est_std_error.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "64bc7052",
   "metadata": {},
   "outputs": [],
   "source": [
    "from pref_voting.profiles_with_ties import *\n",
    "from pref_voting.voting_methods import *\n",
    "from pref_voting.analysis import *\n",
    "from pref_voting.generate_profiles import *\n",
    "from pref_voting.generate_weighted_majority_graphs import *\n",
    "\n",
    "from pref_voting.utility_profiles import UtilityProfile\n",
    "\n",
    "from pref_voting.generate_utility_profiles import *\n",
    "from pref_voting.generate_utility_profiles import *\n",
    "from pref_voting.utility_methods import *\n",
    "\n",
    "from pref_voting.spatial_profiles import SpatialProfile\n",
    "from pref_voting.generate_spatial_profiles import *\n",
    "from pref_voting.utility_functions import *\n",
    "from pref_voting.probabilistic_methods import *\n",
    "from tqdm.notebook import tqdm\n",
    "import nashpy as nash\n",
    "import numpy as np\n",
    "import random2 as random\n",
    "from multiprocess import Pool, cpu_count, current_process\n",
    "from numba import njit, float32"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "58e6c888",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.6\n",
      "0.6\n",
      "0.6\n"
     ]
    }
   ],
   "source": [
    "print(np.mean([1, 1, 1, 0, 0]))\n",
    "\n",
    "print(np.nanmean([1, 1, 1, 0, 0, np.nan]))\n",
    "\n",
    "print(np.nanmean([1, 1, np.nan, np.nan, 0, 0, np.nan, 1]))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "f699b6cb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([nan,  1., nan, nan])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ms=np.array([0, 1, 0, 0.0])\n",
    "np.where(ms != 0, 1, np.nan)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "963cbf38",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.any(np.array([np.nan, np.nan, np.nan]) > 0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "5201631e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.nan < 0.1\n",
    "\n",
    "# test if array has nan\n",
    "np.isnan(np.array([np.nan, np.nan, np.nan])).any()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "6de8df63",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "+---+---+---+---+\n",
      "| 1 | 1 | 1 | 1 |\n",
      "+---+---+---+---+\n",
      "| 2 | 1 | 0 | 0 |\n",
      "| 1 | 0 | 2 | 1 |\n",
      "| 0 | 2 | 1 | 2 |\n",
      "+---+---+---+---+\n"
     ]
    }
   ],
   "source": [
    "prof = generate_profile(3, 4, \"Spatial\", (3, quadratic_utility))\n",
    "\n",
    "prof.display()\n",
    "\n",
    "def gen_samples(voting_methods, num_profiles=1):\n",
    "\n",
    "    profiles = [generate_profile(5, 10, \"Spatial\", (3, quadratic_utility)) for _ in range(num_profiles)]\n",
    "\n",
    "    return np.array([[len(vm(prof)) for prof in profiles] for vm in voting_methods])\n",
    "\n",
    "def is_condorcet_consistent(prof, vm): \n",
    "    cw = prof.condorcet_winner()\n",
    "    return int([cw] == vm(prof)) if cw is not None else np.nan\n",
    "\n",
    "def gen_samples2(voting_methods, num_profiles=1):\n",
    "\n",
    "    profiles = [generate_profile(5, 11, \"Spatial\", (3, quadratic_utility)) for _ in range(num_profiles)]\n",
    "\n",
    "    return np.array([[is_condorcet_consistent(prof, vm) for prof in profiles] for vm in voting_methods])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "da712f89",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[1, 1, 1, 1, 1, 1, 1, 1, 0, 1],\n",
       "       [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gen_samples2([plurality, split_cycle], num_profiles=10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "ad52545e",
   "metadata": {},
   "outputs": [],
   "source": [
    "# def estimated_variance_of_sampling_dist(\n",
    "#     mean_for_each_vm, \n",
    "#     values_for_each_vm):\n",
    "#     # values_for_each_vm is a 2d numpy array\n",
    "\n",
    "#     m = values_for_each_vm.shape[1]\n",
    "\n",
    "#     row_means_reshaped = mean_for_each_vm[:, np.newaxis]\n",
    "#     return  (1/(m*(m-1))) * np.sum((values_for_each_vm - row_means_reshaped)**2, axis=1)\n",
    "\n",
    "# def estimated_std_error(\n",
    "#     mean_for_each_vm, \n",
    "#     values_for_each_vm):\n",
    "#     # values_for_each_vm is a 2d numpy array\n",
    "#     return np.sqrt(estimated_variance_of_sampling_dist(mean_for_each_vm, values_for_each_vm))\n",
    "\n",
    "\n",
    "def estimated_variance_of_sampling_dist( \n",
    "    values_for_each_experiment,\n",
    "    mean_for_each_experiment=None):\n",
    "    # values_for_each_vm is a 2d numpy array\n",
    "\n",
    "    mean_for_each_experiment = np.nanmean(values_for_each_experiment, axis=1) if mean_for_each_experiment is not None else mean_for_each_experiment\n",
    "\n",
    "    num_val_for_each_exp = np.sum(~np.isnan(values_for_each_experiment), axis=1)\n",
    "    \n",
    "    row_means_reshaped = mean_for_each_experiment[:, np.newaxis]\n",
    "    return np.where(\n",
    "        num_val_for_each_exp * (num_val_for_each_exp - 1) != 0.0,\n",
    "        (1 / (num_val_for_each_exp * (num_val_for_each_exp - 1))) * np.nansum(\n",
    "            (values_for_each_experiment - row_means_reshaped) ** 2, \n",
    "            axis=1),\n",
    "            np.nan\n",
    "            )\n",
    "\n",
    "def estimated_std_error(values_for_each_experiment, mean_for_each_experiment=None):\n",
    "    # values_for_each_vm is a 2d numpy array\n",
    "    return np.sqrt(estimated_variance_of_sampling_dist(values_for_each_experiment, mean_for_each_experiment=mean_for_each_experiment))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "e98fec69",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "69579385",
