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<title>Nonparametric statistics</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Nonparametric statistics</span></span>

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<p><b>Nonparametric statistics</b> is a type of statistical analysis that makes minimal assumptions about the underlying <a href="Distribution_(mathematics)" title="Distribution (mathematics)">distribution</a> of the data being studied. Often these models are infinite-dimensional, rather than finite dimensional, as in <a href="Parametric_statistics" title="Parametric statistics">parametric statistics</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Nonparametric statistics can be used for <a href="Descriptive_statistics" title="Descriptive statistics">descriptive statistics</a> or <a href="Statistical_inference" title="Statistical inference">statistical inference</a>. Nonparametric tests are often used when the assumptions of parametric tests are evidently violated.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2></div>

<p>The term "nonparametric statistics" has been defined imprecisely in the following two ways, among others:

</p><p>The first meaning of <i>nonparametric</i> involves techniques that do not rely on data belonging to any particular parametric family of probability distributions.

These include, among others:

</p>

<ul><li>Methods which are <i>distribution-free</i>, which do not rely on assumptions that the data are drawn from a given parametric family of <a href="Probability_distributions" class="mw-redirect" title="Probability distributions">probability distributions</a>.</li>

<li>Statistics defined to be a function on a sample, without dependency on a <a href="Parameter" title="Parameter">parameter</a>.</li></ul>

<p>An example is <a href="Order_statistic" title="Order statistic">Order statistics</a>, which are based on <a href="Ranking#Ordinal_ranking_("1234"_ranking)" title="Ranking">ordinal ranking</a> of observations.

</p><p>The discussion following is taken from <i>Kendall's Advanced Theory of Statistics</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<p>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.

</p><p>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 <i>parametric</i>.

</p><p>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 <i>non-parametric</i>. Hypothesis (d) is also non-parametric but, in addition, it does not even specify the underlying form of the distribution and may now be reasonably termed <i>distribution-free</i>. Notwithstanding these distinctions, the statistical literature now commonly applies the label "non-parametric" to test procedures that we have just termed "distribution-free", thereby losing a useful classification.

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<p>The second meaning of <i>non-parametric</i> involves techniques that do not assume that the <i>structure</i> of a model is fixed. Typically, the model grows in size to accommodate the complexity of the data. In these techniques, individual variables <i>are</i> typically assumed to belong to parametric distributions, and assumptions about the types of associations among variables are also made. These techniques include, among others:

</p>

<ul><li><i><a href="Nonparametric_regression" title="Nonparametric regression">non-parametric regression</a></i>, which is modeling whereby the structure of the relationship between variables is treated non-parametrically, but where nevertheless there may be parametric assumptions about the distribution of model residuals.</li>

<li><i>non-parametric hierarchical Bayesian models</i>, such as models based on the <a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a>, which allow the number of <a href="Latent_variables" class="mw-redirect" title="Latent variables">latent variables</a> 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.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Applications_and_purpose">Applications and purpose</h2></div>

<p>Non-parametric methods are widely used for studying populations that have a ranked order (such as movie reviews receiving one to five "stars"). The use of non-parametric methods may be necessary when data have a <a href="Ranking" title="Ranking">ranking</a> but no clear <a href="Number" title="Number">numerical</a> interpretation, such as when assessing <a href="Preferences" class="mw-redirect" title="Preferences">preferences</a>. In terms of <a href="Level_of_measurement" title="Level of measurement">levels of measurement</a>, non-parametric methods result in <a href="Ordinal_data" title="Ordinal data">ordinal data</a>.

</p><p>As non-parametric 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, non-parametric methods are more <a href="Robust_statistics#Introduction" title="Robust statistics">robust</a>.

</p><p>Non-parametric 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. Non-parametric 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.

</p><p>The wider applicability and increased <a href="Robust_statistics" title="Robust statistics">robustness</a> of non-parametric tests comes at a cost: in cases where a parametric test's assumptions are met, non-parametric tests have less <a href="Statistical_power" class="mw-redirect" title="Statistical power">statistical power</a>. In other words, a larger sample size can be required to draw conclusions with the same degree of confidence.

