option (D) is incorrect because the variance of \( F_n(x) \) does not tend to zero as \( n \to \infty \).
Thus, the correct answer is \( \boxed{(C)} \).