{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Convergence of the empirical cdf" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ ":0: FutureWarning: IPython widgets are experimental and may change in the future.\n" ] } ], "source": [ "import IPython.html.widgets as widgets\n", "from IPython.html.widgets import interact, interactive, fixed\n", "import numpy as np\n", "import seaborn as sns\n", "import statsmodels.api as sm # recommended import according to the docs\n", "import matplotlib.pyplot as plt\n", "from scipy.stats import norm, cauchy, expon, gamma, uniform\n", "\n", "\n", "import mpld3\n", "mpld3.enable_notebook()\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Glivenko Cantelli\n", "Quand $N\\rightarrow \\infty$,\n", " \\begin{equation*}\n", " ||\\widehat F_N-F||_\\infty\\stackrel{p.s.}{\\longrightarrow} 0\n", " \\end{equation*}\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def cdf(sample, x):\n", " n = sample.size\n", " nb = sum(sample <= x)\n", " return nb/float(n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "####Loi uniforme" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La norme infinie entre la cdf théorique et empirique vaut 0.0997010202381\n" ] }, { "data": { "image/png": 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CPc40t3UCcFJRDnPLol8xq1oXSU4K9zjT0hoI99wxeEi0qnWR5KVwjzPNYxDu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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 50\n", "sample = np.random.uniform(0, 1, N)\n", "x= np.linspace(min(sample), max(sample), 10*N)\n", "y = [cdf(sample, ele) for ele in x]\n", "g = [uniform.cdf(ele) for ele in x]\n", "plot1, =plt.plot(x,y, label = 'empirical cdf')\n", "plot2, =plt.plot(x,g, label = 'Uniform cdf')\n", "plt.legend(loc = 2)\n", "diff = [abs(a - b) for a, b in zip(y,g)]\n", "print 'La norme infinie entre la cdf théorique et empirique vaut {}'.format(max(diff))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "####Loi Gaussienne" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La norme infinie entre la cdf théorique et empirique vaut 0.0799291619293\n" ] }, { "data": { "image/png": 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88qk8/fQTfPPNCrKysrn33r9z5ZVX8/DD/+joz/Z6PTz66MP84Q93ccstvyUpKalT0B98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\nJMyFECIB/H9F442WIV5bqAAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 50\n", "sample = np.random.randn(N)\n", "x= np.linspace(min(sample), max(sample), 10*N)\n", "y = [cdf(sample, ele) for ele in x]\n", "g = [norm.cdf(ele) for ele in x]\n", "plt.plot(x,y, label = 'empirical cdf')\n", "plt.plot(x,g, label = 'Gaussian cdf')\n", "plt.legend(loc = 2)\n", "diff = [abs(a - b) for a, b in zip(y,g)]\n", "print 'La norme infinie entre la cdf théorique et empirique vaut {}'.format(max(diff))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "####Loi de Cauchy " ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La norme infinie