Question

Let \( X_1, X_2, \dots, X_n \) be a random sample from a distribution with cumulative distribution function \( F(x) \). Let the empirical distribution function of the sample be \( F_n(x) \). The classical Kolmogorov-Smirnov goodness of fit test statistic is given by \[ T_n = \sqrt{n} D_n = \sqrt{n} \sup_{-\infty<x<\infty} | F_n(x) - F(x) |. \] Consider the following statements: \begin{enumerate} \item The distribution of \( T_n \) is the same for all continuous underlying distribution functions \( F(x) \). \item \( D_n \) converges to 0 almost surely, as \( n \to \infty \). \end{enumerate} Which one of the following statements is/are true?

• A. (I) only
• B. (II) only
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

The Kolmogorov-Smirnov test statistic is based on the maximum difference between the empirical distribution function and the true distribution function. As the sample size increases, this difference tends to zero almost surely.

Answer 1

The Correct Option is C

Step 1: Analyzing Statement (I).
The Kolmogorov-Smirnov statistic \( T_n \) is a measure of the difference between the empirical distribution function \( F_n(x) \) and the true cumulative distribution function \( F(x) \). For all continuous distributions, the limiting distribution of \( T_n \) is the same. Hence, statement (I) is correct.
Step 2: Analyzing Statement (II).
The statistic \( D_n = \sup_{-\infty<x<\infty} | F_n(x) - F(x) | \) converges to 0 almost surely as \( n \to \infty \). This is a well-known property of the empirical distribution function, and is a result from the Glivenko-Cantelli theorem. Thus, statement (II) is also correct.
Step 3: Conclusion.
Both statements (I) and (II) are true, so the correct answer is (C).