</p>

<div class="mw-heading mw-heading2"><h2 id="Non-parametric_models">Non-parametric models</h2></div>

<p><i>Non-parametric models</i> differ from <a href="Parametric_statistics" title="Parametric statistics">parametric</a> models in that the model structure is not specified <i>a priori</i> but is instead determined from data. The term <i>non-parametric</i> 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.

</p>

<ul><li>A <a href="Histogram" title="Histogram">histogram</a> is a simple nonparametric estimate of a probability distribution.</li>

<li><a href="Kernel_density_estimation" title="Kernel density estimation">Kernel density estimation</a> is another method to estimate a probability distribution.</li>

<li><a href="Nonparametric_regression" title="Nonparametric regression">Nonparametric regression</a> and <a href="Semiparametric_regression" title="Semiparametric regression">semiparametric regression</a> methods have been developed based on <a href="Kernel_(statistics)" title="Kernel (statistics)">kernels</a>, <a href="Spline_(mathematics)" title="Spline (mathematics)">splines</a>, and <a href="Wavelet" title="Wavelet">wavelets</a>.</li>

<li><a href="Data_envelopment_analysis" title="Data envelopment analysis">Data envelopment analysis</a> provides efficiency coefficients similar to those obtained by <a href="Multivariate_analysis" class="mw-redirect" title="Multivariate analysis">multivariate analysis</a> without any distributional assumption.</li>

<li><a href="K-nearest_neighbors_algorithm" title="K-nearest neighbors algorithm">KNNs</a> classify the unseen instance based on the K points in the training set which are nearest to it.</li>

<li>A <a href="Support_vector_machine" title="Support vector machine">support vector machine</a> (with a Gaussian kernel) is a nonparametric large-margin classifier.</li>

<li>The <a href="Method_of_moments_(statistics)" title="Method of moments (statistics)">method of moments</a> with polynomial probability distributions.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Methods">Methods</h2></div>

<p><b>Non-parametric</b> (or <b>distribution-free</b>) <b>inferential statistical methods</b> are mathematical procedures for statistical hypothesis testing which, unlike <a href="Parametric_statistics" title="Parametric statistics">parametric statistics</a>, make no assumptions about the <a href="Probability_distribution" title="Probability distribution">probability distributions</a> of the variables being assessed. The most frequently used tests include

</p>

<ul><li><a href="Analysis_of_similarities" title="Analysis of similarities">Analysis of similarities</a></li>

<li><a href="Anderson%E2%80%93Darling_test" title="Anderson–Darling test">Anderson–Darling test</a>: tests whether a sample is drawn from a given distribution</li>

<li><a href="Bootstrapping_(statistics)" title="Bootstrapping (statistics)">Statistical bootstrap methods</a>: estimates the accuracy/sampling distribution of a statistic</li>

<li><a href="Cochran's_Q_test" title="Cochran's Q test">Cochran's Q</a>: tests whether <i>k</i> treatments in randomized block designs with 0/1 outcomes have identical effects</li>

<li><a href="Cohen's_kappa" title="Cohen's kappa">Cohen's kappa</a>: measures inter-rater agreement for categorical items</li>

<li><a href="Friedman_test" title="Friedman test">Friedman two-way analysis of variance (Repeated Measures)</a> by ranks: tests whether <i>k</i> treatments in randomized block designs have identical effects</li>

<li><a href="Empirical_likelihood" title="Empirical likelihood">Empirical likelihood</a></li>

<li><a href="Kaplan%E2%80%93Meier_estimator" title="Kaplan–Meier estimator">Kaplan–Meier</a>: estimates the survival function from lifetime data, modeling censoring</li>

<li><a href="Kendall_tau_rank_correlation_coefficient" class="mw-redirect" title="Kendall tau rank correlation coefficient">Kendall's tau</a>: measures statistical dependence between two variables</li>

<li><a href="Kendall's_W" title="Kendall's W">Kendall's W</a>: a measure between 0 and 1 of inter-rater agreement.</li>

<li><a href="Kolmogorov%E2%80%93Smirnov_test" title="Kolmogorov–Smirnov test">Kolmogorov–Smirnov test</a>: tests whether a sample is drawn from a given distribution, or whether two samples are drawn from the same distribution.</li>

