Part CCSIR NET December 2023kolmogorov-scaling

Kolmogorov scaling

Let $X_{1}, \dots, X_{n}$ be a random sample from an unknown absolutely continuous CDF F, and $F_{0} a$ specified absolutely continuous CDF. For $H_{0}: F = F_{0}$ vs $H_{1}: F \ne F_{0}$ , consider $T_{1,n}$ = sup_x | $F_{n}(x) - F_{0}(x)$ | and $T_{2,n}$ = n·sup_x | $F_{n}(x) - F_{0}(x)$ |, where $F_{n}$ is the empirical CDF. Which of the following statements are true?

A. $T_{1,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$ ✓ Correct
B. $T_{2,n}$ → 0 in probability as $n \to \infty$ under $H_{0}$
C.lim P_F( $T_{2,n}$ > 1) = 1 for all F✓ Correct
D. $T_{2,n}$ converges in distribution to a degenerate random variable under $H_{0}$

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests moments and tails.

See pricing

50 are analysed free — try those first.

The trap it tests

Moments and tails

A moment assumed to exist, or tail behaviour assumed to be tame.

Drill statements like this

Kolmogorov scaling

More on this topic