{ "id": "epxeycbzdf", "slug": "density-estimation", "title": "Density estimation", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics" }, { "id": "physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference" }, { "id": "physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.statistics.stats_statistical_inference.estimation" } ], "geo": [ { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Non-geographic · 1800 to 1945: Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1800", "label": "Non-geographic · 1800 to 1945", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800" }, { "id": "geo.nongeo.t1800.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical" }, { "id": "geo.nongeo.t1800.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math" }, { "id": "geo.nongeo.t1800.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics" }, { "id": "geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1800.physical.math.statistics.stats_statistical_inference" } ] }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Non-geographic · 1946 to 2000: Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.t1946", "label": "Non-geographic · 1946 to 2000", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946" }, { "id": "geo.nongeo.t1946.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical" }, { "id": "geo.nongeo.t1946.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math" }, { "id": "geo.nongeo.t1946.physical.math.statistics", "label": "Statistics and probability", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference", "label": "Statistical inference, estimation, sampling, and testing", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference" }, { "id": "geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation", "label": "Estimation theory and estimator families", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.t1946.physical.math.statistics.stats_statistical_inference.estimation" } ] } ], "excerpt": "Density estimation is the statistical task of recovering a probability density function from observed data, using parametric mixture models or nonparametric methods like kernel density estimation.", "snippet": "Density estimation is the statistical task of recovering a probability density function from observed data, using parametric mixture models or nonparametric methods like kernel density estimation.", "node": "physical.math.statistics.stats_statistical_inference.estimation", "markdown": "# Density estimation\n\nDensity estimation is the statistical task of recovering the probability density function \\( f \\) of a random variable from observed data; nonparametric density estimation does so without assuming a fixed parametric form for \\( f \\). Approaches divide into parametric methods, which fit a family such as a finite mixture model by maximum likelihood, and nonparametric methods, chiefly histograms and kernel density estimation (KDE), which let the data determine the shape of the density.[1](https://ned.ipac.caltech.edu/level5/March02/Silverman/paper.pdf)\n\n| Key fact | Value |\n|---|---|\n| Kernel estimate | \\( \\hat{f}_h(x) = \\frac{1}{n \\cdot h} \\sum_{i=1}^{n} K\\!\\left( \\frac{x - X_i}{h} \\right) \\); the bandwidth \\( h \\) is critical, the kernel shape has little practical impact[2](https://eliassi.org/Sheather_StatSci_2004.pdf) |\n| Silverman's rule of thumb | \\( h_{\\mathrm{SROT}} = 0.9 \\cdot A \\cdot n^{-1/5} \\), with \\( A = \\min\\{ \\text{sample SD}, \\text{sample IQR}/1.34 \\} \\)[2](https://eliassi.org/Sheather_StatSci_2004.pdf) |\n| Bias comparison | Histogram bias is order \\( h \\); a symmetric kernel centered at each data point gives leading bias of order \\( h^2 \\)[2](https://eliassi.org/Sheather_StatSci_2004.pdf) |\n| Minimax squared \\( L_2 \\) error | \\( n^{-4/(4+d)} \\) in \\( d \\) dimensions, versus the parametric rate \\( d/n \\)[3](https://proceedings.mlr.press/v54/mcdonald17a/mcdonald17a.pdf) |\n| Bandwidth-selector convergence | Cross-validation selectors \\( n^{1/10} \\); Sheather–Jones plug-in \\( n^{5/14} \\)[4](https://egarpor.github.io/NP-EAFIT/dens-bwd.html) |\n| Score-debiased KDE (2025) | AMISE \\( O(n^{-8/(d+8)}) \\) versus \\( O(n^{-4/(d+4)}) \\) for standard KDE[5](https://papers.nips.cc/paper_files/paper/2025/file/fd8548947a14f68375d7e6235ab493ec-Paper-Conference.pdf) |\n| Software | scikit-learn `KernelDensity` (six kernels, Ball Tree/KD Tree); R `density()` with `bw.nrd0`, `bw.SJ`; `ks::kde()`; SAS PROC KDE[6](https://scikit-learn.org/stable/modules/density.html) • [7](https://cswr.nrhstat.org/density) • [2](https://eliassi.org/Sheather_StatSci_2004.pdf) |\n\n## How it works\n\nThe kernel estimate places a scaled copy of a smooth function, the kernel \\( K \\), at every observation and averages them: \\( \\hat{f}_h(x) = \\frac{1}{n \\cdot h} \\sum_{i=1}^{n} K\\!