{
  "id": "1812.08619",
  "version": "v1",
  "published": "2018-12-20T14:58:56.000Z",
  "updated": "2018-12-20T14:58:56.000Z",
  "title": "Limitations Of Richardson Extrapolation For Kernel Density Estimation",
  "authors": [
    "Ruben G. Ascoli"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper develops the process of using Richardson Extrapolation to improve the Kernel Density Estimation method, resulting in a more accurate (lower Mean Squared Error) estimate of a probability density function for a distribution of data in $R_d$ given a set of data from the distribution. The method of Richardson Extrapolation is explained, showing how to fix conditioning issues that arise with higher-order extrapolations. Then, it is shown why higher-order estimators do not always provide the best estimate, and it is discussed how to choose the optimal order of the estimate. It is shown that given n one-dimensional data points, it is possible to estimate the probability density function with a mean squared error value on the order of only $n^{-1}\\sqrt{\\ln(n)}$. Finally, this paper introduces a possible direction of future research that could further minimize the mean squared error.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-20T14:58:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62G07"
    ],
    "keywords": [
      "richardson extrapolation",
      "probability density function",
      "limitations",
      "kernel density estimation method",
      "one-dimensional data points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}