entre la cdf théorique et empirique vaut 0.124709475499\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N = 50\n", "sample = np.random.standard_cauchy(N)\n", "x= np.linspace(min(sample), max(sample), 10*N)\n", "y = [cdf(sample, ele) for ele in x]\n", "g = [cauchy.cdf(ele) for ele in x]\n", "plt.plot(x,y, label = 'empirical cdf')\n", "plt.plot(x,g, label = 'Cauchy cdf')\n", "plt.legend(loc = 4)\n", "diff = [abs(a - b) for a, b in zip(y,g)]\n", "print 'La norme infinie entre la cdf théorique et empirique vaut {}'.format(max(diff))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "La norme infinie entre la cdf théorique et empirique vaut 0.0399482382128\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "@interact(N=(50, 1000))\n", "def generate(N):\n", " sample = np.random.uniform(0,1,N)\n", " x= np.linspace(min(sample), max(sample), 10*N)\n", " y = [cdf(sample, ele) for ele in x]\n", " g = [uniform.cdf(ele) for ele in x]\n", " plt.plot(x,y)\n", " plt.plot(x,g) \n", " diff = [abs(a - b) for a, b in zip(y,g)]\n", " print 'La norme infinie entre la cdf théorique et empirique vaut {}'.format(max(diff))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###Construction d'intervalle de confiance pour la convergence ponctuelle\n", "Pour tout $x\\in\\mathbb{R}$, quand $N\\rightarrow \\infty$,\n", "\n", "$$\\widehat F_N(x) \\stackrel{p.s.}{\\longrightarrow} F(x).$$\n", "De plus, on a un controle des oscillation de $\\widehat F_N(x)$ autour de $F(x)$ par le TCL : quand $N\\rightarrow \\infty$, \n", "$$\\widehat F_N(x) \\approx F(x) + \\frac{\\sigma}{\\sqrt{N}}\\mathcal{N}(0,1)$$\n", "où $\\sigma^2=F(x)(1-F(x))$ ou (par Slutsky) $\\sigma^2=\\widehat F_N(x)(1-\\widehat F_N(x))$.\n", "On peut vérifier ce résultat en générant pour $j=1,\\ldots,k$ un $N$-échantillons $(X_1^{(j)},\\ldots,X_{N}^{(j)})$ et en en traçant un histogramme des valeurs \n", "$$\\widehat F_N^{(j)}(x)=\\frac{1}{N}\\sum_{i=1}^N I(X_i^{(j)}0)$$" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "N=500, k=1000\n" ] }, { "data": { "image/png": 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hnzNdCwpnXtM+DNr6JxUnT4JvftA7JMXNNT9/mGan93Cgcj2212gB8Rf1Dsmt\nba5Qm4uNm1Bh0wbif/mRmM7ahJ3FafJA1cHWCYLSkhi0ZRop3r5MbvWo3U1I3dWCpg9yMSSCcrP+\nxDvquN7hKG7MKzWFl7fNwmjwYlKHp1QfOBukJsUyvHRr0r19KPHZ53z0w9+8PGZxsao6VwWTEzy6\nfQ4lkxP4TbThanAJvcNxuAwfv5vVDe+8VWyrGxTrKv06kQqJMSxs8gBnIu/QOxyPEVfxLuY160dk\ncgJPH91U7CYPVAWTg1WIu0CvfUu5FBbJnzVb6R2O02ypUp/4Fi3x+3sdfiuX6x2O4oa8Dx2k/Ixp\n/BsS4RGDFrubuc36cimsDL33Lqbm1Wi9w3EpVTA5ktnMC2t+wteUyeR2Q0nzLroPeU0mI9se6Y/J\n2xv/t9/g1OGDpKerHuuKhdFI6JsvYzAa+brVo6T5BugdkcdJ9/Xn+y4v4G02MXLzHxgyMvQOyWVU\nweRAXU9sp8G5g+yo1oytRfhuCbQm5KO3xDC7dgcCLl7g2LNvFas6cCV/gT98i++e3Vzt0pVdle7W\nOxyPtb9qQ1bWu4/qcRepOPVXvcNxGTXthYP4xMXx/K4FpPgG8PO9zxSLh7xB4WWY23EYHc4dYmDU\nNo4cO6rmbCoAIcReIMHy8pSU8ik943EU74P/EPzFxxjLluPM62/BrFv742SNxp2dJ3elcLbJ7YbQ\n6NQuKvw2mYQBg8ms10DvkJxOFUwOUnX8WK1Xe4enuBIWqXc4LpPmG8B3XV7gk7kfUO3Tj0i+rzv4\nFt0qTEcRQgQASCmLVm/T1FTCXhyGISOD6+N/IDP89sY/rpzAryhI8Q/iqzYDGLPqe0Kfe4q4VRsg\nOFjvsJxKVeXZKfvglHGTfiLyrxUcLV2VZQ176B2ay/1TpQHLarYmOOo4Qd9/o3c4nqIBECSE+EsI\nsdYytYzHC/7oPXyOHSVl6NNk3Nslz/UKOxp3cbO7Ym3+feRxfKKOax1vizi775iKajWErbKGIKrk\n68/khZ+Q4uXDZ/cMxuTlrXdoupjQrA9d4o4T9NXnpN/bmcwGjfQOyd0lAWOklL8KIWoCK4QQtaSU\neQ7d7vZTq8+cCZN+htq1CfzuGwKDg22eYlyxLvG9UXBCEjhjGoE9uxVoIGVPGwPSroKpyFZDFFBQ\nWCSj/p5MeFoSYxp051yJchTX0/C6jx/bnnmB9p9+RMCTgzg4dTqmoCAqV1ZjSOfhONqUMkgpo4QQ\nMUB54EID+IaOAAAgAElEQVReG7jT1Oo5By0OPH2Kuk89iSE4mPhJf2BMNkHydZunGFesi0lMp9SP\nv1KyczsMzzxLXMVqGO+ua3W74jS1epGshiiohw+vo8mZvey5ozFzqzXVOxxdpSbG8NYxAzPrdibw\nXDQxz75Z7HqrF9BQYCyAEKICEAb8m+8WbiT7oMUff7eOEs8PxzslhRNvv4exZi29wyuyTNWqc338\njxiSkwgf/BiGq1f1Dskp7C2YsqohugLPAdOFEEX2eVXOyc5OnowiZcUynt29kNjgknzTdUSxaIVn\nTVB4GWZ1epaosjXofmI7fS+f1Dskd/YrECaE2AjMBIbmV43njoLCy1AirAyfbPqdqgmXmHV3J2I7\n5f1cSXGM9F69SXprFN7nogkfOgDS0vQOyeHsfcZU4GoIT54o8Pjx4zcmOwMonRTPTws/wwx8cf+b\nxAeXhCunnRKLp8n09uXLnq8zbvqbvLp1Jv+dvR9aNlYTBeYgpcwEBukdR6GYzYxY9QMNzh1ka42W\n/NL0QT7RO6ZiIvn1kXjLYwQsmk/oS89x/cdJ4F10nm/bWzANBeoDL9paDeHJEwXGxibeaEUUkJ7C\np8u/JiI9he9aPMSRinUcHoOn+69Eeb68/w1Gz/+Qci++CG3acMXfpomOC0RNFKgjs5kXds3n3qN/\nc6xcLcZ1fxVTYtGsVnJLBgOxX31DyOmThC2YR4LJxOmR79youfH0kcjtrX7z+GoIe3gbMxm5dAw1\nL51gcdWGzK/dQe+Q3Nb+qg2Z0KwffjEx0KMHhvg4vUNSHMVspsr343n48DqiS1Xifw++Q5qvv95R\nFTvRV6/Qp0o3jpeqTNlFCzg37C3e/nlbkXi2a9cdU5Gohiggg9nE8DU/0tTS2OHzht3xV8+V8jWv\nTgcequxFuXmzCe//MPGzF0JIcW23WESYTASPfpegP//gbHhZ3n34f1wLcvzdsGIbU2QVPnzkYz6f\n/Q4PH15HsLcfYxv11DusQlMjP9jCaOTNzdPpfGI7UWVr8Pn9b2K8eFTvqNyfwcCZ196kXKAfvtOm\nEf5EfxJ+n1Hke60XWWlphL78PAHz55JSpSqvtXqW9OCSN95WQw05T37f7bWgcEY9/DEfzh9Nj39W\nEnD9CoYnPbuVsCqYrMnIoPrHo4k8sZ3jZWvyfr8PSPUL1Dsqz+HlBZMnk3Y1Dv+VyyjxcG8Sps/G\nXFL19PckhitXCBv2BH5bN5PRrAWH//cpsbOO3dJvTw015DzWvtv44BK8/cgnvLfwE+49vYfrLz5L\n+p9zMJUrr1fIhVJkm3g7giE+jvDH+hG5cjmHI+/kvYdGkxSgqqIKzNeXa7/+Tmrfh/HdvZMSD/bA\nS11Jewyf7dso2aktfls3k3Z/b+LnLs51DDxQQw05k7XvNtk/mA/6fsCq6s0IPfQPJTq3w3f7Vp2i\nLRxVMOWQ1Wfp4pq/CO7cDr9Nf3OhaXPe6DqcZH9VBWU3X1+u/ziR5KefxefoEUre1x7fv9fpHZWS\nn7Q0gj7/mBJ9euB15TKJ733EtV9/h0BVY+Cu0n39+eyeJzjz0mt4Xb1CeO/uMHKkx/V1UgVTDmdO\nn2bD8+9x16D+BEafZWbdzvQo0YxUNdFZgWXVix8/flzrmHz6JHGjP+H62G8xJCYS/lhfgj/50ONO\nmuLAZ9cOaNyY4HFfYipXnoR5S0gZ8YrqSO4BTCYjO9u05fCESaRVqAhffklw+5awfq3eodlMPWPK\nxvu4pNIbIxi5fzvXAkIZ0/MNdtRoQcCZvXqH5pFu1Iuv1OrFE+P+5c3HG1OldRuCf5xIzffeJmj8\nWHxXLiPxq2/JbNFS54iLj5xj3WW5IyOTkmM+w3/JQgASBw/l6OAhGIND4GTUjfVUowb3dfN5VBkC\nOr7Kc7sX0uvYJoIe7UNaj14kvTsaY42aeR4D7tAHShVMgNel/wj87msCJ0/EkJnJtkp1+an7q8SG\nRugdmsfLqhcHSE64xNhZB248wA3o9DpPbZ3BQ3I3JXvdR9r9vUl69wOM1WroGXKxkDXWXdZoJjWv\nRtN3/3LKXDyCl9FIRpOm+I7/hkPegbesl0U1anBv2c+7ST1eZ1m15nxychmRy5fgt2IpsR07cbhL\nV97blXjLvk1OuMz4Nx+gus4Tfhbrgsn76BH8Jv9C0IxpeKWnk1qhIkeeGcaos5GEqELJKbKfMADf\ntR1EySF9aT57BqFLF+G3bDGp3XqS+sJLZDZvoaqOnCgyIISu5w5z79H13H1B6/6QUKkyl54fTmzH\nTpSKCCV6/+Hb9hloFxmK5zgQEMz9dR6ha4V/GfjPSmqtW8N969ZQuXRV1jXswUbR1q2eods77YUX\n8CPasERpwNNSSo8YsdPr7Bn8V63Af/5cfPfsAuBCUAlmNH2Iv2q04NKWKCIqFZ8ZaPWWmhjDqP0