   "metadata": {},
   "outputs": [],
   "source": [
    "def means_with_estimated_standard_error(\n",
    "        generate_samples, \n",
    "        max_std_error, \n",
    "        initial_trials=1000, \n",
    "        step_trials=1000,\n",
    "        min_num_trials=10_000, \n",
    "        max_num_trials=None,\n",
    "        verbose=False\n",
    "        ):\n",
    "    \"\"\"\n",
    "    For each list of numbers produced by generate_samples, returns the means, the [estimated standard error](https://en.wikipedia.org/wiki/Standard_error) of the means, the variance of the samples, and the total number of trials.  \n",
    "\n",
    "    Uses the estimated_variance_of_sampling_dist (as described in [https://berkeley-stat243.github.io/stat243-fall-2023/units/unit9-sim.html](https://berkeley-stat243.github.io/stat243-fall-2023/units/unit9-sim.html)) and estimated_std_error functions. \n",
    "    \n",
    "    Args:\n",
    "        generate_samples (function): A function that generates samples. It should take a single argument num_profiles and return a 2d numpy array of samples.\n",
    "        max_std_error (float): The desired estimated standard error for the mean of each sample.\n",
    "        initial_trials (int, default=1000): The number of samples to initially generate.\n",
    "        step_trials (int, default=1000): The number of samples to generate in each step.\n",
    "        min_num_trials (int, default=10000): The minimum number of trials to run.\n",
    "        max_num_trials (int, default=None): If not None, then the maximum number of trials to run.\n",
    "        verbose (bool, default=False): If True, then print progress information.\n",
    "\n",
    "    Returns:\n",
    "        A tuple (means, est_std_errors, variances, num_trials) where means is an array of the means of the samples, est_std_errors is an array of estimated standard errors of the samples,  variances is an array of the variances of the samples, and num_trials is the total number of trials.\n",
    "\n",
    "    \"\"\"\n",
    "    \n",
    "    # samples is a 2d numpy array\n",
    "    samples = generate_samples(num_profiles = initial_trials)\n",
    "    \n",
    "    means = np.nanmean(samples, axis=1)\n",
    "    variances = np.nanvar(samples, axis=1)\n",
    "    est_std_errors = estimated_std_error( \n",
    "        samples, \n",
    "        mean_for_each_experiment=means)\n",
    "        \n",
    "    if verbose:\n",
    "        print(\"Initial number of trials:\", initial_trials)\n",
    "        print(f\"Remaining estimated standard errors greater than {max_std_error}:\", np.sum(est_std_errors > max_std_error))\n",
    "        print(f\"Estimated standard errors that are still greater than {max_std_error}:\\n\",est_std_errors[est_std_errors > max_std_error])\n",
    "\n",
    "    num_trials = initial_trials\n",
    "    \n",
    "    while (np.isnan(est_std_errors).any() or np.any(est_std_errors > max_std_error) or (num_trials < min_num_trials)) and (max_num_trials is None or num_trials < max_num_trials):\n",
    "        if verbose:\n",
    "            print(\"Number of trials:\", num_trials)\n",
    "            print(f\"Remaining estimated standard errors greater than {max_std_error}:\", np.sum(est_std_errors > max_std_error))\n",
    "            print(f\"Estimated standard errors that are still greater than {max_std_error}:\\n\",est_std_errors[est_std_errors > max_std_error])\n",
    "\n",
    "        new_samples = generate_samples(num_profiles = step_trials)\n",
    "\n",
    "        samples = np.concatenate((samples, new_samples), axis=1)\n",
    "\n",
    "        num_trials += step_trials\n",
    "\n",
    "        means = np.nanmean(samples, axis=1)\n",
    "        variances = np.nanvar(samples, axis=1)\n",
    "        est_std_errors = estimated_std_error(\n",
    "            samples, \n",
    "            mean_for_each_experiment=means)\n",
    "\n",
    "    return means, est_std_errors, variances, num_trials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "1ff20478",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Initial number of trials: 1000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0151113  0.00387506]\n",
      "Number of trials: 1000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0151113  0.00387506]\n",
      "Number of trials: 2000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.01084164 0.00228446]\n",
      "Number of trials: 3000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00890166 0.00208502]\n",
      "Number of trials: 4000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00770203 0.00180247]\n",
      "Number of trials: 5000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00688602 0.00161013]\n",
      "Number of trials: 6000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00628394 0.00144981]\n",
      "Number of trials: 7000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00582014 0.00131311]\n",
      "Number of trials: 8000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00543897 0.00126772]\n",
      "Number of trials: 9000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00513935 0.00118469]\n",
      "Number of trials: 10000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00487504 0.00113626]\n",
      "Number of trials: 11000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00465117 0.00111776]\n",
      "Number of trials: 12000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00445162 0.00105785]\n",
      "Number of trials: 13000\n",
      "Remaining estimated standard errors greater than 0.001: 2\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00427585 0.00101524]\n",
