Bootstrap assignment.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Assignment 3\n",
    "By Nitin Reddy and Siddharth Sundararajan"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Jackknife function\n",
    "Jackknife<-function(v1,statfunc=sd, alpha = 0.05)\n",
    "{\n",
    "  n1<-length(v1)\n",
    "  jackvec<-NULL\n",
    "  mu0<-statfunc(v1)\n",
    "  for(i in 1:n1){\n",
    "    mua<-statfunc(v1[-i])\n",
    "    jackvec<-c(jackvec, n1*(mu0)-(n1-1)*mua)\n",
    "  }\n",
    "  jackbias<-mean(jackvec)-mu0\n",
    "  jacksd<-sd(jackvec)\n",
    "  JLB<-mean(jackvec)-(jacksd/sqrt(n1))*qnorm(1-alpha/2)\n",
    "  JUB<-mean(jackvec)+(jacksd/sqrt(n1))*qnorm(1-alpha/2)\n",
    "  list(mu0=mu0,jackbias=jackbias,jacksd=jacksd, \n",
    "       jackknife.confidence.interval = c(JLB, JUB))\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$mu0</dt>\n",
       "\t\t<dd>500.5</dd>\n",
       "\t<dt>$jackbias</dt>\n",
       "\t\t<dd>0</dd>\n",
       "\t<dt>$jacksd</dt>\n",
       "\t\t<dd>288.819436095749</dd>\n",
       "\t<dt>$jackknife.confidence.interval</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>482.599114827485</li>\n",
       "\t<li>518.400885172515</li>\n",
       "</ol>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$mu0] 500.5\n",
       "\\item[\\$jackbias] 0\n",
       "\\item[\\$jacksd] 288.819436095749\n",
       "\\item[\\$jackknife.confidence.interval] \\begin{enumerate*}\n",
       "\\item 482.599114827485\n",
       "\\item 518.400885172515\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$mu0\n",
       ":   500.5\n",
       "$jackbias\n",
       ":   0\n",
       "$jacksd\n",
       ":   288.819436095749\n",
       "$jackknife.confidence.interval\n",
       ":   1. 482.599114827485\n",
       "2. 518.400885172515\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$mu0\n",
       "[1] 500.5\n",
       "\n",
       "$jackbias\n",
       "[1] 0\n",
       "\n",
       "$jacksd\n",
       "[1] 288.8194\n",
       "\n",
       "$jackknife.confidence.interval\n",
       "[1] 482.5991 518.4009\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "Jackknife(1:1000, statfunc = mean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Observations\n",
    "We observe that the bias is zero. This happens because the statistic we are building the estimator for the mean."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Bootstrap function\n",
    "my.bootstrapci.ml<-function(vec0,nboot=10000,alpha=0.05)\n",
    "{\n",
    "  #extract sample size, mean and standard deviation from the \n",
    "  #original data\n",
    "  n0<-length(vec0)\n",
    "  mean0<-mean(vec0)\n",
    "  sd0<-sqrt(var(vec0))\n",
    "  #create a vector to store the location of the bootstrap \n",
    "  #studentized deviation vector\n",
    "  bootvec<-NULL\n",
    "  bootbiasvec<-NULL\n",
    "  #create the bootstrap distribution using a for loop\n",
    "  for( i in 1:nboot){\n",
    "    vecb<-sample(vec0,replace=T)\n",
    "    #create mean and standard deviation to studentize\n",
    "    meanb<-mean(vecb)\n",
    "    sdb<-sqrt(var(vecb))\n",
    "    #note since resampling full vector we can use n0 for sample size of vecb\n",
    "    bootvec<-c(bootvec,(meanb-mean0)/(sdb/sqrt(n0)))\n",
    "    #Calculation the vector that stores the bias of each bootstap sample\n",
    "    bootbiasvec <- c(bootbiasvec, meanb-mean0)\n",
    "  }\n",
    "  \n",
    "  bootbias <- mean(bootbiasvec)  \n",
    "  bootsd <- mean(bootvec) #***************** Not sure about this ****************#\n",
    "  #Calculate lower and upper quantile of the bootstrap distribution\n",
    "  lq<-quantile(bootvec,alpha/2)\n",
    "  uq<-quantile(bootvec,1-alpha/2)\n",
    "  #incorporate into the bootstrap confidence interval (what algebra supports this?)\n",
    "  #and output result\n",
    "  LB<-mean0-(sd0/sqrt(n0))*uq[[1]]\n",
    "  UB<-mean0-(sd0/sqrt(n0))*lq[[1]]\n",
    "  #since I have the mean and standard deviation calculate the \n",