\\left( \\frac{x - X_i}{h} \\right) \\). The kernel determines the shape of these bumps and the window width \\( h \\) their width.[1](https://ned.ipac.caltech.edu/level5/March02/Silverman/paper.pdf) Equivalently, the estimate is an equiprobable mixture of \\( n \\) similar-shaped densities centered at the data points.[8](https://luc.devroye.org/Devroye-ACourseInDensityEstimation-1987.pdf)\n\nThe bandwidth dominates the kernel choice. Published comparisons agree that the value of \\( h \\) is of critical importance while the kernel's shape has little practical impact; the Gaussian kernel is standard and yields an infinitely differentiable estimate.[2](https://eliassi.org/Sheather_StatSci_2004.pdf) • [7](https://cswr.nrhstat.org/density) The reason is the bias–variance tradeoff. A histogram with bin width \\( h \\) has bias of order \\( h \\); centering a symmetric kernel at each data point cancels that term, giving leading bias of order \\( h^2 \\), while variance behaves as \\( 1/(n \\cdot h) \\).[2](https://eliassi.org/Sheather_StatSci_2004.pdf) The asymptotically optimal mean integrated squared error decomposes as \\( \\mathrm{AMISE}(\\hat{f}(\\cdot;h)) = \\frac{1}{4} \\mu_2(K)^2 R(f'') \\cdot h^4 + \\frac{R(K)}{n \\cdot h} \\), where \\( R(f'') = \\int (f''(x))^2 \\, dx \\) is unknown, so it must be estimated.[4](https://egarpor.github.io/NP-EAFIT/dens-bwd.html)\n\n## How it is done\n\n1. Choose the route: histogram for a quick view, KDE for a smooth estimate, or a mixture model when a compact parametric summary is needed.\n2. Pick a kernel (Gaussian is the default) and a bandwidth. Silverman's rule of thumb, \\( h_{\\mathrm{SROT}} = 0.9 \\cdot A \\cdot n^{-1/5} \\) with \\( A = \\min\\{ \\text{sample SD}, \\text{sample IQR}/1.34 \\} \\), is R's default via `bw.nrd0()`; the plain normal-reference rule \\( h = 1.06 \\cdot S \\cdot n^{-1/5} \\) was found to be usually unacceptably large, and Silverman's rule is known to oversmooth multimodal densities.[2](https://eliassi.org/Sheather_StatSci_2004.pdf) • [7](https://cswr.nrhstat.org/density)\n3. For data-driven selection, use least-squares cross-validation (minimizing a leave-one-out criterion), biased cross-validation, or a plug-in selector. The Sheather–Jones direct plug-in converges at rate \\( n^{5/14} \\), much faster than cross-validation selectors at \\( n^{1/10} \\), and is regarded as a solid default.[4](https://egarpor.github.io/NP-EAFIT/dens-bwd.html) • [9](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1991.tb01857.x) • [2](https://eliassi.org/Sheather_StatSci_2004.pdf)\n4. Handle boundaries if the support is bounded: options include doing nothing, log-transforming the data, data reflection, boundary kernels, local likelihood, or dividing by the edge correction \\( w_h(x_i) = \\int_a^b K_h(x_i - x) \\, dx \\).[10](https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf) • [11](https://mdporter.github.io/SYS6018/lectures/density.pdf)\n5. Compute and validate. In R, `density()` with `bw = \"sj\"` gives the Sheather–Jones selector, also available as `bw.SJ()`; the `ks` package's `kde()` supports multivariate KDE with selectors `hpi()`, `hlscv()`, `hscv()`, and `hucv()`.[7](https://cswr.nrhstat.org/density) • [11](https://mdporter.github.io/SYS6018/lectures/density.pdf) scikit-learn's `KernelDensity` implements gaussian, tophat, epanechnikov, exponential, linear, and cosine kernels, uses Ball Tree or KD Tree for efficient queries, and accepts Scott's or Silverman's bandwidth or a manual value.[6](https://scikit-learn.org/stable/modules/density.html)\n\n## Origin\n\nThe first published paper dealing explicitly with density estimation was [Murray Rosenblatt](https://www.edgepedia.org/murray-rosenblatt)'s \"Remarks on Some Nonparametric Estimates of a Density Function\" (The Annals of Mathematical Statistics, 1956), which discussed both the naive and the general kernel estimator.