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  "Title": "Distribution Functions and Parameter Estimates for the Triangle\nDistribution",
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    },
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    "dtriangle",
    "pltriangle",
    "ptriangle",
    "qltriangle",
    "qqtriangle",
    "qtriangle",
    "rltriangle",
    "rtriangle",
    "standard_triangle_mle",
    "triangle_cdfe",
    "triangle_mle",
    "triangle_mom"
  ],
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    {
      "page": "compare_triangle_fit",
      "title": "Compare multiple triangle distributions fits",
      "topics": [
        "compare_triangle_fit"
      ]
    },
    {
      "page": "ltriangle",
      "title": "The Log-Triangle Distribution",
      "topics": [
        "dltriangle",
        "ltriangle",
        "pltriangle",
        "qltriangle",
        "rltriangle"
      ]
    },
    {
      "page": "qqtriangle",
      "title": "Quantile-Quantile Plot for Triangle Distributed Data",
      "topics": [
        "qqtriangle"
      ]
    },
    {
      "page": "standard_triangle_mle",
      "title": "Maximum likelihood estimate of the standard triangle distribution mode",
      "topics": [
        "standard_triangle_mle"
      ]
    },
    {
      "page": "mle-utils",
      "title": "Utility Methods for S3 class triangle_mle",
      "topics": [
        "AIC.triangle_mle",
        "BIC.triangle_mle",
        "coef.triangle_mle",
        "confint.triangle_mle",
        "logLik.triangle_mle",
        "print.triangle_mle",
        "profile.triangle_mle",
        "summary.triangle_mle",
        "vcov.triangle_mle"
      ]
    },
    {
      "page": "triangle",
      "title": "The Triangle Distribution",
      "topics": [
        "dtriangle",
        "ptriangle",
        "qtriangle",
        "rtriangle",
        "triangle"
      ]
    },
    {
      "page": "triangle_cdfe",
      "title": "Triangle parameter estimates using a non-linear fit of the empirical CDF",
      "topics": [
        "triangle_cdfe"
      ]
    },
    {
      "page": "triangle_mle",
      "title": "Maximum likelihood estimate of the triangle distribution parameters",
      "topics": [
        "triangle_mle"
      ]
    },
    {
      "page": "triangle_mom",
      "title": "Triangle distribution method of moments estimate",
      "topics": [
        "triangle_mom"
      ]
    }
  ],
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  ],
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      "filename": "triangledistributionmath.html",
      "title": "Triangle Distribution Math",
      "author": "Rob Carnell",
      "engine": "knitr::rmarkdown",
      "headings": [
        "Triangle Notation",
        "Triangle Probability Density Funciton (PDF)",
        "Triangle Cumulative Distribution Function (CDF)",
        "Triangle Mean",
        "Triangle Variance",
        "Method of Moments Estimation",
        "Type 1",
        "Type 2",
        "Maximum Likelihood Estimation",
        "Maximizing the Likelihood with respect to $c$ (given $a$ and $b$)",
        "Case 1:  $c$ is between the first and second to last order statistic $r \\ \\epsilon \\ (1, \\dots, n-1)$",
        "Side note on $z=c^r(1-c)^{n-r}$ being unimodal",
        "Case 2:  $c$ is between 0 and the first order statistic $r = 0$",
        "Case 3:  $c$ is between the last order statistic $r = n$ and 1",
        "All Cases",
        "Negative Log Likelihood",
        "Case 1:  $a = c \\lt b$",
        "Case 2:  $a \\lt c = b$",
        "Case 3:  $a \\lt c \\lt b$",
        "Gradient of the negative Log Likelihood Given $c$:",
        "Hessian of the negative Log Likelihood Given $c$:",
        "MLE Variance - Covariance",
        "$r^{th}$ order statistic",
        "Expected value of the $r^{th}$ order statistic",
        "Expected Value of $r^{th}$ order statistic squared",
        "Variance of the $r^{th}$ order statistic",
        "Numerical Stability of Variance and Expected value of $r^{th}$ order statistic",
        "Logarithmic Triangle distribution"
      ],
      "created": "2018-12-16 20:54:57",
      "modified": "2026-04-26 23:49:46",
      "commits": 10
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