Nonparametric statistics is a type of statistical analysis that makes minimal assumptions about the underlying distribution of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in parametric statistics. Nonparametric statistics can be used for descriptive statistics or statistical inference. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.
Definitions
The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others:
The first meaning of nonparametric involves techniques that do not rely on data belonging to any particular parametric family of probability distributions. These include, among others: Methods which are distribution-free, which do not rely on assumptions that the data are drawn from a given parametric family of probability distributions. Statistics defined to be a function on a sample, without dependency on a parameter. An example is order statistics, which are based on ordinal ranking of observations.
The discussion following is taken from Kendall's Advanced Theory of Statistics.Stuart A., Ord J.K, Arnold S. (1999), Kendall's Advanced Theory of Statistics: Volume 2A—Classical Inference and the Linear Model, sixth edition, §20.2–20.3 (Arnold).
Statistical hypotheses concern the behavior of observable random variables... For example, the hypothesis (a) that a normal distribution has a specified mean and variance is statistical; so is the hypothesis (b) that it has a given mean but unspecified variance; so is the hypothesis (c) that a distribution is of normal form with both mean and variance unspecified; finally, so is the hypothesis (d) that two unspecified continuous distributions are identical.
It will have been noticed that in the examples (a) and (b) the distribution underlying the observations was taken to be of a certain form (the normal) and the hypothesis was concerned entirely with the value of one or both of its parameters. Such a hypothesis, for obvious reasons, is called parametric.
Hypothesis (c) was of a different nature, as no parameter values are specified in the statement of the hypothesis; we might reasonably call such a hypothesis nonparametric. Hypothesis (d) is also nonparametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed distribution-free. Notwithstanding these distinctions, the statistical literature now commonly applies the label "nonparametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification.
The second meaning of nonparametric involves techniques that do not assume that the structure of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables are typically assumed to belong to parametric distributions, and assumptions about the types of associations among variables are also made. These techniques include, among others: nonparametric regression, which is modeling whereby the structure of the relationship between variables is treated nonparametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals. nonparametric hierarchical Bayesian models, such as models based on the Dirichlet process, which allow the number of latent variables to grow as necessary to fit the data, but where individual variables still follow parametric distributions and even the process controlling the rate of growth of latent variables follows a parametric distribution.
Applications and purpose Nonparametric methods are widely used for studying populations that have a ranked order (such as movie reviews receiving one to five "stars"). The use of nonparametric methods may be necessary when data have a ranking but no clear numerical interpretation, such as when assessing preferences. In terms of levels of measurement, nonparametric methods result in ordinal data.
As nonparametric methods make fewer assumptions, their applicability is much more general than the corresponding parametric methods. In particular, they may be applied in situations where less is known about the application in question. Also, due to the reliance on fewer assumptions, nonparametric methods are more robust.
Nonparametric methods are sometimes considered simpler to use and more robust than parametric methods, even when the assumptions of parametric methods are justified. This is due to their more general nature, which may make them less susceptible to misuse and misunderstanding. Nonparametric methods can be considered a conservative choice, as they will work even when their assumptions are not met, whereas parametric methods can produce misleading results when their assumptions are violated.
The wider applicability and increased robustness of nonparametric tests comes at a cost: in cases where a parametric test's assumptions are met, nonparametric tests have less statistical power. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.
Nonparametric models Nonparametric models differ from parametric models in that the model structure is not specified à priori but is instead determined from data. The term nonparametric is not meant to imply that such models completely lack parameters but that the number and nature of the parameters are flexible and not fixed in advance. Histogram: a simple nonparametric estimate of a probability distribution. Kernel density estimation: method to estimate a probability distribution, often based on local averaging. Smoothing splines: regression method based on splines. Data envelopment analysis: provides efficiency coefficients similar to those obtained by multivariate analysis without any distributional assumption. k-nearest neighbors (kNN): classifies the unseen instance based on the k points in the training set which are nearest to it. Support vector machine (with a Gaussian kernel): a nonparametric large-margin classifier. Method of moments: estimator for a single value, such as the mean or the variance of a distribution.