[12](https://doi.org/10.1214/aoms/1177728190) Emanuel Parzen's \"On Estimation of a Probability Density Function and Mode\" (The Annals of Mathematical Statistics, 1962) gave KDE the form used today, and the method is also called the Parzen–Rosenblatt window method.[13](https://doi.org/10.1214/aoms/1177704472) • [7](https://cswr.nrhstat.org/density) Later building blocks followed: the nearest-neighbor estimator of D. O. Loftsgaarden and C. P. Quesenberry (The Annals of Mathematical Statistics, 1965),[14](https://doi.org/10.1214/aoms/1177700079) the variable kernel method of [Leo Breiman](https://www.edgepedia.org/leo-breiman), William Meisel, and Edward Purcell (Technometrics, 1977),[15](https://doi.org/10.1080/00401706.1977.10489521) Ian S. Abramson's square-root-law variable bandwidth (The [Annals of Statistics](https://www.edgepedia.org/annals-of-statistics), 1982),[16](https://doi.org/10.1214/aos/1176345986) and Bernard W. Silverman's monograph *Density Estimation for Statistics and Data Analysis* (1986), the source of the rule of thumb bearing his name, which was reviewed in Technometrics (1987).[1](https://ned.ipac.caltech.edu/level5/March02/Silverman/paper.pdf) • [17](https://doi.org/10.2307/1269475)\n\n## Variants\n\n**Adaptive bandwidths.** M.C. Jones (Australian Journal of Statistics, 1990) distinguished two meanings of \"variable kernel density estimate\": a different bandwidth for each data point (the sample-point form) versus a bandwidth that is a function of the estimation location (the balloon form).[18](https://doi.org/10.1111/j.1467-842x.1990.tb01031.x) Abramson's square-root law sets the bandwidth at each sample point inversely proportional to the square root of a pilot density estimate there.[16](https://doi.org/10.1214/aos/1176345986) • [19](https://hal.science/hal-03775536v1/document)\n\n**Boundary and conditional methods.** Jonathan C. Marshall and Martin L. Hazelton (Journal of Multivariate Analysis, 2009) proposed a linear boundary kernel that reduces the asymptotic boundary bias of adaptive estimators and works on irregular boundaries.[20](https://doi.org/10.1016/j.jmva.2009.09.003) B. U. Park and colleagues (Journal of Nonparametric Statistics, 2003) introduced adaptive variable-location estimators achieving bias improvement by an order of magnitude at boundaries as well as higher-order interior bias.[21](https://doi.org/10.1080/10485250306041) The diffusion estimator improves asymptotic accuracy, handles domain boundaries naturally, adapts its smoothing to the data, and always yields a bona fide density.[22](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0259111)\n\n**Score-debiased KDE.** SD-KDE, reported by Elliot L. Epstein and colleagues in 2025, shifts each data point by one step along an estimated score function, then performs KDE with a modified bandwidth, achieving AMISE of order \\( O(n^{-8/(d+8)}) \\) instead of the \\( O(n^{-4/(d+4)}) \\) of standard KDE.[5](https://papers.nips.cc/paper_files/paper/2025/file/fd8548947a14f68375d7e6235ab493ec-Paper-Conference.pdf) • [23](https://doi.org/10.48550/arxiv.2504.19084) It works with empirical scores obtained from a vanilla KDE via the gradient of the log of the estimate, requires no learned diffusion model, and can be viewed as a one-step Euler–Maruyama discretization of [Langevin dynamics](https://www.edgepedia.org/langevin-dynamics); its guarantees assume an exact score oracle, typically unavailable in practice.[5](https://papers.nips.cc/paper_files/paper/2025/file/fd8548947a14f68375d7e6235ab493ec-Paper-Conference.pdf)\n\n## Applications\n\nThe resulting estimate supports regression, classification, clustering, unsupervised prediction, and anomaly detection, and nonparametric generative modeling in which new samples are drawn from the fitted density.[10](https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf) • [6](https://scikit-learn.org/stable/modules/density.html) [Diffusion](https://www.edgepedia.org/diffusion) models can be regarded as an implicit approach to nonparametric density estimation: no direct estimator of \\( p_0 \\) is constructed, but the estimated score function enables sampling via score-based MCMC.[24](https://www.jmlr.org/papers/volume27/25-0121/25-0121.pdf) • [25](https://arxiv.org/pdf/2408.05807) For visualization, the mode tree of Michael C. Minnotte and David W. Scott (Journal of Computational and Graphical Statistics, 1993) summarizes how modes appear and merge across bandwidths.[26](https://doi.org/10.1080/10618600.1993.10474599)\n\n## Limitations and alternatives\n\n**Failure modes.