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.
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
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