G\ngpo+S7vI/Ty2bxm1VywlcMVSMmvUJK3PQ6T17quNWF2ECilnnUPWqme8Tp3Eb/0axNLFzN++DT9j\nJgD7qjZgSqW72VGmBkEnQuDEDkDdGRUlQSXKsu/Oxuxr2INGZ/dz39Y/aXXpBLXXTOCZdRM5VPlu\nNparhf+5ilCthq7nm713TA8CflLK1pYpL8ZalrkXkwmvM6fx3b1T+9u+FZ9j2pWh2WAgrVMXznS+\njxeP+RMYUYUAIDAxRt+Yi6GsK/K9pSqxsWpDvm/uT/W/VuD/13KCx3xG8JjPSCtTloRmzbnesDFJ\ntQQpd1YjPNyjJyMs9DkUHx9HbOytx2t09Fm+mrmf4LDSlE5KoErCf9xx/jCVQlIIO3Ma/8s373RO\nlyjPxrqd2HBXe66ERXL5zF6Cwsvecnek7oyKIIOBfXc04i/MVPHxp9eFI7SJ2kajswdodPYAPDKH\n9FIRJNarz/V69UmuWYvwzt3wjXTdBbu9BVMbYCWAlHKHEMKl3YzTU1OJ2hvFtX+v4n39Or6xsfjF\nxhBpNOEXG4P36VN4nzyB16mTeKWm3NjO6OdPbMvWxN/Tnri27cgoU4bo6LMYj3vMNDhFnslk5HBE\nBNf/7x28RrxCqY1/47d6JWF79lFm2RLKLFsCQIbBi9SqVfCuXJW08hVIK1+BoC5dMTT3mAYUhT6H\nlv88hSv7LlIq5RolU65TKuUapWPO8KfRSKVrV/E3pt+yfmxgKDuqNGBXpTqs8vIlvWaL26rolOLl\nanAJ5rR4mDktHqb09SvctWsBTa6coUHcBSI3rKfUhvUApJeOJOHwCZfdRdlbMIUB17K9NgohvJw1\nkGvK4McJ3LqVgIx0/DLS8c/MsLqNMTCQhHLl2JbizdEyd3KwVEV2pybjFRJBQJQXRG0GIOHSKUqU\nvzmxWcr1WODWL9+WZa5cpyjnH/dvFB9PPEZAyM3OgwlUpWTPztQzplP7yhlqxJ7nzn+jqHbuAqXO\nnLmxnnHiBGJPXdRGm3B/hTqHDPFxjBj3IV5m823vJfn4cTa8LBfCIjkfVoZ9QFSFu0gsd7N6Jub8\nUQITLt+yna37viDrumJZcc3b0fEkAnsiqzG/WjOCwiIpmxTLXVfOUvm/KO7t2YoQF1bt2VswXQOy\nzw9g7YQyFGo6gRVLC7yJN1AS6GH5U4o+b8CDng4W7hyKDAVT7qsHA7Usf1DMRltWigR7Ly23YPm9\nF0K0BP5xWESKUjyoc0hR8mDvHdMCoIsQYovl9VAHxaMoxYU6hxQlDwZzLnXUiqIoiqIXj3hKrCiK\nohQfqmBSFEVR3IoqmBRFURS3ogomRVEUxa0UahBXW8b7EkIEAauBJ6WU0rLNJLRuFiZgmJRSFiRd\nIcTjwMtAJnAQeAGtp1h+29iTpg8wGagK+AMfSymXFDZWKaXZ8l4ZYA/QSUp5vLBpCiHeBnoBvsD3\nUsrfHPS95rm/bEizHzASMAPTpZTf2njc2JOuL/nsL3vSzPZervvKEfKLSwhRFpiZbfWGwEgp5S/2\nxmVvftaOL0fnCUwEfiWf3wp78rO83wcYhbavJ0spfyrs+IV25pnvMeuMPLO959BjJ7/8CnrsFPaO\n6cZ4X8D/oY33lf1DNAU2AndaAgW4DwiWUrYFPgI+KUi6QohA4H9AB0sa4cD9lm3884rFzjQHAFek\nlO2AbsD3DooVywH5M5DkiDSFEB2AVpZtOgDVHBSrtf2VX5rewGdAJ6AV8IIQIgLr+8redAeS//4q\naJqlLO/lt68cIc+4pJSXpJQdpZQd0U74PWg/2IWJq8D52Xh8OfozdsX6b0WB87MYB3RBGxrqdSFE\nCWw7Lh2dp7Vj1tF5hoNzjp288rPn2ClswXTLeF9AzvG+/NA+SParnBQgXAhhQPvxS+d2+aWbivYh\nUy2vfSzL2gAr8omloGmmAHOA9y3LvNDuJAoba9bgfWOACUBuA/XZ8/nvAw4KIRYCS4DFDorV2v7K\nM00ppRG4S0p5HW1QBm/L9tb2lT3ppgGzyX9/2RMr5L+vHMHaeYTl+/8WeD7rjrsQcdmTX1esH1+O\nztOW3wp788sASgBBaLUCZmw7Lh2RZ6AlTxPWj1lH5+nsYyfn9wp2HDuFLZhyHe8r64WUcquU8nyO\nbbYAAcAxtBL7u4KkK6U0SymvAAghRqBdUa22Fosdaa6RUiZJKROFEKFohdQ7Doh1jRBiCNpV0irL\nNjkHobLn80cCTYCHgOeA6Y6IFev7y9oxYBJC9AX2AevRrtCs7St70k22YX8VOE0b9pUj2PJ99AIO\nSSmjAAoZV4HzA0pj/fhydJ62/FbYm99YtDuzg8ASKWWCjTE6Is9Dljyv2fgb49A8nXzs5Pa9FvjY\nKWzBVNDxvgDeArZIKQVaXfJvQoic8/jmm64QwksI8RVatUs/G2OxJ02EEJWBdcDvUsrs9eCFSXco\nWq//9dm+g7KFTPMqsEpKmSm1+uJUIURpB8RqbX9ZPQaklPOBimh16INt2cbOdK3tL3vStLavHMGW\n72MA8Eu214WJy578bDm+HJ2nLb8VBc5PCFEFGI72XOcOoKwQ4iEbY3R0nrb8xjg6T6ccO/nkV+Bj\np7AFkz3jfQVzs8SNQ3sY5l3AdH9G++Hok63qydo2BU7TsrNWAW9JKafm8XkKnK6Usr2UsoOlTn0/\nMFhKeakwaQKb0eqoEUJUQPuec04uZU+61vZXnmkKIcKEEBuEEH6WqpkkwGhDHHala8P+KnCaNuwr\nR7Dl+2gqpdyW9aKQcRU4P2w7vhydpy2/FfbkF4B2HKZZflQvo1U/FXb8wgLnaeNvjEPzdOKxk9f3\nWuBjp1BDElnqfrNaaIBWEjcBQqSUE7Ottx54Vkp53PLAbwra7Z0v8E3Oq4T80gV