      "Number of trials: 14000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00412316]\n",
      "Number of trials: 15000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00398405]\n",
      "Number of trials: 16000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00385389]\n",
      "Number of trials: 17000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00374006]\n",
      "Number of trials: 18000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00363167]\n",
      "Number of trials: 19000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00353228]\n",
      "Number of trials: 20000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0034442]\n",
      "Number of trials: 21000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00336229]\n",
      "Number of trials: 22000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00328561]\n",
      "Number of trials: 23000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00321145]\n",
      "Number of trials: 24000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00314388]\n",
      "Number of trials: 25000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00308093]\n",
      "Number of trials: 26000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00301767]\n",
      "Number of trials: 27000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00295936]\n",
      "Number of trials: 28000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00290707]\n",
      "Number of trials: 29000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00285773]\n",
      "Number of trials: 30000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00281068]\n",
      "Number of trials: 31000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00276446]\n",
      "Number of trials: 32000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00272056]\n",
      "Number of trials: 33000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00267953]\n",
      "Number of trials: 34000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00264078]\n",
      "Number of trials: 35000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0026018]\n",
      "Number of trials: 36000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00256564]\n",
      "Number of trials: 37000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00253073]\n",
      "Number of trials: 38000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00249605]\n",
      "Number of trials: 39000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0024645]\n",
      "Number of trials: 40000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00243302]\n",
      "Number of trials: 41000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00240293]\n",
      "Number of trials: 42000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00237388]\n",
      "Number of trials: 43000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0023458]\n",
      "Number of trials: 44000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00231921]\n",
      "Number of trials: 45000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00229396]\n",
      "Number of trials: 46000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00226933]\n",
      "Number of trials: 47000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00224542]\n",
      "Number of trials: 48000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00222135]\n",
      "Number of trials: 49000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00219925]\n",
      "Number of trials: 50000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00217694]\n",
      "Number of trials: 51000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00215559]\n",
      "Number of trials: 52000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00213489]\n",
      "Number of trials: 53000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00211412]\n",
      "Number of trials: 54000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00209476]\n",
      "Number of trials: 55000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00207605]\n",
      "Number of trials: 56000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00205736]\n",
      "Number of trials: 57000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00203955]\n",
      "Number of trials: 58000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00202254]\n",
      "Number of trials: 59000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00200532]\n",
      "Number of trials: 60000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00198873]\n",
      "Number of trials: 61000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00197207]\n",
      "Number of trials: 62000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00195614]\n",
      "Number of trials: 63000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00194066]\n",
      "Number of trials: 64000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00192539]\n",
      "Number of trials: 65000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00191038]\n",
      "Number of trials: 66000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00189585]\n",
      "Number of trials: 67000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00188183]\n",
      "Number of trials: 68000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00186789]\n",
      "Number of trials: 69000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00185461]\n",
      "Number of trials: 70000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0018413]\n",
      "Number of trials: 71000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00182825]\n",
      "Number of trials: 72000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00181568]\n",
      "Number of trials: 73000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00180362]\n",