    "  #normal confidence interval here as well\n",
    "  NLB<-mean0-(sd0/sqrt(n0))*qnorm(1-alpha/2)\n",
    "  NUB<-mean0+(sd0/sqrt(n0))*qnorm(1-alpha/2)\n",
    "  list(bootbias = bootbias, bootsd = bootsd, bootstrap.confidence.interval=c(LB,UB)\n",
    "       ,normal.confidence.interval=c(NLB,NUB))\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$bootbias</dt>\n",
       "\t\t<dd>0.0665394999999999</dd>\n",
       "\t<dt>$bootsd</dt>\n",
       "\t\t<dd>0.00724731581752771</dd>\n",
       "\t<dt>$bootstrap.confidence.interval</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>482.192992705963</li>\n",
       "\t<li>517.934622763378</li>\n",
       "</ol>\n",
       "</dd>\n",
       "\t<dt>$normal.confidence.interval</dt>\n",
       "\t\t<dd><ol class=list-inline>\n",
       "\t<li>482.599114827485</li>\n",
       "\t<li>518.400885172515</li>\n",
       "</ol>\n",
       "</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$bootbias] 0.0665394999999999\n",
       "\\item[\\$bootsd] 0.00724731581752771\n",
       "\\item[\\$bootstrap.confidence.interval] \\begin{enumerate*}\n",
       "\\item 482.192992705963\n",
       "\\item 517.934622763378\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\item[\\$normal.confidence.interval] \\begin{enumerate*}\n",
       "\\item 482.599114827485\n",
       "\\item 518.400885172515\n",
       "\\end{enumerate*}\n",
       "\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$bootbias\n",
       ":   0.0665394999999999\n",
       "$bootsd\n",
       ":   0.00724731581752771\n",
       "$bootstrap.confidence.interval\n",
       ":   1. 482.192992705963\n",
       "2. 517.934622763378\n",
       "\n",
       "\n",
       "\n",
       "$normal.confidence.interval\n",
       ":   1. 482.599114827485\n",
       "2. 518.400885172515\n",
       "\n",
       "\n",
       "\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$bootbias\n",
       "[1] 0.0665395\n",
       "\n",
       "$bootsd\n",
       "[1] 0.007247316\n",
       "\n",
       "$bootstrap.confidence.interval\n",
       "[1] 482.1930 517.9346\n",
       "\n",
       "$normal.confidence.interval\n",
       "[1] 482.5991 518.4009\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "my.bootstrapci.ml(1:1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Observations\n",
    "It is seen that the normal based confidence interval is same as the Jackknife based confidence interval. This reason was seen above. The pivotal based confidence interval is observed to be around the normal based confidence interval but comparitively closer in this case. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Simulation function\n",
    "simulation <- function(mu.val=3, n=30, nsim=1000)\n",
    "{\n",
    "  #create coverage indicator vectors for bootstrap and normal\n",
    "  cvec.boot<-NULL\n",
    "  cvec.norm<-NULL\n",
    "  cvec.jack<-NULL\n",
    "  #calculate real mean\n",
    "  mulnorm<-(exp(mu.val+1/2))\n",
    "  #run simulation\n",
    "  for(i in 1:nsim){\n",
    "    if((i/100)==floor(i/100)){\n",
    "      print(i)\n",
    "      #let me know computer hasnt died\n",
    "    }\n",
    "    #sample the simulation vector\n",
    "    vec.sample<-rlnorm(n,mu.val)\n",
    "    #bootstrap it\n",
    "    boot.list<-my.bootstrapci.ml(vec.sample)\n",
    "    #jackknife it\n",
    "    jack.list <- Jackknife(vec.sample, statfunc=mean, alpha = 0.05)\n",
    "    #fetch confidence intervals\n",
    "    boot.conf<-boot.list$bootstrap.confidence.interval\n",
    "    jack.conf<-jack.list$jackknife.confidence.interval\n",
    "    norm.conf<-boot.list$normal.confidence.interval\n",
    "    \n",
    "    #calculate if confidence intervals include mu\n",
    "    #count up the coverage by the bootstrap interval\n",
    "    cvec.boot<-c(cvec.boot,(boot.conf[1]<mulnorm)*(boot.conf[2]>mulnorm))\n",
    "    #count up the coverage by the jackknife interval\n",
    "    cvec.jack<-c(cvec.jack,(jack.conf[1]<mulnorm)*(jack.conf[2]>mulnorm))\n",
