{ "cells": [ { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "# 12. Non-parametric Estimation and Testing\n", "\n", "## [Mathematical Statistical and Computational Foundations for Data Scientists](https://lamastex.github.io/scalable-data-science/360-in-525/2018/04/)\n", "\n", "©2018 Raazesh Sainudiin. [Attribution 4.0 International (CC BY 4.0)](https://creativecommons.org/licenses/by/4.0/)" ] }, { "cell_type": "markdown", "metadata": { "deletable": true, "editable": true }, "source": [ "### Topics\n", "\n", "- Non-parametric Estimation\n", "- Glivenko-Cantelli Theorem\n", "- Dvoretsky-Kiefer-Wolfowitz Inequality\n", "- Hypothesis Testing\n", "- Permutation Testing\n", "- Permutation Testing with Shells Data\n", " \n", "\n", "## Inference and Estimation: The Big Picture\n", "\n", "The Big Picture is about inference and estimation, and especially inference and estimation problems where computational techniques are helpful. \n", "\n", "
\n", " | Point estimation | \n", "Set estimation | \n", "
\n",
" Parametric \n", "\n", " | \n",
"\n",
" MLE of finitely many parameters | \n",
"\n",
" Confidence intervals, | \n",
"
\n",
" Non-parametric | \n",
"about to see ... | \n", "about to see ... | \n", "
\n",
" One/Many-dimensional Integrals | \n",
"coming up ... | \n", "coming up ... | \n", "
'true state of nature' | \n", "Do not reject $H_0$ | \n",
"Reject $H_0$ | \n",
"
\n",
" $H_0$ is true \n", " | \n",
"\n",
" OK \n", " | \n",
"\n",
" Type I error \n", " | \n",
"
\n",
" $H_0$ is false \n", " | \n",
"Type II error | \n", "OK | \n", "
Permutation | \n", "$t$ | \n", "$\\mathbf{P}_0(T=t)$ | \n", "
\n", " | \n", " | Probability under Null | \n", "
(52, 54, 58) | \n", "5 | \n", "$\\frac{1}{6}$ | \n", "
(52, 58, 54) | \n", "1 | \n", "$\\frac{1}{6}$ | \n", "
(54, 52, 58) | \n", "5 | \n", "$\\frac{1}{6}$ | \n", "
(54, 58, 52) | \n", "4 | \n", "$\\frac{1}{6}$ | \n", "
(58, 52, 54) | \n", "1 | \n", "$\\frac{1}{6}$ | \n", "
(58, 54, 52) | \n", "4 | \n", "$\\frac{1}{6}$ | \n", "