nonparametrics

Nonparametrics, in statistics, refers to methods that do not assume a specific parametric form for the population distribution or for the underlying relationship between variables. These approaches aim to be robust to model misspecification and to be applicable when the true distribution is unknown or difficult to specify. Nonparametric methods can be distribution-free or rely on minimal assumptions about functional form, such as monotonicity or continuity.

Nonparametric inference includes hypothesis tests and confidence intervals that do not depend on a known distribution. Classic examples include sign tests, Wilcoxon signed-rank tests, Mann-Whitney U tests, Kruskal-Wallis tests, and Friedman tests. Correlations measures such as Spearman's rho and Kendall's tau are also rank-based. In many settings, permutation and bootstrap resampling are used to obtain p-values or intervals without strong distributional assumptions.

Nonparametric estimation covers density estimation, regression, and related function estimation. Kernel density estimation and histograms estimate a distribution's shape without assuming a parametric form; kernel regression, locally weighted regression (LOESS), and spline-based methods estimate smooth relationships. Isotonic regression and other shape-constrained techniques impose qualitative features like monotonicity. These methods often require tuning choices (for example, bandwidths) and can be computationally intensive.

Compared with parametric methods, nonparametric techniques offer flexibility and robustness at the cost of potentially lower efficiency when a parametric model is correct and larger sample sizes are needed. They may be less interpretable and increasingly sensitive to data quality and tuning parameters. Nonparametric Bayesian methods are a related area that combines flexible models with probabilistic inference, while bootstrap and permutation tools are common companions used to assess variability in nonparametric procedures.