<li><a href="Kruskal%E2%80%93Wallis_one-way_analysis_of_variance" class="mw-redirect" title="Kruskal–Wallis one-way analysis of variance">Kruskal–Wallis one-way analysis of variance</a> by ranks: tests whether > 2 independent samples are drawn from the same distribution.</li>

<li><a href="Kuiper's_test" title="Kuiper's test">Kuiper's test</a>: tests whether a sample is drawn from a given distribution, sensitive to cyclic variations such as day of the week.</li>

<li><a href="Logrank_test" title="Logrank test">Logrank test</a>: compares survival distributions of two right-skewed, censored samples.</li>

<li><a href="Mann%E2%80%93Whitney_U" class="mw-redirect" title="Mann–Whitney U">Mann–Whitney U</a> or Wilcoxon rank sum test: tests whether two samples are drawn from the same distribution, as compared to a given alternative hypothesis.</li>

<li><a href="McNemar's_test" title="McNemar's test">McNemar's test</a>: tests whether, in 2 × 2 contingency tables with a dichotomous trait and matched pairs of subjects, row and column marginal frequencies are equal.</li>

<li><a href="Median_test" title="Median test">Median test</a>: tests whether two samples are drawn from distributions with equal medians.</li>

<li><a href="Pitman_permutation_test" class="mw-redirect" title="Pitman permutation test">Pitman's permutation test</a>: a statistical significance test that yields exact <i>p</i> values by examining all possible rearrangements of labels.</li>

<li><a href="Rank_product" title="Rank product">Rank products</a>: detects differentially expressed genes in replicated microarray experiments.</li>

<li><a href="Siegel%E2%80%93Tukey_test" title="Siegel–Tukey test">Siegel–Tukey test</a>: tests for differences in scale between two groups.</li>

<li><a href="Sign_test" title="Sign test">Sign test</a>: tests whether matched pair samples are drawn from distributions with equal medians.</li>

<li><a href="Spearman's_rank_correlation_coefficient" title="Spearman's rank correlation coefficient">Spearman's rank correlation coefficient</a>: measures statistical dependence between two variables using a monotonic function.</li>

<li><a href="Squared_ranks_test" title="Squared ranks test">Squared ranks test</a>: tests equality of variances in two or more samples.</li>

<li><a href="Tukey%E2%80%93Duckworth_test" title="Tukey–Duckworth test">Tukey–Duckworth test</a>: tests equality of two distributions by using ranks.</li>

<li><a href="Wald%E2%80%93Wolfowitz_runs_test" title="Wald–Wolfowitz runs test">Wald–Wolfowitz runs test</a>: tests whether the elements of a sequence are mutually independent/random.</li>

<li><a href="Wilcoxon_signed-rank_test" title="Wilcoxon signed-rank test">Wilcoxon signed-rank test</a>: tests whether matched pair samples are drawn from populations with different mean ranks.</li>

<li>Universal Linear Fit Identification: A Method Independent of Data, Outliers and Noise Distribution Model and Free of Missing or Removed Data Imputation.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>Early nonparametric statistics include the <a href="Median" title="Median">median</a> (13th century or earlier, use in estimation by <a href="Edward_Wright_(mathematician)" title="Edward Wright (mathematician)">Edward Wright</a>, 1599; see <a href="Median#History" title="Median">Median § History</a>) and the <a href="Sign_test" title="Sign test">sign test</a> by <a href="John_Arbuthnot" title="John Arbuthnot">John Arbuthnot</a> (1710) in analyzing the <a href="Human_sex_ratio" title="Human sex ratio">human sex ratio</a> at birth (see <a href="Sign_test#History" title="Sign test">Sign test § History</a>).<sup id="cite_ref-Conover1999_5-0" class="reference"><a href="#cite_note-Conover1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Sprent1989_6-0" class="reference"><a href="#cite_note-Sprent1989-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="CDF-based_nonparametric_confidence_interval" title="CDF-based nonparametric confidence interval">CDF-based nonparametric confidence interval</a></li>

<li><a href="Parametric_statistics" title="Parametric statistics">Parametric statistics</a></li>

<li><a href="Resampling_(statistics)" title="Resampling (statistics)">Resampling (statistics)</a></li>

