Non-parametric Statistics - Statistics Previous Year Questions

Let \(X_1, X_2, X_3\) and \(X_4\) be a random sample from a population having probability density function \(f_0(x) = f(x - \theta), -\infty < x < \infty\), where \(\theta \in (-\infty, \infty)\) and \(f(-x) = f(x)\), for all \(x \in (-\infty, \infty)\). For testing \(H_0: \theta = 0\) against \(H_1: \theta < 0\), let \(T^+\) denote the Wilcoxon Signed-rank statistic. Then under \(H_0\), \[ 32 \times P(T^+ \le 5) = ____________________ \]

Let \( X_1, X_2, ..., X_{10} \) be a random sample from a probability density function \( f_{\theta}(x) = f(x - \theta), \; -\infty < x < \infty \), where \( -\infty < \theta < \infty \) and \( f(-x) = f(x) \) for \( -\infty < x < \infty \). For testing \( H_0: \theta = 1.2 \) against \( H_1: \theta \neq 1.2 \), let \( T^+ \) denote the Wilcoxon Signed-rank test statistic. If \( \eta \) denotes the probability of the event \( \{T^+ < 50\} \) under \( H_0 \), then \( 32 \eta \) equals ________ (round off to 2 decimal places).

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 \(\{0.13, 0.12, 0.78, 0.51\}\) 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 ______________

Let \(\{1, 6, 5, 3\}\) and \(\{11, 7, 15, 4\}\) be realizations of two independent random samples of size 4 from two separate populations having cumulative distribution functions \(F(\cdot)\) and \(G(\cdot)\), respectively, and probability density functions \(f(\cdot)\) and \(g(\cdot)\), respectively. To test \(H_0: F(t) = G(t)\) for all \(t\), against \(H_1: F(t) \ge G(t)\) with strict inequality for some \(t\), let \(U_{MW}\) denote the Mann-Whitney \(U\)-test statistic. Let, under \(H_0\), \(P(U_{MW} > 12) \le 0.10\), \(P(U_{MW} > 14) \le 0.05\), \(P(U_{MW} > 15) \le 0.025\) and \(P(U_{MW} > 16) \le 0.01\). Then, based on the given data, which of the following statements is/are true?

Let \((1, 3), (2, 4), (7, 8)\) be three independent observations. Then the sample Spearman rank correlation coefficient based on the above observations is __________ (rounded off to two decimal places).