Short answers, pulled from the story.
Nonparametric statistics makes minimal assumptions about the underlying distribution of the data, whereas parametric statistics assumes data belong to a specific parametric family of distributions with a fixed number of parameters. Nonparametric models can grow in size and complexity to fit the data, rather than being specified in advance.
Commonly used nonparametric tests include the Kolmogorov-Smirnov test, the Mann-Whitney U test, the Kruskal-Wallis one-way analysis of variance by ranks, the Kaplan-Meier survival estimator, Spearman's rank correlation coefficient, and the Wilcoxon signed-rank test.
John Arbuthnot introduced the sign test in 1710, using it to analyze the human sex ratio at birth. The test checks whether matched pair samples are drawn from distributions with equal medians.
Nonparametric tests have less statistical power than parametric tests when the assumptions of the parametric test are genuinely met. This means a larger sample size is required to draw conclusions with the same degree of confidence when using nonparametric methods.
A minimax optimal estimator achieves the fastest possible convergence rate toward the true target function in the worst-case scenario over a class of functions. Under certain smoothness assumptions, a minimal convergence rate exists that no estimator can surpass, and any estimator matching that rate is called minimax optimal.
Nonparametric methods are appropriate when data has a ranked order but no clear numerical interpretation, when the assumptions of parametric tests are evidently violated, or when less is known about the data distribution. They can also be preferred as a conservative choice because they produce valid results even when parametric assumptions fail.