{ "__type": "IngestedDoc", "__tag": 4010, "_content": {}, "_ordered_sections": [], "item_file": null, "item_line": null, "item_type": null, "aliases": [], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "see_also": [], "signature": null, "references": null, "qa": "tutorial:stats:continuous", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [], "title": [ { "__type": "Text", "__tag": 4046, "value": "Continuous Statistical Distributions" } ], "level": 0, "target": "continuous-random-variables" }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "All distributions will have location (L) and Scale (S) parameters along with any shape parameters needed, the names for the shape parameters will vary. Standard form for the distributions will be given where " }, { "__type": "InlineMath", "__tag": 4057, "value": "L=0.0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "S=1.0." }, { "__type": "Text", "__tag": 4046, "value": " The nonstandard forms can be obtained for the various functions using (note " }, { "__type": "InlineMath", "__tag": 4057, "value": "U" }, { "__type": "Text", "__tag": 4046, "value": " is a standard uniform random variate)." } ] }, { "__type": "Code", "__tag": 4050, "value": "====================================== ============================================================================= =========================================================================================================================================\nFunction Name Standard Function Transformation\n====================================== ============================================================================= =========================================================================================================================================\nCumulative Distribution Function (CDF) :math:`F\\left(x\\right)` :math:`F\\left(x;L,S\\right)=F\\left(\\frac{\\left(x-L\\right)}{S}\\right)`\nProbability Density Function (PDF) :math:`f\\left(x\\right)=F^{\\prime}\\left(x\\right)` :math:`f\\left(x;L,S\\right)=\\frac{1}{S}f\\left(\\frac{\\left(x-L\\right)}{S}\\right)`\nPercent Point Function (PPF) :math:`G\\left(q\\right)=F^{-1}\\left(q\\right)` :math:`G\\left(q;L,S\\right)=L+SG\\left(q\\right)`\nProbability Sparsity Function (PSF) :math:`g\\left(q\\right)=G^{\\prime}\\left(q\\right)` :math:`g\\left(q;L,S\\right)=Sg\\left(q\\right)`\nHazard Function (HF) :math:`h_{a}\\left(x\\right)=\\frac{f\\left(x\\right)}{1-F\\left(x\\right)}` :math:`h_{a}\\left(x;L,S\\right)=\\frac{1}{S}h_{a}\\left(\\frac{\\left(x-L\\right)}{S}\\right)`\nCumulative Hazard Function (CHF) :math:`H_{a}\\left(x\\right)=` :math:`\\log\\frac{1}{1-F\\left(x\\right)}` :math:`H_{a}\\left(x;L,S\\right)=H_{a}\\left(\\frac{\\left(x-L\\right)}{S}\\right)`\nSurvival Function (SF) :math:`S\\left(x\\right)=1-F\\left(x\\right)` :math:`S\\left(x;L,S\\right)=S\\left(\\frac{\\left(x-L\\right)}{S}\\right)`\nInverse Survival Function (ISF) :math:`Z\\left(\\alpha\\right)=S^{-1}\\left(\\alpha\\right)=G\\left(1-\\alpha\\right)` :math:`Z\\left(\\alpha;L,S\\right)=L+SZ\\left(\\alpha\\right)`\nMoment Generating Function (MGF) :math:`M_{Y}\\left(t\\right)=E\\left[e^{Yt}\\right]` :math:`M_{X}\\left(t\\right)=e^{Lt}M_{Y}\\left(St\\right)`\nRandom Variates :math:`Y=G\\left(U\\right)` :math:`X=L+SY`\n(Differential) Entropy :math:`h\\left[Y\\right]=-\\int f\\left(y\\right)\\log f\\left(y\\right)dy` :math:`h\\left[X\\right]=h\\left[Y\\right]+\\log S`\n(Non-central) Moments :math:`\\mu_{n}^{\\prime}=E\\left[Y^{n}\\right]` :math:`E\\left[X^{n}\\right]=L^{n}\\sum_{k=0}^{N}\\left(\\begin{array}{c} n\\\\ k\\end{array}\\right)\\left(\\frac{S}{L}\\right)^{k}\\mu_{k}^{\\prime}`\nCentral Moments :math:`\\mu_{n}=E\\left[\\left(Y-\\mu\\right)^{n}\\right]` :math:`E\\left[\\left(X-\\mu_{X}\\right)^{n}\\right]=S^{n}\\mu_{n}`\nmean (mode, median), var :math:`\\mu,\\,\\mu_{2}` :math:`L+S\\mu,\\, S^{2}\\mu_{2}`\nskewness :math:`\\gamma_{1}=\\frac{\\mu_{3}}{\\left(\\mu_{2}\\right)^{3/2}}` :math:`\\gamma_{1}`\nkurtosis :math:`\\gamma_{2}=\\frac{\\mu_{4}}{\\left(\\mu_{2}\\right)^{2}}-3` :math:`\\gamma_{2}`\n====================================== ============================================================================= =========================================================================================================================================", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Overview" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Non-central moments are defined using the PDF" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\mu_{n}^{\\prime}=\\int_{-\\infty}^{\\infty}x^{n}f\\left(x\\right)dx." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note, that these can always be computed using the PPF. Substitute " }, { "__type": "InlineMath", "__tag": 4057, "value": "x=G\\left(q\\right)" }, { "__type": "Text", "__tag": 4046, "value": " in the above equation and get" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\mu_{n}^{\\prime}=\\int_{0}^{1}G^{n}\\left(q\\right)dq" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which may be easier to compute numerically. Note that " }, { "__type": "InlineMath", "__tag": 4057, "value": "q=F\\left(x\\right)" }, { "__type": "Text", "__tag": 4046, "value": " so that " }, { "__type": "InlineMath", "__tag": 4057, "value": "dq=f\\left(x\\right)dx." }, { "__type": "Text", "__tag": 4046, "value": " Central moments are computed similarly " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu=\\mu_{1}^{\\prime}" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In particular" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Skewness is defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\gamma_{1}=\\sqrt{\\beta_{1}}=\\frac{\\mu_{3}}{\\mu_{2}^{3/2}}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "while (Fisher) kurtosis is" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\gamma_{2}=\\frac{\\mu_{4}}{\\mu_{2}^{2}}-3," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "so that a normal distribution has a kurtosis of zero." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Moments" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The median, " }, { "__type": "InlineMath", "__tag": 4057, "value": "m_{n}" }, { "__type": "Text", "__tag": 4046, "value": " is defined as the point at which half of the density is on one side and half on the other. In other words, " }, { "__type": "InlineMath", "__tag": 4057, "value": "F\\left(m_{n}\\right)=\\frac{1}{2}" }, { "__type": "Text", "__tag": 4046, "value": " so that" } ] }, { "__type": "Math", "__tag": 4058, "value": "m_{n}=G\\left(\\frac{1}{2}\\right)." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In addition, the mode, " }, { "__type": "InlineMath", "__tag": 4057, "value": "m_{d}" }, { "__type": "Text", "__tag": 4046, "value": " , is defined as the value for which the probability density function reaches it's peak" } ] }, { "__type": "Math", "__tag": 4058, "value": "m_{d}=\\arg\\max_{x}f\\left(x\\right)." } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Median and mode" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To fit data to a distribution, maximizing the likelihood function is common. Alternatively, some distributions have well-known minimum variance unbiased estimators. These will be chosen by default, but the likelihood function will always be available for minimizing." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineMath", "__tag": 4057, "value": "f\\left(x;\\boldsymbol{\\theta}\\right)" }, { "__type": "Text", "__tag": 4046, "value": " is the PDF of a random-variable where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\boldsymbol{\\theta}" }, { "__type": "Text", "__tag": 4046, "value": " is a vector of parameters ( " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "e.g." } ] }, { "__type": "Text", "__tag": 4046, "value": " " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "S" }, { "__type": "Text", "__tag": 4046, "value": " ), then for a collection of " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " independent samples from this distribution, the joint distribution the random vector " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mathbf{x}" }, { "__type": "Text", "__tag": 4046, "value": " is" } ] }, { "__type": "Math", "__tag": 4058, "value": "f\\left(\\mathbf{x};\\boldsymbol{\\theta}\\right)=\\prod_{i=1}^{N}f\\left(x_{i};\\boldsymbol{\\theta}\\right)." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The maximum likelihood estimate of the parameters " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\boldsymbol{\\theta}" }, { "__type": "Text", "__tag": 4046, "value": " are the parameters which maximize this function with " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mathbf{x}" }, { "__type": "Text", "__tag": 4046, "value": " fixed and given by the data:" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Where" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that if " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\boldsymbol{\\theta}" }, { "__type": "Text", "__tag": 4046, "value": " includes only shape parameters, the location and scale-parameters can be fit by replacing " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_{i}" }, { "__type": "Text", "__tag": 4046, "value": " with " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\left(x_{i}-L\\right)/S" }, { "__type": "Text", "__tag": 4046, "value": " in the log-likelihood function adding " }, { "__type": "InlineMath", "__tag": 4057, "value": "N\\log S" }, { "__type": "Text", "__tag": 4046, "value": " and minimizing, thus" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If desired, sample estimates for " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "S" }, { "__type": "Text", "__tag": 4046, "value": " (not necessarily maximum likelihood estimates) can be obtained from samples estimates of the mean and variance using" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu_{2}" }, { "__type": "Text", "__tag": 4046, "value": " are assumed known as the mean and variance of the " }, { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "untransformed" } ] }, { "__type": "Text", "__tag": 4046, "value": " distribution (when " }, { "__type": "InlineMath", "__tag": 4057, "value": "L=0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "S=1" }, { "__type": "Text", "__tag": 4046, "value": " ) and" } ] }, { "__type": "Math", "__tag": 4058, "value": "" } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Fitting data" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "We will use" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\overline{y\\left(\\mathbf{x}\\right)}=\\frac{1}{N}\\sum_{i=1}^{N}y\\left(x_{i}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " should be clear from context as the number of samples " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_{i}" } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Standard notation for mean" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Documentation for ranlib, rv2, cdflib" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Eric Weisstein's world of mathematics http://mathworld.wolfram.com/, http://mathworld.wolfram.com/topics/StatisticalDistributions.html" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Documentation to Regress+ by Michael McLaughlin item Engineering and Statistics Handbook (NIST), https://www.itl.nist.gov/div898/handbook/" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Documentation for DATAPLOT from NIST, https://www.itl.nist.gov/div898/software/dataplot/distribu.htm" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Norman Johnson, Samuel Kotz, and N. Balakrishnan Continuous Univariate Distributions, second edition, Volumes I and II, Wiley & Sons, 1994." } ] } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "In the tutorials several special functions appear repeatedly and are listed here." } ] }, { "__type": "Code", "__tag": 4050, "value": "=============================================================== ====================================================================================== =============================================================================================================================\nSymbol Description Definition\n=============================================================== ====================================================================================== =============================================================================================================================\n:math:`\\gamma\\left(s, x\\right)` lower incomplete Gamma function :math:`\\int_0^x t^{s-1} e^{-t} dt`\n:math:`\\Gamma\\left(s, x\\right)` upper incomplete Gamma function :math:`\\int_x^\\infty t^{s-1} e^{-t} dt`\n:math:`B\\left(x;a,b\\right)` incomplete Beta function :math:`\\int_{0}^{x} t^{a-1}\\left(1-t\\right)^{b-1} dt`\n:math:`I\\left(x;a,b\\right)` regularized incomplete Beta function :math:`\\frac{\\Gamma\\left(a+b\\right)}{\\Gamma\\left(a\\right)\\Gamma\\left(b\\right)} \\int_{0}^{x} t^{a-1}\\left(1-t\\right)^{b-1} dt`\n:math:`\\phi\\left(x\\right)` PDF for normal distribution :math:`\\frac{1}{\\sqrt{2\\pi}}e^{-x^{2}/2}`\n:math:`\\Phi\\left(x\\right)` CDF for normal distribution :math:`\\int_{-\\infty}^{x}\\phi\\left(t\\right) dt = \\frac{1}{2}+\\frac{1}{2}\\mathrm{erf}\\left(\\frac{x}{\\sqrt{2}}\\right)`\n:math:`\\psi\\left(z\\right)` digamma function :math:`\\frac{d}{dz} \\log\\left(\\Gamma\\left(z\\right)\\right)`\n:math:`\\psi_{n}\\left(z\\right)` polygamma function :math:`\\frac{d^{n+1}}{dz^{n+1}}\\log\\left(\\Gamma\\left(z\\right)\\right)`\n:math:`I_{\\nu}\\left(y\\right)` modified Bessel function of the first kind\n:math:`\\mathrm{Ei}(\\mathrm{z})` exponential integral :math:`-\\int_{-x}^\\infty \\frac{e^{-t}}{t} dt`\n:math:`\\zeta\\left(n\\right)` Riemann zeta function :math:`\\sum_{k=1}^{\\infty} \\frac{1}{k^{n}}`\n:math:`\\zeta\\left(n,z\\right)` Hurwitz zeta function :math:`\\sum_{k=0}^{\\infty} \\frac{1}{\\left(k+z\\right)^{n}}`\n:math:`\\,{}_{p}F_{q}(a_{1},\\ldots,a_{p};b_{1},\\ldots,b_{q};z)` Hypergeometric function :math:`\\sum_{n=0}^{\\infty} {\\frac{(a_{1})_{n}\\cdots(a_{p})_{n}}{(b_{1})_{n}\\cdots(b_{q})_{n}}} \\,{\\frac{z^{n}}{n!}}`\n=============================================================== ====================================================================================== =============================================================================================================================", "execution_status": null } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "References" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Alpha Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_alpha" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Anglit Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_anglit" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Arcsine Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_arcsine" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Beta Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_beta" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Beta Prime Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_betaprime" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Bradford Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_bradford" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Burr Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_burr" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Burr12 Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:continuous_burr12" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Cauchy Distribution", 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