    "    #count up the coverage by the normal theory interval\n",
    "    cvec.norm<-c(cvec.norm,(norm.conf[1]<mulnorm)*(norm.conf[2]>mulnorm))\n",
    "  }\n",
    "  #calculate and output coverage probability estimates\n",
    "  list(boot.coverage=(sum(cvec.boot)/nsim), jack.coverage=(sum(cvec.jack)/nsim),\n",
    "       norm.coverage=(sum(cvec.norm)/nsim))\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 100\n",
      "[1] 200\n",
      "[1] 300\n",
      "[1] 400\n",
      "[1] 500\n",
      "[1] 600\n",
      "[1] 700\n",
      "[1] 800\n",
      "[1] 900\n",
      "[1] 1000\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$boot.coverage</dt>\n",
       "\t\t<dd>0.912</dd>\n",
       "\t<dt>$jack.coverage</dt>\n",
       "\t\t<dd>0.801</dd>\n",
       "\t<dt>$norm.coverage</dt>\n",
       "\t\t<dd>0.801</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$boot.coverage] 0.912\n",
       "\\item[\\$jack.coverage] 0.801\n",
       "\\item[\\$norm.coverage] 0.801\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$boot.coverage\n",
       ":   0.912\n",
       "$jack.coverage\n",
       ":   0.801\n",
       "$norm.coverage\n",
       ":   0.801\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$boot.coverage\n",
       "[1] 0.912\n",
       "\n",
       "$jack.coverage\n",
       "[1] 0.801\n",
       "\n",
       "$norm.coverage\n",
       "[1] 0.801\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "simulation(mu.val = 4, n = 10, nsim = 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 100\n",
      "[1] 200\n",
      "[1] 300\n",
      "[1] 400\n",
      "[1] 500\n",
      "[1] 600\n",
      "[1] 700\n",
      "[1] 800\n",
      "[1] 900\n",
      "[1] 1000\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$boot.coverage</dt>\n",
       "\t\t<dd>0.909</dd>\n",
       "\t<dt>$jack.coverage</dt>\n",
       "\t\t<dd>0.861</dd>\n",
       "\t<dt>$norm.coverage</dt>\n",
       "\t\t<dd>0.861</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$boot.coverage] 0.909\n",
       "\\item[\\$jack.coverage] 0.861\n",
       "\\item[\\$norm.coverage] 0.861\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$boot.coverage\n",
       ":   0.909\n",
       "$jack.coverage\n",
       ":   0.861\n",
       "$norm.coverage\n",
       ":   0.861\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$boot.coverage\n",
       "[1] 0.909\n",
       "\n",
       "$jack.coverage\n",
       "[1] 0.861\n",
       "\n",
       "$norm.coverage\n",
       "[1] 0.861\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "simulation(mu.val = 4, n = 30, nsim = 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 100\n",
      "[1] 200\n",
      "[1] 300\n",
      "[1] 400\n",
      "[1] 500\n",
      "[1] 600\n",
      "[1] 700\n",
      "[1] 800\n",
      "[1] 900\n",
      "[1] 1000\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$boot.coverage</dt>\n",
       "\t\t<dd>0.949</dd>\n",
       "\t<dt>$jack.coverage</dt>\n",
       "\t\t<dd>0.92</dd>\n",
       "\t<dt>$norm.coverage</dt>\n",
       "\t\t<dd>0.92</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$boot.coverage] 0.949\n",
       "\\item[\\$jack.coverage] 0.92\n",
       "\\item[\\$norm.coverage] 0.92\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$boot.coverage\n",
       ":   0.949\n",
       "$jack.coverage\n",
       ":   0.92\n",
       "$norm.coverage\n",
       ":   0.92\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$boot.coverage\n",
       "[1] 0.949\n",
       "\n",
       "$jack.coverage\n",
       "[1] 0.92\n",
       "\n",
       "$norm.coverage\n",
       "[1] 0.92\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "simulation(mu.val = 4, n = 100, nsim = 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Observations\n",
    "It is observed that as you increase the sample size the coverage rate improves. A significant increase is observed when n is increased from 30 to 100."