      "Number of trials: 74000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00179201]\n",
      "Number of trials: 75000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00178046]\n",
      "Number of trials: 76000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00176884]\n",
      "Number of trials: 77000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00175742]\n",
      "Number of trials: 78000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00174642]\n",
      "Number of trials: 79000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00173523]\n",
      "Number of trials: 80000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00172443]\n",
      "Number of trials: 81000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00171375]\n",
      "Number of trials: 82000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0017031]\n",
      "Number of trials: 83000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00169278]\n",
      "Number of trials: 84000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00168245]\n",
      "Number of trials: 85000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00167248]\n",
      "Number of trials: 86000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0016632]\n",
      "Number of trials: 87000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00165397]\n",
      "Number of trials: 88000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00164466]\n",
      "Number of trials: 89000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00163561]\n",
      "Number of trials: 90000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00162669]\n",
      "Number of trials: 91000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00161795]\n",
      "Number of trials: 92000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00160925]\n",
      "Number of trials: 93000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00160061]\n",
      "Number of trials: 94000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00159194]\n",
      "Number of trials: 95000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00158337]\n",
      "Number of trials: 96000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00157518]\n",
      "Number of trials: 97000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0015666]\n",
      "Number of trials: 98000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00155852]\n",
      "Number of trials: 99000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0015505]\n",
      "Number of trials: 100000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00154287]\n",
      "Number of trials: 101000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00153522]\n",
      "Number of trials: 102000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00152757]\n",
      "Number of trials: 103000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00152021]\n",
      "Number of trials: 104000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00151291]\n",
      "Number of trials: 105000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00150559]\n",
      "Number of trials: 106000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00149851]\n",
      "Number of trials: 107000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00149132]\n",
      "Number of trials: 108000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00148447]\n",
      "Number of trials: 109000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00147746]\n",
      "Number of trials: 110000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00147074]\n",
      "Number of trials: 111000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00146437]\n",
      "Number of trials: 112000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00145793]\n",
      "Number of trials: 113000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0014513]\n",
      "Number of trials: 114000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00144482]\n",
      "Number of trials: 115000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00143864]\n",
      "Number of trials: 116000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00143243]\n",
      "Number of trials: 117000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00142648]\n",
      "Number of trials: 118000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00142035]\n",
      "Number of trials: 119000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00141434]\n",
      "Number of trials: 120000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00140853]\n",
      "Number of trials: 121000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00140262]\n",
      "Number of trials: 122000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00139676]\n",
      "Number of trials: 123000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00139114]\n",
      "Number of trials: 124000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0013853]\n",
      "Number of trials: 125000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00137986]\n",
      "Number of trials: 126000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00137442]\n",
      "Number of trials: 127000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00136921]\n",
      "Number of trials: 128000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00136383]\n",
      "Number of trials: 129000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00135849]\n",
      "Number of trials: 130000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00135304]\n",
      "Number of trials: 131000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00134786]\n",
      "Number of trials: 132000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0013428]\n",
