Empirical Distributions and Goodness of Fit - Non-parametric Statistics - Statistics Previous Year Questions

Let \(X_1, X_2, \ldots, X_n\) 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:
(I) The distribution of \(T_n\) is the same for all continuous underlying distribution functions \(F(x)\).
(II) \(D_n\) converges to 0 almost surely, as \(n \to \infty\).
Which of the above statements is/are true?

Let \( X_1, X_2, ..., X_8 \) be a random sample taken from a distribution with the probability density function \( f_X(x) = \begin{cases} \frac{x}{8}, & 0 < x < 4, \\ 0, & \text{elsewhere}. \end{cases} \) Let \( F_8(x) \) be the empirical distribution function of the sample. If \( \alpha \) is the variance of \( F_8(2) \), then \( 128\alpha \) (in integer) is equal to ________

Let \(\{0.13, 0.12, 0.78, 0.51\}\) be a realization of a random sample of size \(4\) from a population with cumulative distribution function \(F(\cdot)\). Consider testing \[H_0: F = F_0 \quad \text{against} \quad H_1: F \neq F_0,\] where \[F_0(x) = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x < 1 \\ 1 & \text{if } x \geq 1. \end{cases}\] Let \(D\) denote the Kolmogorov-Smirnov test statistic. If \(P(D > 0.669) = 0.01\) under \(H_0\) and \[\psi = \begin{cases} 1 & \text{if } H_0 \text{ is accepted at level } 0.01 \\ 0 & \text{otherwise}, \end{cases}\] then based on the given data, the observed value of \(D + \psi\) (rounded off to two decimal places) equals ______________

For a random sample \(X_1, \ldots, X_n\) \(n \geq 2\), from a population with distribution function \(F_X\), let \(Z_n(x)\) be the proportion of sample values less than or equal to \(x\), \(x \in \mathbb{R}\). Which of the following statements is/are true?