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  "body": "\n## Definition\n\nLet $\\mathcal{P}$ be a [[statistical model|K3AMMSY8DD]], a set of probability distributions on a sample space.\nThe model $\\mathcal{P}$ is *nonparametric* if it is not a [[parametric model|MSC0MR8VMH]].\nThat is, it cannot be indexed in a natural way by a parameter in a subset of $\\R^k$ for a fixed $k$.\n*Nonparametric statistics* is the study of inference in nonparametric models.\nIt also covers *distribution-free* procedures: tests or intervals whose law under the null hypothesis is the same for all distributions in a nonparametric class.\n\n## Remarks\n\n\"Natural\" means that the indexing map has the regularity required of a parametric model.\nAny set of the size of $\\R$ is in bijection with $\\R$, so a bare bijection does not count.\nExample: the set of all distributions on $\\R$ with a continuous distribution function is nonparametric.\nExample: the sign test of $H_0 \\colon$ the median is $0$ is distribution-free.\nUnder $H_0$, the number of positive observations is binomial with parameters $n$ and $1/2$, for every continuous law with median $0$.\nNon-example: the family $\\set{\\mathcal{N}(\\mu, \\sigma^2) : \\mu \\in \\R, \\sigma \u003e 0}$ is parametric, with $k = 2$.\nSemiparametric models mix both kinds, with a finite-dimensional parameter of interest and an infinite-dimensional nuisance part.\n",
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