      "Number of trials: 133000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00133771]\n",
      "Number of trials: 134000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00133273]\n",
      "Number of trials: 135000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0013279]\n",
      "Number of trials: 136000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00132305]\n",
      "Number of trials: 137000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00131822]\n",
      "Number of trials: 138000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00131364]\n",
      "Number of trials: 139000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0013089]\n",
      "Number of trials: 140000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00130441]\n",
      "Number of trials: 141000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00129983]\n",
      "Number of trials: 142000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00129533]\n",
      "Number of trials: 143000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00129088]\n",
      "Number of trials: 144000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00128663]\n",
      "Number of trials: 145000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00128211]\n",
      "Number of trials: 146000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0012778]\n",
      "Number of trials: 147000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00127347]\n",
      "Number of trials: 148000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00126911]\n",
      "Number of trials: 149000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00126473]\n",
      "Number of trials: 150000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00126042]\n",
      "Number of trials: 151000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00125632]\n",
      "Number of trials: 152000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00125217]\n",
      "Number of trials: 153000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00124794]\n",
      "Number of trials: 154000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00124395]\n",
      "Number of trials: 155000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0012399]\n",
      "Number of trials: 156000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.001236]\n",
      "Number of trials: 157000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00123203]\n",
      "Number of trials: 158000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00122808]\n",
      "Number of trials: 159000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00122424]\n",
      "Number of trials: 160000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00122053]\n",
      "Number of trials: 161000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00121663]\n",
      "Number of trials: 162000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00121298]\n",
      "Number of trials: 163000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00120933]\n",
      "Number of trials: 164000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00120559]\n",
      "Number of trials: 165000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0012019]\n",
      "Number of trials: 166000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00119818]\n",
      "Number of trials: 167000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00119463]\n",
      "Number of trials: 168000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00119102]\n",
      "Number of trials: 169000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0011874]\n",
      "Number of trials: 170000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00118388]\n",
      "Number of trials: 171000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00118042]\n",
      "Number of trials: 172000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00117707]\n",
      "Number of trials: 173000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00117372]\n",
      "Number of trials: 174000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00117028]\n",
      "Number of trials: 175000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00116695]\n",
      "Number of trials: 176000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00116366]\n",
      "Number of trials: 177000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00116046]\n",
      "Number of trials: 178000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00115723]\n",
      "Number of trials: 179000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00115412]\n",
      "Number of trials: 180000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00115092]\n",
      "Number of trials: 181000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00114758]\n",
      "Number of trials: 182000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00114445]\n",
      "Number of trials: 183000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00114136]\n",
      "Number of trials: 184000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00113829]\n",
      "Number of trials: 185000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00113507]\n",
      "Number of trials: 186000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00113202]\n",
      "Number of trials: 187000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00112884]\n",
      "Number of trials: 188000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00112581]\n",
      "Number of trials: 189000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00112292]\n",
      "Number of trials: 190000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0011199]\n",
      "Number of trials: 191000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00111694]\n",