Nonparametric tests Nonparametric (or distribution-free) inferential statistical methods are mathematical procedures for statistical hypothesis testing which, unlike parametric statistics, make no assumptions about the probability distributions of the variables being assessed. The most frequently used tests include Analysis of similarities Anderson–Darling test: tests whether a sample is drawn from a given distribution Statistical bootstrap methods: estimates the accuracy/sampling distribution of a statistic Chi-squared test Cochran's Q: tests whether k treatments in randomized block designs with 0/1 outcomes have identical effects Cohen's kappa: measures inter-rater agreement for categorical items Friedman two-way analysis of variance (Repeated Measures) by ranks: tests whether k treatments in randomized block designs have identical effects Empirical likelihood Kaplan–Meier: estimates the survival function from lifetime data, modeling censoring Kendall's tau: measures statistical dependence between two variables Kendall's W: a measure between 0 and 1 of inter-rater agreement. Kolmogorov–Smirnov test: tests whether a sample is drawn from a given distribution, or whether two samples are drawn from the same distribution. Kruskal–Wallis one-way analysis of variance by ranks: tests whether > 2 independent samples are drawn from the same distribution. Kuiper's test: tests whether a sample is drawn from a given distribution, sensitive to cyclic variations such as day of the week. Logrank test: compares survival distributions of two right-skewed, censored samples. Mann–Whitney U or Wilcoxon rank sum test: tests whether two samples are drawn from the same distribution, as compared to a given alternative hypothesis. McNemar's test: tests whether, in 2 × 2 contingency tables with a dichotomous trait and matched pairs of subjects, row and column marginal frequencies are equal. Median test: tests whether two samples are drawn from distributions with equal medians. Pitman's permutation test: a statistical significance test that yields exact p values by examining all possible rearrangements of labels. Rank products: detects differentially expressed genes in replicated microarray experiments. Siegel–Tukey test: tests for differences in scale between two groups. Sign test: tests whether matched pair samples are drawn from distributions with equal medians. Spearman's rank correlation coefficient: measures statistical dependence between two variables using a monotonic function. Squared ranks test: tests equality of variances in two or more samples. Tukey–Duckworth test: tests equality of two distributions by using ranks. Wald–Wolfowitz runs test: tests whether the elements of a sequence are mutually independent/random. Wilcoxon signed-rank test: tests whether matched pair samples are drawn from populations with different mean ranks. Universal Linear Fit Identification: A Method Independent of Data, Outliers and Noise Distribution Model and Free of Missing or Removed Data Imputation.
Mathematical Statistics In mathematical statistics, nonparametric models are considered models that do not rely on a parametric assumption of the unknown data distribution (in density estimation problems) or of the regression function (in regression problems). While the goal of any parametric model is the estimation of a finite number of parameters , nonparametric models aim to directly estimate the data distribution/regression function.
For the mathematical analysis, however, parametric and nonparametric approaches fit into the same setting: Assuming that the function that is to be estimated (data distribution or regression function) belongs to a set of functions parametrized by a set , one searches for a (measurable) function that estimates the "true" parameter based on data points . The key difference between parametric and nonparametric approaches is that in the former for some , while in the latter is typically the set of possible target functions itself, for example, the set of continuous functions or differentiable functions.
Relevant questions in this field regard the construction of reasonable estimators, consistency, rates of convergence and their optimality, and adaptive estimation.
Consistency As in parametric statistics, a desirable property for an estimator is that it converges to the target function as the sample size goes to infinity, that is, the approximation error converges to zero. Usually, the approximation is measured in terms of -norm distance between and . Since the estimator is a function of the randomly drawn data , the approximation is a random variable as well, and so we distinguish two different modes of convergence:
Weak consistency: .
Strong consistency: almost surely.
If an estimator is consistent for all square-integrable , then it is called universally consistent.
Many common nonparametric estimators are weakly universally consistent under suitable choice of model hyperparameters, for example, the Nadarya-Watson estimator, kNNs and certain local polynomial estimators.
Minimax optimal rates of convergence A central topic in the statistical analysis of nonparametric estimators is their speed of convergence towards the true target function and whether the speed is optimal, i.e., the convergence is as fast as possible. The most common way to measure the speed of convergence of an estimator is the minimax convergence rate, which considers the expected loss of the estimator in the worst case scenario. Under certain assumptions on the smoothness of , one can show that there is a minimal convergence rates that no estimator can undercut, and so any estimator achieving this minimal rate is called optimal.