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Question 4\n",
    "For the standard deviation of the normal distribution, estimate the bias of the the sample standard deviation when dividing by n, compare the bootstrap and the jacknife (1000 simulations)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Jackknife function for part 4\n",
    "Jackknife_sd<-function(v1){\n",
    "      n1<-length(v1)\n",
    "      jackvec<-NULL\n",
    "      mu0<-sd(v1)/n1\n",
    "      for(i in 1:n1){\n",
    "        mua<-sd(v1[-i])/(n1-1)\n",
    "        jackvec<-c(jackvec, n1*(mu0)-(n1-1)*mua)\n",
    "          }\n",
    "      jackbias<-mean(jackvec)-mu0\n",
    "      return (jackbias)\n",
    "  } "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Bootstrap function for part 4\n",
    "bootstrap_sd<-function(vec0,nboot=10000){\n",
    "      #extract sample size, mean and standard deviation from the original data\n",
    "      n<-length(vec0)\n",
    "      mean0<-sd(vec0)/n\n",
    "      bootvec<-NULL\n",
    "      bootbiasvec<-NULL\n",
    "      #create the bootstrap distribution using a for loop\n",
    "      for( i in 1:nboot){\n",
    "        vecb<-sample(vec0,replace=T)\n",
    "        #create mean and standard deviation to studentize\n",
    "        meanb<-sd(vecb)/n\n",
    "        #note since resampling full vector we can use n0 for sample size of vecb\n",
    "        bootvec<-c(bootvec,meanb)\n",
    "        #Calculation the vector that stores the bias of each bootstap sample\n",
    "        bootbiasvec <- c(bootbiasvec, meanb-mean0)\n",
    "          }\n",
    "      return(mean(bootbiasvec))  \n",
    "\n",
    "    }  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Simulation function for part 4\n",
    "simulation_q4 <- function(mu=3, sd= 2, n=30 , nsim=4)\n",
    "{\n",
    "  #create coverage indicator vectors for bootstrap and normal\n",
    "  bvec.boot<-NULL\n",
    "  bvec.jack<-NULL\n",
    "    \n",
    "  #run simulation\n",
    "  for(i in 1:nsim){\n",
    "        if((i/100)==floor(i/100)){\n",
    "          print(i)\n",
    "        #let me know computer hasnt died\n",
    "        }\n",
    "        #sample the simulation vector\n",
    "        vec.sample<-rnorm(n,mean = mu, sd = sd)\n",
    "        #bootstrap bias\n",
    "        bvec.boot<- c(bvec.boot, bootstrap_sd(vec.sample))\n",
    "        #jackknife bias\n",
    "        bvec.jack <- c(bvec.jack, Jackknife_sd(vec.sample))\n",
    "          }\n",
    "  #return\n",
    "  list(boot_bias = bvec.boot, jack_bias = bvec.jack)  \n",
    "  \n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 100\n",
      "[1] 200\n",
      "[1] 300\n",
      "[1] 400\n",
      "[1] 500\n",
      "[1] 600\n",
      "[1] 700\n",
      "[1] 800\n",
      "[1] 900\n",
      "[1] 1000\n"
     ]
    }
   ],
   "source": [
    "Output_4 <- simulation_q4(mu=3, sd= 2, n=30 , nsim=1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SEkFYSUG0JSQUi5ISQVhJQbQlJBSLkh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\nQAAhAQIICRBASIAAQgIEEBIg4B+/5dsyuqAgXQAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Histogram of Output_4$boot_bias\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(Output_4$boot_bias)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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5DMEcIPiOSfqYPWu8MxfLuD/YJI/uGI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\nQAQQCSACiAQQAUQCiAAiAUQAkQAi8A+PeeXQ8uG+HAAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Histogram of Output_4$jack_bias\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(Output_4$jack_bias)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Observations\n",
    "It is observed that the bias from the jackknife is approximately normally distributed. Similarly, the bias from the bootstrap is also approximately normally distributed but is skewed to the left."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.5.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}