{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Maximum Likelihood estimation of a power law"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
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   "source": [
    "using MaximumLikelihoodPower\n",
    "const MLE = MaximumLikelihoodPower;  # make a short name"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The power-law distribution is known as the Pareto distribution. We get $10^6$ samples from the Pareto distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "seed = 12; α = 0.5;\n",
    "data = MLE.Example.makeparetodata(α, seed);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can recover the exponent $\\alpha$ from the data by minimizing the\n",
    "[Kolmogorov-Smirnov](https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test) statistic.\n",
    "We compute the Kolmogorov-Smirnov statistic for $101$ equally-spaced values of $\\alpha$.\n",
    "\n",
    "`scanKS` returns the value that minimizes the statistic and the two bracketing values. The error is $1/1000$, about as good as can be expected with $10^6$ data points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Array{Float64,1}:\n",
       " 0.4991\n",
       " 0.4992\n",
       " 0.4993"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scanKS(data, range(.495, length=101, stop=.505))"
   ]
  }
 ],
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