** Near a boundary of the sample space, an uncorrected KDE has nonvanishing pointwise bias, typically of order \\( O(1) \\), and the density is underestimated; fixes include reflection, transformations, boundary kernels, and local likelihood.[10](https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf) • [27](https://iis.uibk.ac.at/public/piater/courses/IIS/modules/KDE/KDE.notes.pdf) For long-tailed distributions, a fixed bandwidth produces spurious noise in the tails, and smoothing enough to remove it masks essential detail in the main part of the distribution, motivating adaptive bandwidths.[1](https://ned.ipac.caltech.edu/level5/March02/Silverman/paper.pdf)\n\n**Computation.** Naive evaluation at \\( m \\) points for \\( n \\) samples costs \\( O(m \\cdot n) \\) kernel evaluations, \\( O(n^2) \\) when evaluation points coincide with data points, and bandwidth-selection objectives must be evaluated many times during optimization.[28](https://ar5iv.labs.arxiv.org/html/1511.07482) FFT-based computation for Gaussian kernels and the fastKDE method, which combines self-consistent KDE with nuFFT-based computation and requires no user-specified bandwidth, reduce scaling to linear in the number of samples, \\( O(N \\cdot q^D + M^D \\log(M^D)) \\); storing a \\( D \\)-dimensional grid with \\( M \\) points per dimension requires \\( M^D \\) values.[29](https://escholarship.org/content/qt9g56181p/qt9g56181p.pdf?t=p7qvyp)\n\n**Alternatives.** Histograms are sensitive to bin width, bin origin, and coordinate directions and produce step functions; KDE removes the bin anchor and centers a block on each data point instead.[27](https://iis.uibk.ac.at/public/piater/courses/IIS/modules/KDE/KDE.notes.pdf) • [6](https://scikit-learn.org/stable/modules/density.html) For small samples, finite mixtures suffer kernel collapsing under EM and Parzen windows are preferred; for large datasets, mixtures offer lower computational cost with comparable performance.[30](https://www.atlantis-press.com/article/1564.pdf)\n\n**Convergence.** Rosenblatt showed in 1956 that no nonparametric estimator can be unbiased and that no error measure can achieve the parametric rate \\( O(n^{-1}) \\).[31](https://www.stat.rice.edu/~scottdw/ss.nh.pdf) It takes an exponential-in-dimension number of samples to estimate the density, the curse of dimensionality; under the manifold hypothesis, rates depend on the manifold dimension \\( d_M \\) rather than the ambient dimension, mitigating the problem.[32](https://proceedings.mlr.press/v70/jiang17b/jiang17b.pdf)\n\n## References\n\n1. [Density Estimation for Statistics and Data Analysis (B.W. Silverman, 1986; online edition)](https://ned.ipac.caltech.edu/level5/March02/Silverman/paper.pdf)\n2. [Density Estimation Based on Kernel Methods (Sheather, Statistical Science 2004)](https://eliassi.org/Sheather_StatSci_2004.pdf)\n3. [Minimax density estimation for growing dimension (McDonald et al., ICML 2017)](https://proceedings.mlr.press/v54/mcdonald17a/mcdonald17a.pdf)\n4. [Bandwidth selection | A Short Course on Nonparametric Curve Estimation](https://egarpor.github.io/NP-EAFIT/dens-bwd.html)\n5. [SD-KDE: Score-Debiased Kernel Density Estimation (NeurIPS 2025)](https://papers.nips.cc/paper_files/paper/2025/file/fd8548947a14f68375d7e6235ab493ec-Paper-Conference.pdf)\n6. [2.8. Density Estimation, scikit-learn documentation](https://scikit-learn.org/stable/modules/density.html)\n7. [Chapter 2 Density estimation | Computational Statistics with R](https://cswr.nrhstat.org/density)\n8. [A Course in Density Estimation (L. Devroye, 1987)](https://luc.devroye.org/Devroye-ACourseInDensityEstimation-1987.pdf)\n9. [A Reliable Data-Based Bandwidth Selection Method for Kernel Density Estimation (Sheather & Jones, JRSS-B, 1991)](https://rss.onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1991.tb01857.x)\n10. [Density Estimation (Wasserman, lecture notes, CMU 10-725)](https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf)\n11. [Density Estimation course notes (SYS6018)](https://mdporter.github.io/SYS6018/lectures/density.pdf)\n12. [Murray Rosenblatt (1956). Remarks on Some Nonparametric Estimates of a Density Function. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177728190)\n13. [Emanuel Parzen (1962). On Estimation of a Probability Density Function and Mode. