Rank, Sign and Distribution-free Tests - 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).

Two independent random samples, each of size 7, from two populations yield the following values:

Population 1	18	20	16	20	17	18	14
Population 2	17	18	14	20	14	13	16

If Mann-Whitney \( U \) test is performed at 5% level of significance to test the null hypothesis \( H_0: \) Distributions of the populations are same, against the alternative hypothesis \( H_1: \) Distributions of the populations are not same, then the value of the test statistic \( U \) (in integer) for the given data, is ______

While calculating Spearman's rank correlation coefficient, based on \(n\) observations \(\{(x_i, y_i), i = 1, 2, ..., n\}\) from a paired data, it is found that \(x_i\) are distinct for all \(i \ge 2\), \(x_1 = x_2\), and \(\sum_{i=1}^n d_i^2 = 19.5\), where \(d_i = \text{rank}(x_i) - \text{rank}(y_i)\). Then the minimum possible value of \(n^2 - n\) (in integer) is ______

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).

Let \(X_1, X_2, \ldots, X_5\) be random observations from a continuous distribution. Let \(\theta_p\) be \(p\)-th population quantile. Consider the following hypotheses
\(H_0: \theta_{\frac{1}{2}} = 2.5\) against \(H_1: \theta_{\frac{1}{2}} > 2.5\).
Let \(X_{(r)}\) denote the \(r\)-th order statistic of the given observations. Then which of the following is a critical region of a level 0.05 test?