<li><a href="Semiparametric_model" title="Semiparametric model">Semiparametric model</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><cite class="citation journal cs1"><span class="id-lock-subscription" title="Paid subscription required"><a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/0-387-30623-4">"All of Nonparametric Statistics"</a></span>. <i>Springer Texts in Statistics</i>. 2006. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F0-387-30623-4">10.1007/0-387-30623-4</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-25145-5</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFPearceDerrick2019" class="citation journal cs1">Pearce, J; Derrick, B (2019). <a rel="nofollow" class="external text" href="https://doi.org/10.31273%2Freinvention.v12i2.339">"Preliminary testing: The devil of statistics?"</a>. <i>Reinvention: An International Journal of Undergraduate Research</i>. <b>12</b> (2). <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.31273%2Freinvention.v12i2.339">10.31273/reinvention.v12i2.339</a></span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Stuart A., Ord J.K, Arnold S. (1999), <i>Kendall's Advanced Theory of Statistics: Volume 2A—Classical Inference and the Linear Model</i>, sixth edition, §20.2–20.3 (<a href="Edward_Arnold_(publisher)" title="Edward Arnold (publisher)">Arnold</a>).</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFAdikaramHusseinEffenbergerBecker2015" class="citation journal cs1">Adikaram, K. K. L. B.; Hussein, M. A.; Effenberger, M.; Becker, T. (16 November 2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4646355">"Universal Linear Fit Identification: A Method Independent of Data, Outliers and Noise Distribution Model and Free of Missing or Removed Data Imputation"</a>. <i>PLOS ONE</i>. <b>10</b> (11): e0141486. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2015PLoSO..1041486A">2015PLoSO..1041486A</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1371%2Fjournal.pone.0141486">10.1371/journal.pone.0141486</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1932-6203">1932-6203</a>. <a href="PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4646355">4646355</a></span>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26571035">26571035</a>.</cite></span>

</li>

<li id="cite_note-Conover1999-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Conover1999_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFConover1999" class="citation cs2">Conover, W.J. (1999), "Chapter 3.4: The Sign Test", <i>Practical Nonparametric Statistics</i> (Third ed.), Wiley, pp. <span class="nowrap">157–</span>176, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-16068-7</bdi></cite></span>

</li>

<li id="cite_note-Sprent1989-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sprent1989_6-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFSprent1989" class="citation cs2">Sprent, P. (1989), <i>Applied Nonparametric Statistical Methods</i> (Second ed.), Chapman & Hall, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-412-44980-3</bdi></cite></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="General_references">General references</h2></div>

<ul><li>Bagdonavicius, V., Kruopis, J., Nikulin, M.S. (2011). "Non-parametric tests for complete data", ISTE & WILEY: London & Hoboken. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84821-269-5</bdi>.</li>

<li><cite id="CITEREFCorderForeman2014" class="citation book cs1">Corder, G. W.; Foreman, D. I. (2014). <i>Nonparametric Statistics: A Step-by-Step Approach</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-118-84031-3</bdi>.</cite></li>

<li><a href="Jean_D._Gibbons" title="Jean D. Gibbons">Gibbons, Jean Dickinson</a>; Chakraborti, Subhabrata (2003). <i>Nonparametric Statistical Inference</i>, 4th Ed. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8247-4052-1</bdi>.</li>

<li><cite id="CITEREFHettmanspergerMcKean1998" class="citation book cs1">Hettmansperger, T. P.; McKean, J. W. (1998). <i>Robust Nonparametric Statistical Methods</i>. Kendall's Library of Statistics. Vol. 5. London: <a href="Edward_Arnold_(publisher)" title="Edward Arnold (publisher)">Edward Arnold</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-340-54937-8</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1604954">1604954</a>.</cite> also <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-19479-4</bdi>.</li>

<li>Hollander M., Wolfe D.A., Chicken E. (2014). <i>Nonparametric Statistical Methods</i>, John Wiley & Sons.</li>

<li>Sheskin, David J. (2003) <i>Handbook of Parametric and Nonparametric Statistical Procedures</i>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58488-440-1</bdi></li>

<li><a href="Larry_A._Wasserman" title="Larry A. Wasserman">Wasserman, Larry</a> (2007). <i>All of Nonparametric Statistics</i>, Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-25145-6</bdi>.</li></ul>

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