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177704472)\n14. [D. O. Loftsgaarden, C. P. Quesenberry (1965). A Nonparametric Estimate of a Multivariate Density Function. The Annals of Mathematical Statistics.](https://doi.org/10.1214/aoms/1177700079)\n15. [Leo Breiman, William Meisel, Edward Purcell (1977). Variable Kernel Estimates of Multivariate Densities. Technometrics.](https://doi.org/10.1080/00401706.1977.10489521)\n16. [Ian S. Abramson (1982). On Bandwidth Variation in Kernel Estimates-A Square Root Law. The Annals of Statistics.](https://doi.org/10.1214/aos/1176345986)\n17. [Khosrow Dehnad, Bernard Silverman (1987). Density Estimation for Statistics and Data Analysis. Technometrics.](https://doi.org/10.2307/1269475)\n18. [M.C. Jones (1990). VARIABLE KERNEL DENSITY ESTIMATES AND VARIABLE KERNEL DENSITY ESTIMATES. Australian Journal of Statistics.](https://doi.org/10.1111/j.1467-842x.1990.tb01031.x)\n19. [Optimal bandwidth selection for variable kernel density estimation](https://hal.science/hal-03775536v1/document)\n20. [Jonathan C. Marshall, Martin L. Hazelton (2009). Boundary kernels for adaptive density estimators on regions with irregular boundaries. Journal of Multivariate Analysis.](https://doi.org/10.1016/j.jmva.2009.09.003)\n21. [B. U. Park and colleagues (2003). Adaptive variable location kernel density estimators with good performance at boundaries. Journal of nonparametric statistics.](https://doi.org/10.1080/10485250306041)\n22. [Semiparametric maximum likelihood probability density estimation (PLOS One)](https://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0259111)\n23. [Epstein, Elliot L. and colleagues (2025). SD-KDE: Score-Debiased Kernel Density Estimation. arXiv (Cornell University).](https://doi.org/10.48550/arxiv.2504.19084)\n24. [Nonparametric Estimation of a Factorizable Density using Diffusion Models (JMLR, vol. 27)](https://www.jmlr.org/papers/volume27/25-0121/25-0121.pdf)\n25. [Kernel density estimation in high dimension: three statistical regimes (glassy phases) (arXiv, 2024)](https://arxiv.org/pdf/2408.05807)\n26. [Michael C. Minnotte, David W. Scott (1993). The Mode Tree: A Tool for Visualization of Nonparametric Density Features. Journal of Computational and Graphical Statistics.](https://doi.org/10.1080/10618600.1993.10474599)\n27. [Kernel Density Estimation - An Introduction (Piater, Universität Innsbruck)](https://iis.uibk.ac.at/public/piater/courses/IIS/modules/KDE/KDE.notes.pdf)\n28. [FFT-Based Fast Bandwidth Selector for Multivariate Kernel Density Estimation (arXiv 1511.07482)](https://ar5iv.labs.arxiv.org/html/1511.07482)\n29. [A fast and objective multidimensional kernel density estimation method: fastKDE (O'Brien et al., Computational Statistics and Data Analysis, 2016)](https://escholarship.org/content/qt9g56181p/qt9g56181p.pdf?t=p7qvyp)\n30. [A comparative study of various probability density estimation methods for data analysis](https://www.atlantis-press.com/article/1564.pdf)\n31. [Multi-dimensional Density Estimation (D. Scott, textbook chapter)](https://www.stat.rice.edu/~scottdw/ss.nh.pdf)\n32. [Uniform Convergence Rates for Kernel Density Estimation (Jiang, ICML 2017)](https://proceedings.mlr.press/v70/jiang17b/jiang17b.pdf)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Statistical inference, estimation, sampling, and testing › Estimation theory and estimator families*\n\n*Initially written Sep 29, 2026 · Edited: Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [ "https://www.stat.cmu.edu/~larry/=sml/densityestimation.pdf", "https://www.stat.rice.edu/~scottdw/ss.nh.pdf" ], "url": "https://www.edgepedia.org/density-estimation", "markdown_url": "https://www.edgepedia.org/density-estimation.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Density estimation\", Edgepedia (EdgeChat), https://www.edgepedia.org/density-estimation. Edgepedia Community License 1.0.", "credit_md": "\"[Density estimation](https://www.edgepedia.org/density-estimation)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/density-estimation](https://www.edgepedia.org/density-estimation). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Density estimation\", Edgepedia (EdgeChat), https://www.edgepedia.org/density-estimation. Edgepedia Community License 1.0.", "speakable": "Density estimation is the statistical task of recovering a probability density function from observed data, using parametric mixture models or nonparametric methods like kernel density estimation." }