2W/42ZtvkG7R6\ny1u2kbe2dCtomuPRHtQ9wq3PyLpn+9G2K10p5cLcvpvCfH4p5SIhxBdAR7QLjrctVXw32Pm9biCf\n/WXtGBBCDAOeQqt7PgCMsKyX576yM92XgK+Bh8ljf9kTq7z5PCfXfeUINsQVCfwlpWycx/YFisve\n/KwdX47O05bfikLk9yrQH+3Z6glgGNqPar7HpYPzfAb4inyOWWd8TillZrbtHX3s5JpfQY8dNVae\noiiK4lZUB1tFURTFraiCSVEURXErqmBSFEVR3IoqmBRFURS3ogomRVEUxa2ogklRFEVxK6pgUhRF\nUdyKKpgURVEUt6IKJkVRFMWtqIJJURRFcSuqYFIURVHciiqYFEVRFLeiCiZFURTFrfjoHUBRJ4TY\ngzZPSda00NOklGN1DClPQoggYBLa5GFewEgp5SJ71hNCNEebQqOClDLW2bErRYcQYhQwCO33aZqU\n8sNc1vFDm9G2rWXRCrQ5jQoysZ/LqHOrYNQdkxMJIYKBakB9KWUjy59bFkoWo4FrUso6QBfgRyFE\nxYKuZ5mdcgLaHDqKYjMhRA+0KbgbA3WBjkKIh3NZdTgQIaW8G21uoNZoc6e5q9Goc8tm6o4pGyHE\nE8D73JwEazfwiZRyWo71tgBBOTbfLKUckWNZcyARWC6EKA+sAUblNwmYEOJTIDQrLSFEN2C0lLKl\n5UqyN9odWDDwhpRyoRBiNNAKKAcckFIOzpZebeDPXLL6Rkr5W45lDwKPA0gpzwkhVqGd7F/bup4Q\nwgv4A3gbWJnX51SKBiecM32A6VLKFMt2U4CBwJzsK0kpxwkhvrW8LI02U2q+s6Kqc8tzqIIpGynl\nb0KILsCXaAfohpwnmGW9NjYmGQKsA15EmxV1OvAZ8Go+20wEdgghXrXMNDkU+EUIUQXoBLSTUqYJ\nIR4DPgKyZsOtDNTNWZUhpTwKNLIx3srAuWyvzwOVCrjeR8AOKeUqIYSN2SqeygnnTCUg++ymF8j9\nGMQyM+pnaHdPu9Cm8M6POrc8hCqYbvcc2jz2yWjVCbcRQmwFAnMs3iKlHJ59gZRyCbAk23afAvPJ\np2CSUp4WQhwAegsh1gH3ok3xnGy5Oh0khKgOtES7ssuyPbf6dSFEHbQCMafxUsqpOZblVrVrzGVZ\nrusJIXqi3SV2tUzBDGDIZV2laHHYOYPtxyAAUsq3hRDvoRU6E4Ah+ayrzi0PoQqm25UD/NHqcCsC\np3OuIKVsbUtCQoheQLyUcpNlkRfanZM1k4DBQFlgvuXEaQwsAsYCfwEb0E7ELEm5JSSlPILtV3XR\nQAXgsuV1JWCvjevtQ7sCzbnNOiHEk1LKPTbGoHgeh50z3Dy2slREu2u4hRCiNXBFShlluXP6Dfg2\n53q5UOeWB1AFUzZCCF9gBvAe4A3MEEK0tdz226Mi8K4Qoj1agfQaMNOG7RYA44AqwDDLsnuAXVLK\nb4QQ3sCPlhgdaRHwDPCCEKIS0BWt+sCm9aSUt/wwCCFMQMei2nJIcco5swj4QAjxC9odxRPAlFzW\nuxdoKYToDZiBAcBaG9JX55YHsFowWW5xh1heBgIN0JpojgdMwCHgRSml2UkxutKnwEUp5WQAIcSD\nwMfA/9mZ3s9orfL2on3X67AcjEKIkUCQlPKDnBtJKdOFEDOBTlLK3ZbFM4B+QohDaA95ZwKPCyFC\n0E5MR3z/HwATLHl4oz0APm2JdyKwW0r5c37r5VAUjgmHEEK8DfRCu6v4HtgCTMXzzyGHnjNSyqVC\niHrATsAPWCil/MOS9rNAUynlMOAL4BvgANp3uAmtUYA6t4oAg9ls++cTQnwP7Ec7wcZKKTcKISYA\nf0kpF+a/tZKdEKIy2o+RvYWe4iGEEB2A16SUD1i6ELyF1k9FnUNOoM4tz2dzPyYhRFOgjpRyEtBE\nSrnR8tYKoLMzgiviamNbnbji+e4DDgohFqI1hlmMOoecSZ1bHq4gz5hGAVk9sLO3BkkEwh0WUTEh\npVyldwyKy0SiNQO+H61qdwnqHHIadW55PpsKJiFECaCWlHKDZVH2ppOhQHx+25vNZrPBUGRbNipF\nizMO1KvAUUuDgONCiFS0hjFZ1DmkFDWFOlhtvWNqx60tXvYJIdpbCqruWGkNYzAYuHLlup0hWhcZ\nGerU9F2Rh/oM7pFHZGSoM5LdDLwMjBNCVEAbAWGtO51DuXHF/tQzPz3yLA6fMSvPwrC1YKoFnMz2\n+nVgomUgxSPA3EJFoShFmJRymRCinRBiJ9pz3ReAM6hzSFFyZVPBJKX8KsfrKKCDMwJSlKJISjky\nl8UdXB2HongCNbq4oiiK4lbUyA+Konik9PR0zp07e9vyypWr4ufnp0NEiqOogklRFI907txZXh6z\nmKDwMjeWJSdcZvybD1C9ek0dI1MKSxVMiqK4vdzujqKjzxIUXoaQkrnNt6d4MlUwKYri9nK7O4o5\nf5SISrV1jEpxFlUwuSlVf64ot8p5d5SccEnHaBRnsmV08aI6KrJbU/XnilJwJmMm0dHWL+iyLvzi\n4kKIjU3Mcz1FH/kWTJZRkVtJKVtnGxW5LzAq26jIvbk5BbHiQKr+XFEKJjUxhrGzYgkK//fGstwu\n6NSFn3uzdseUfVTkMOBN4KkcoyLfhyqYFEVxE7Ze0KkLP/dlrWBSoyIrigMIIfYCCZaXp4DPUFXi\nipIrawVToUdFzuKkwTFdlr4r8sieflxcSK7rlCoVUqg41H5wPSFEAICUsmO2ZYtRVeKKkitrBVOh\nR0XOoka1Llj62R/IZhcbm2h3HGp0cdvSd4IGQJAQ4i+0c+4doLGqEleU3OVbMKlRkRXFIZKAMVLK\nX4UQNYGVOd5XVeKKko3V5uJqVGRFKbTjwAnQRuYXQsQAjbK9b1OVuB5VmK7OM6/88qratlXOKnBn\nVZXbojjsx8JSHWwVxfmGAvWBFy1V4qHAqoJWiRf1Cebyyy+vqm1b5awCd0ZVuS3URIG2UQWTojjf\nr8AUIUTWM6WhQAyqSlxRcqUKJkVxMkur1kG5vNXBxaEUaWUSLtH92CYaJ1yiUnICYOZKWCT7S1Yk\nILo8qI6zHkMVTIqieLRSibEM3DKdTofX4YXWFSwhMAwzBirFXaTR2QPw6HJSe/cl6b0PMVWpqnPE\nijWqYFIUxWO1jv6H/9s8ndC0RM5EVOGPqg3ZLdpgKi8ACE5NpN6h1bx8eSchi+bjt3Y1iZ9/BU2a\n6hy5kh81tbqiKB5p4PEtfLL2Z/wy05lw7zO8POhrlt7RkGsBN1vcJQWEsKZ6cw5N/oNr304AIGz4\ns1Se8D0Gs0mv0BUrVMGkKIpnMZsZtPkPRhxay+WgErze/0uWN+yBycs7720MBtIeG0Dcmo1kVqtO\nxd+n8PK2WWBWo0C5I5uq8tQ4X4qiuIs+exbxyM55nA2J4K3ur5AceYfN25qqVSd+2RoCH+hGb7mZ\nzNDSTG03xGmxKvaxeseUfZwvy99TwDi0cb7aoQ3q2tu5YSqKokD7oxt4cuNUroZEMLztQC6HlCpw\nGuaICI5+8z3R4WXpt3sh3f75ywmRKoVhyx2TGudLURSXyW325ujos9wZe4ERq38gyS+ID/q+z+XE\nGILszCOzVCne6vIiPy/7imfWTeRsRBV2BYUVPnjFIWwpmBwyzldRGOZDjS6uf/quykPRT26T+KWc\nOcDsvYvwz0znqwdeI7p0VUiMKVQ+l0Ij+KLnm/xv3gf839IvGdLzDZtmv1Wcz5aCySHjfHn6MB9q\ndHH903dFHs4q9IQQZYA9QCe0Z7NTUc9o85RzEr+31vxElYRLLGjSm+01Wjosn4NV6vF724EM3fQ7\nr236nZEZjxJUIv/ZbxXns6VV3lBgLEDOcb4s73cHNuaxraIUe0IIX+BntNoHA+oZbYG0jNpO93MH\nOVq6Kr+1zW0AjcJZ2KQ3/1SqS4d/JQ9dPklIyYo3/rLftSmuY0vB9CsQZhnnayZaQfUK8KEQYiva\nXZca50tR8jYGmABkXYrnfEbbWZeoPEBoyjVeWPsTaV7efH7PYIzejh8TwOTlzTfdXua6rz/Dd8wh\n8toVh+ehFIwt016ocb48UG4PkOPiQggOjlD15S4khBgCXJFSrhJCvI12h2TItoqaiykfw9b/Ssnk\neL6r24noEuUo3OQXebsSFsnX9bvy/p7FPLN+Ip/0HuWknBRbqCGJiqjcHiCr+nJdDAXMQojOQEPg\nNyAy2/s2PaOFojmPT3p6OmfOnAEgLk67oUxI0O5YGpw9QMdjG5DlavJnzVYEFCIfW+ZjWlalAQ9c\nOErLkztpGbWd7TVb5rptYRXF/ehoqmAqisxmwnds471Dq7k79gKlkuJI8/HnbGhpykyPx/DcCMxl\nVN25K0gps57FIoRYDzwHjCnoXExQNOdjOnky6rYLqJjzRylbvibPrp+ICQM/dn4eU7JNZXeebJqP\nyWBgXKvH+HXxZzy7fiIHqtTnmjGT/fsP37a+vS311HxMtlEFUxHju30rwaPeIvLQP9QGkn0DiA2J\nwD8zjWYXj8L3RzFP/ImUZ18k6ZU3IMRZlSNKHszA66i5mG7I2QIvOeESfY7+TeXY8yxv0I1TZarB\nmb0uieVciXLMadaP/ttn8eiO2XxZpT5jZ8USFK5a6rmSKpiKCqOR4I9HE/TDeMwGA1e7dGW0b13O\n1Wp9Ywwx73+P8XX5GKrM/JOgb8fht2wx1yb+hrFuPX1jLyaklB2zveygVxzuLiLlOkP2LedaQCh/\ntB7g8vznNetL58PreGDfUqZFVCa+wl23FJyK86lBXIuCxETCBj9G0A/jyaxeg/ilqzjx0accLlvt\nloEtEwJCufTwY8Ru3UPy8yPwOXmCEj27cHX6b5w8GXXbX3p6uo4fSimuXji8jqDMNH5vO5DEQNc/\nG0n39WfqPYPxNWYy4uAal+evqDsmz5ecTPigR/Hbson0jp24NnEq5rBwOBmV9zZBQSR9+AkZLVoR\n+sxQarz2Mh91fIrNVRveTFZVVyg6qHI1mh7RBzhZsgKr6+rXin6TaEuvfUu59+JRFv0XxSl1x+RS\nNt0xCSHKCCHOCSFqCSFqCCE2CyE2CiF+FEIYrKegOEVGBuFD+uO3ZRNp9/cmYdpsrVCyUXqP+zn6\nzfdkePvy/oYptEyMUx0LFV0N2PonXsCvjR/IfxoLZzMYmNThKQBe2Dlfzd3kYraMLq56rbupkHdH\n4vf3OtK6dOXaT7+Cr2+B07jeqDHv3zsMzPDuok+pFHPO8YEqig1q/hdF6xPb+adUJbZVrqt3OBwv\nX4uVlesiYqJpd2yT3uEUK7bcMale624oYOqvBE6ZRGadulz7eQoUotPsnoq1Gd91BMHpyYxa8gWB\n6SkOjFRRbDNwy3QAJtx9LxjcoyLmpzr3kuHlTf+tM/A2ZuodTrGR7zMmR/ZaV6NaFyz9fEcXjz4O\no96E0qXxWbaE8AoRNzopZsnqpJidyZhJQsKVW9LOWm9D7fbUuHSSB/cu5qVV3/Fe6/52dSwsCvtB\ncb265w7S+Ox+9lVtwN7IO+yezsLR/g0uwRLRlr5HN9Dl0BpWNuimd0jFgrXGDw7rta5GtS5Y+nmN\nLh5/4QqZw4bgk5lJ/I+TyAiO4OTeQ7l2UoyoVPuWbVMTY3j/l20EhZ/Mdb2p9wymxqUTtD2+lR6R\nd7f1/kYAACAASURBVBIb26pAn7mo7AfFxcxmBm+eBsAfbQZCqms7g1ozrUE3ekRt57Hts1hfpyO5\nn5mKI+VblSelbC+l7GDpf7EfGAysVCOL66fqt1/