Nonparametric statistics

Nonparametric statistics comprises methods that avoid specifying a fixed parametric form for a population distribution or statistical model. Such methods often rely on ranks, resampling, or flexible estimates instead.

Statistical model: A mathematical description of how data are generated, including assumptions about parameters, distributions, or relationships. Nonparametric methods still use models, but avoid fixing a finite-dimensional parametric form.
Mann–Whitney U test: A rank-based test that compares two independent samples through their relative ordering. It replaces a two-sample mean comparison with a test based on pooled ranks.
Parametric statistics: Statistical methods that represent populations or relationships with models indexed by a finite set of parameters. It is the main counterpart: parametric inference gains structure by specifying a model form.
Nonparametric regression: Regression methods that estimate relationships without restricting them to a fixed finite-dimensional functional form. It applies the nonparametric idea to estimating how outcomes vary with predictors.
Curse of dimensionality: The rapid growth in data or sampling requirements as the number of variables increases. Flexible estimation becomes difficult when smoothing must cover many dimensions.
Probability distribution: A function describing the probabilities or probability densities of possible outcomes of a random variable. Avoiding a specified distribution is central to many nonparametric procedures.
Wilcoxon signed-rank test: A test of paired differences that ranks their absolute magnitudes and accounts for their signs. It offers a paired-sample alternative that does not require normally distributed differences.
Semiparametric statistics: Methods combining a finite-dimensional parametric component with an unspecified or infinite-dimensional component. It occupies the middle ground between fully specified parametric models and nonparametric approaches.
Generalized additive model: A regression model in which predictors contribute through smooth functions combined through a link function. Smooth terms provide flexible effects while retaining an interpretable model structure.
Bandwidth selection: The choice of smoothing scale that controls the bias and variability of a nonparametric estimator. Kernel methods can change substantially with this tuning choice.

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