List of top Non-parametric Statistics Questions

Two independent random samples, each of size 7, from two populations yield the following values: \[ \begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Population 1} & 18 & 20 & 16 & 20 & 17 & 18 & 14 \\ \hline \text{Population 2} & 17 & 18 & 14 & 20 & 14 & 13 & 16 \\ \hline \end{array} \] 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 on $n$ paired observations, it is found that $x_i$ are distinct for all $i\ge 2$, with a single tie $x_1=x_2$, 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 $\,n^3-n\,$ (in integer) is $\;__________$.

Let \( X_1, X_2, \dots, X_8 \) 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 \( X_1, X_2, \dots, X_n \) 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?