jc/IEyc8NJ6PDvTeWZ3VSzPoLDM19Zs/81jN6\n+zC2x6sk+gfx4o65+F847/TPoygtzx+i9r+SrTVaElXO/VqBxgWGsbhxLyKS4ui5fxkmYybR0WdV\n9wonKmhzcdVrXUfNzh+m7Op5ZNapS9I7Hzglj6uhkfzScRivrRxP9Y9Hk7aiHXjr2DpKKdpMJp7e\nswQTBqa37q93NHma3/RBuh9YyUM75zMtvJwaDcLJbC6YVK91fflnpPLq1pmYvL058va7JJ+PvvFe\nbrNuFsb62h1ocnQD7ffvI3HyL6QMe96h6Rc3QghvYCJQC+3i7jkgDTVZIBFrV1M97gLrancgunQV\nvcPJU1JACPOa9WXI5t8ZELWVP1o/rkaDcCLVwdZDDNg6g/JJsUyt1Zbf1sXAupvTSuf2PKlQDAa+\nafUYbWJPEfTZx6Td3xtT+QqOS7/4uR8wSSnbWqrBP7UsHyWl3CiEmIDW7WKhbhHqISODSr9MIMPL\nmz9bP653NFYtbdSTXvuW8PiJHSxt1JMMVTA5jRqSyANUv3SSB/Yu4VxwSWa06HfLM6L8nicVRnxg\nKNEvjsAr8Toh7/6fw9MvTqSUi4BnLS/vAOKAJsWt20V6evotz2SufzeOwPPnWFarDZfCy+odnlVp\nvv7MbvEwgcYM+v+zSu9wijR1x+TmvExGXlzzI95mE1806km6jx+umubvcq8Hqbx2Df5LFuK35i/S\nO3d1Uc5Fj5TSKISYCjwIPAx0yfa2Td0uPGken+zzLGU5ffq0pVVoGXwzM5g+7wdSvHz4wwVNsG2Z\nj8kWq+p1off22TxwbBPLW/fnStjNRsq2dq/wpP2oF1UwubnOh9dR89JJ1tduz64y1VzWv8NkzCT6\n/DkCX3qV+kMG4P/GKxz7cw6VatRSM+DaSUo5RAhRFtgJt8x7Z1O3C0+axyeveZYiKtUmpGRFeu9Z\nTGRyPL/Xak1sULjTZqbNYtN8TDbI9PZlYu32fLBnEY9un833972YZx65UfMx2UZV5bmxoPQUBm6Z\nRopvAFPvecKleacmxjB21gFeWRfD7Ls6EHDxAnteePe26doV64QQgywd1AFSACOwu6h3u8ira0Jg\negoP75xLkl8Qf9RsrXOUBbeySj3OhJej8+G1lI+7qHc4RZIqmNzYgH/+omRyAnOb9SU2xPHPkazJ\n+mGZ3+FJ4gPDGXp8M75Xbh9RQrFqLtBQCLEBWAm8DAwHPhRCbEWruSg23S4e2LuY8JRrLGjam2v+\n7jLGg+1MBi+mNO6Jt9nEgK0z9A6nSLJalaeauurD/+IFHjq8niuhpVnYRN9xcpP9g5nWpj/D10yg\nyk/fQ0vPu8rVk5QyBXg0l7c6uDgU3YWlJtJn9yLiA8NZ3PgBuHhU75DssqlqQ06UqUZ7uYm5zftx\nyKfgAygrebPljulGU1fgXbSmrmNRI4w7VZXvx+NnymTKPU+Q7uuvdzisrtuZE6UqErl8KT779ugd\njuKhHju4huD0ZOa06EeKX6De4djNbPDShk9Cm6pDcSyrBZNq6up6vtu28P/tnXd8k9X+x99Jmo60\npYWyCpRRxiN7yFIRcICK4AVFRRFx4ERFVDYiiggo4wKCl3EB9SIiClf86RVlIwiyoQKntAXaQlld\ndKdt8vsjaSmFtkma9knS8369+no1zzjfz0me73P294Rs20JErSbsUnqoLQcAk1bH510HA1imj5tl\nA1liHzWz0nj05HauBITwv3buHwz1UOOO/F2/Jd2j/6Ll5TNqy/EobBpjKjLVdT6wGgcjjEtswGTC\n/33LOPnibo+5TPh/gKOhLUjsfS/6/fvw2VBlhkQkTuL5U7vwyc/l2zuGkOvlATM7NRq+sraaRhza\nqLIYz8KekETlmuoqt72wMf1Vq+DYEa4NGMDJmk0qfBqtvVybPJGQPX9QbfpUGDYEDDcOXnvC7yBx\nPnVTEhh49hBx1WqzpdU9Zd/gJpxo0JqDjTpy+7nDnNj/F8hYeU7BlskPw4AGQogZFJvqKoTYgWWq\n65ay0pHbXtiQ/pkEaoyfgNbPj6gXX4Ufz1aYPUe55F+dOq+MxLBwHhnTZpD5ztjCc57yO0icz9A9\na/Aym1jRqT/5Os9aPvl1j6Hcfu4wYUsWkf3k0y7Vy+Gu2NKVJ6e6VhKGz+ehu3SRzJGjMNapq7ac\nmygI9//3PwZirF4D33/OJnbvHhnuX1IqTS7H0PvUTk4Fh7KjcUe15Tid6DrN2NGoA4F/R+D9269q\ny/EIyqy6yKmulURsLIbFC8mvG0rmyFFw0fUW7lk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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from scipy.stats import norm, gamma\n", "#paramètres\n", "a, k, N, bins = 2, 1000, 500, 30\n", "print 'N={}, k={}'.format(N,k)\n", "#données\n", "for i, x in enumerate([0.3,1,3,5]):\n", " HatFx = []\n", " for j in range(k):\n", " data0 = gamma.rvs(a=a, size=N)\n", " HatFx.append(cdf(data0,x))\n", " #Gaussienne\n", " Fx = gamma.cdf(x, a=a)\n", " var = Fx*(1-Fx)\n", " X = np.linspace(min(HatFx), max(HatFx), 1000)\n", " y = norm.pdf(X, loc = Fx, scale = np.sqrt(var/float(N)))\n", " #plot\n", " ax = plt.subplot(2,2,i)\n", " plt.tight_layout()\n", " ax.hist(HatFx, bins = bins, normed = True)\n", " ax.plot(X, y, 'r')\n", " ax.set_title('x = {}, var = {}'.format(x, \"%.2f\" % var))\n", " \n", "\n", "\n", "\n", "#plot\n", "#plt.hist(HatFx, bins = 50, normed = True)\n", "#plt.plot(X, y, 'r')\n", "#plt.title('N={}, k={}, var = {}'.format(N,k, var))" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "##Comparaison de la cdf empirique $\\widehat F_n$ et de $F_{\\hat \\theta_n}$ pour un estimateur $\\hat\\theta_n$ dans le modèle exponentiel\n", "\n", "Soit $X_1,\\ldots,X_n\\stackrel{i.i.d.}{\\sim} {\\rm Exp}(\\theta)$ pour un certain $\\theta>0$. L'estimateur du maximum de vraisemblance (qui est aussi l'estimateur des moments d'ordre $1$) est :\n", "$$\\hat\\theta_n=\\frac{1}{\\bar X_n}$$\n", "La fonction de répartition d'une ${\\rm Exp}(\\theta)$ est\n", "$$F_\\theta(x)=\\mathbb{P}[ {\\rm Exp}(\\theta)\\leq x]=\n", "\\left\\{\n", "\\begin{array}{cc}\n", "0 & \\mbox{ si } x\\leq0\\\\\n", "1-\\exp(-\\theta x) & \\mbox{ sinon.}\n", "\\end{array}\n", "\\right.$$" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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7nqIexbKlvkKI7JVRy70t4GI2m+sDI0gLcgBMJpMCLAHeNpvNDYFtQJlHVdH8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\nQoi8I8+Huyb4EsYVS3CUKEn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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "La norme infinie entre la cdf théorique et empirique vaut 0.0849\n", "La norme infinie entre la cdf théorique et la cdf F_hatTheta vaut 0.0404\n" ] } ], "source": [ "import numpy as np\n", "from scipy.stats import expon\n", "import matplotlib.pylab as plt\n", "import mpld3\n", "mpld3.enable_notebook()\n", "%matplotlib inline\n", "####param\n", "n, theta = 100, 2\n", "x = np.linspace(0,2,100)\n", "#######empirical cdf\n", "def cdf(sample, x):\n", " n = sample.size\n", " nb = sum(sample <= x)\n", " return nb/float(n)\n", "sample = expon.rvs(size=n ,scale = 1/float(theta))\n", "hatF = []\n", "for ele in x:\n", " hatF.append(cdf(sample, ele))\n", "########theoretical cdf\n", "F = expon.cdf(x, loc = 0, scale = 1/float(theta))\n", "##########F_{\\hat\\theta_n}\n", "hatTheta = 1/float(np.mean(sample))\n", "F_hatTheta = expon.cdf(x, loc = 0, scale = 1/float(hatTheta))\n", "########Plot\n", "plt.plot(x, F, 'r', label = 'theoretical cdf')\n", "plt.plot(x, hatF, 'b', label = 'empirical cdf')\n", "plt.plot(x, F_hatTheta, 'g', label = 'cdf hatTheta')\n", "plt.legend(loc=4)\n", "plt.show()\n", "diff = [abs(a - b) for a, b in zip(F, hatF)]\n", "print 'La norme infinie entre la cdf théorique et empirique vaut {}'.format(\"%.4f\" % max(diff))\n", "diff = [abs(a - b) for a, b in zip(F, F_hatTheta)]\n", "print 'La norme infinie entre la cdf théorique et la cdf F_hatTheta vaut {}'.format(\"%.4f\" % max(diff))" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": false }, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }