Let X_1, \ldots , X_ n \stackrel{iid}{\sim } \mathbf{P} for some distribution \, \mathbf{P}\,, and let F denote its cdf. Let F_ n denote the empirical cdf. Then it holds for every t \in \mathbb {R} that
\sqrt{n} (F_ n(t) - F(t)) \xrightarrow [n \to \infty ]{(d)} \mathcal{N}(0, \sigma ^2)
for any fixed t and for some asymptotic variance \sigma ^2.
What theorem implies the above convergence statement?
Central limit theorem.
Law of large numbers.
Glivenko-Cantelli theorem.
unanswered
Which of the following is \sigma ^2? Note that \sigma ^2 is dependent on t.
F(t)
1 - F(t)
\sqrt{F(t)(1 - F(t))}
F(t)(1 - F(t))
unanswered
What is the asymptotic variance \sigma ^2 of F_ n(0) in terms of the values of the cdf F? (Enter F(x) for F(x) for any numerical value x.)
\, \sigma ^2 = \,