{
  "id": "1911.01985",
  "version": "v1",
  "published": "2019-11-05T18:25:15.000Z",
  "updated": "2019-11-05T18:25:15.000Z",
  "title": "Behavior of Fréchet mean and Central Limit Theorems on spheres",
  "authors": [
    "Do Tran"
  ],
  "categories": [
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Jacobi fields are used to compute higher derivatives of the Fr\\'{e}chet function on spheres with absolutely continuous rotationally symmetric probability distribution. Consequences include (i) a practical condition to test if the mode of the symmetric distribution is a local Fr\\'{e}chet mean; (ii) a Central Limit Theorem on spheres with practical assumptions and an explicit limiting distribution; and (iii) an answer to the question of whether the smeary effect can occur on spheres with absolutely continuous and rotationally symmetric distributions: with the method presented here, it can in dimension at least 4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-05T18:25:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F05",
      "62H11"
    ],
    "keywords": [
      "central limit theorem",
      "fréchet mean",
      "symmetric distribution",
      "continuous rotationally symmetric probability distribution",
      "higher derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}