      "Number of trials: 192000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00111403]\n",
      "Number of trials: 193000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00111108]\n",
      "Number of trials: 194000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00110806]\n",
      "Number of trials: 195000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00110518]\n",
      "Number of trials: 196000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00110238]\n",
      "Number of trials: 197000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00109969]\n",
      "Number of trials: 198000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00109692]\n",
      "Number of trials: 199000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00109416]\n",
      "Number of trials: 200000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00109155]\n",
      "Number of trials: 201000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00108886]\n",
      "Number of trials: 202000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0010862]\n",
      "Number of trials: 203000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00108349]\n",
      "Number of trials: 204000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00108087]\n",
      "Number of trials: 205000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00107823]\n",
      "Number of trials: 206000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00107563]\n",
      "Number of trials: 207000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00107299]\n",
      "Number of trials: 208000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00107031]\n",
      "Number of trials: 209000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0010678]\n",
      "Number of trials: 210000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00106534]\n",
      "Number of trials: 211000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00106275]\n",
      "Number of trials: 212000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00106026]\n",
      "Number of trials: 213000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00105776]\n",
      "Number of trials: 214000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00105516]\n",
      "Number of trials: 215000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00105267]\n",
      "Number of trials: 216000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00105018]\n",
      "Number of trials: 217000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00104771]\n",
      "Number of trials: 218000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.0010453]\n",
      "Number of trials: 219000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00104291]\n",
      "Number of trials: 220000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00104053]\n",
      "Number of trials: 221000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00103814]\n",
      "Number of trials: 222000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00103586]\n",
      "Number of trials: 223000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00103364]\n",
      "Number of trials: 224000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00103126]\n",
      "Number of trials: 225000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00102896]\n",
      "Number of trials: 226000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00102667]\n",
      "Number of trials: 227000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00102441]\n",
      "Number of trials: 228000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00102211]\n",
      "Number of trials: 229000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00101987]\n",
      "Number of trials: 230000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00101768]\n",
      "Number of trials: 231000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00101549]\n",
      "Number of trials: 232000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00101325]\n",
      "Number of trials: 233000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00101103]\n",
      "Number of trials: 234000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00100893]\n",
      "Number of trials: 235000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00100679]\n",
      "Number of trials: 236000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00100464]\n",
      "Number of trials: 237000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00100253]\n",
      "Number of trials: 238000\n",
      "Remaining estimated standard errors greater than 0.001: 1\n",
      "Estimated standard errors that are still greater than 0.001:\n",
      " [0.00100048]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(array([0.65162032, 1.        , 0.98602056, 1.        ]),\n",
       " array([0.00099835, 0.        , 0.00024601, 0.        ]),\n",
       " array([0.22701128, 0.        , 0.01378402, 0.        ]),\n",
       " 239000)"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "voting_methods = [plurality, split_cycle, bracket_voting, stable_voting]\n",
    "gsamples = partial(gen_samples2, voting_methods)\n",
    "\n",
    "means_with_estimated_standard_error(\n",
    "    gsamples, \n",
    "    0.001, \n",
    "    verbose=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "c7eae733",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.25"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.var([1]*1000 + [0]*1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a9b53063",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.1"
  },
  "vscode": {
   "interpreter": {
    "hash": "3e0b5a01b8f727ee5e6561dee9387cd7de3c110c3a4eff66a475afe21c23d281"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}