{
  "id": "2005.06905",
  "version": "v1",
  "published": "2020-05-14T12:12:57.000Z",
  "updated": "2020-05-14T12:12:57.000Z",
  "title": "Optimal Berry-Esséen bound for Maximum likelihood estimation of the drift parameter in $ α$-Brownian bridge",
  "authors": [
    "Khalifa Es-Sebaiy",
    "Jabrane Moustaaid"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Let $T>0,\\alpha>\\frac12$. In the present paper we consider the $\\alpha$-Brownian bridge defined as $dX_t=-\\alpha\\frac{X_t}{T-t}dt+dW_t,~ 0\\leq t< T$, where $W$ is a standard Brownian motion. We investigate the optimal rate of convergence to normality of the maximum likelihood estimator (MLE) for the parameter $ \\alpha $ based on the continuous observation $\\{X_s,0\\leq s\\leq t\\}$ as $t\\uparrow T$. We prove that an optimal rate of Kolmogorov distance for central limit theorem on the MLE is given by $\\frac{1}{\\sqrt{\\vert\\log(T-t)\\vert}}$, as $t\\uparrow T$. First we compute an upper bound and then find a lower bound with the same speed using Corollary 1 and Corollary 2 of \\cite{kp-JVA}, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-14T12:12:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum likelihood estimation",
      "optimal berry-esséen bound",
      "brownian bridge",
      "drift parameter",
      "optimal rate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}