Fitting data to a custom Pareto distribution

Question

data = {1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 1.865, 1.944,
    1.958, 1.966, 1.997, 2.006, 2.021, 2.027, 2.055, 2.063, 2.098, 
   2.14, 2.179, 2.224, 2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 
   2.301, 2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478, 2.490, 
   2.511, 2.514, 2.535, 2.554, 2.566, 2.57, 2.586, 2.629, 2.633, 
   2.642, 2.648, 2.684, 2.697, 2.726, 2.770, 2.773, 2.800, 2.809, 
   2.818, 2.821, 2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090, 
   3.096, 3.128, 3.233, 3.433, 3.585, 3.585};

I constructed a basic Pareto distribution and fitted it to the data to obtain the best-fit value of the distribution parameter $\lambda$:

dist = ProbabilityDistribution[{"CDF", 1 - x^(-\[Lambda])}, {x, 0, \[Infinity]}, Assumptions -> {\[Lambda] > 0}]


mle = FindDistributionParameters[data, dist]

This returned {\[Lambda] -> 1.14273}

I then wanted to plot fitted vs predicted values so I tried ProbabilityScalePlot[data, dist] but it returned an error:

I ultimately want to calculate this DistributionFitTest[data, dist, {"TestDataTable", "AndersonDarling"}]:

I would like to use my custom distribution, rather than a built-in from Mathematica, and i want to plot its empirical CDF.

Answer 1

data = {1.312, 1.314, 1.479, 1.552, 1.700, 1.803, 1.861, 
        1.865, 1.944, 1.958, 1.966, 1.997, 2.006, 2.021,
        2.027, 2.055, 2.063, 2.098, 2.14, 2.179, 2.224,
        2.240, 2.253, 2.270, 2.272, 2.274, 2.301, 2.301,
        2.359, 2.382, 2.382, 2.426, 2.434, 2.435, 2.478,
        2.490, 2.511, 2.514, 2.535, 2.554, 2.566, 2.57,
        2.586, 2.629, 2.633, 2.642, 2.648, 2.684, 2.697,
        2.726, 2.770, 2.773, 2.800, 2.809, 2.818, 2.821,
        2.848, 2.88, 2.954, 3.012, 3.067, 3.084, 3.090,
        3.096, 3.128, 3.233, 3.433, 3.585, 3.585};

dist[λ_] = ProbabilityDistribution[{"CDF", 1 - x^(-λ)}, {x, 1, ∞}];

Λ = λ /. FindDistributionParameters[data, dist[λ]]
(*    1.14273    *)

ProbabilityPlot[data, dist[Λ]]

DistributionFitTest[data, dist[Λ], {"TestDataTable", "AndersonDarling"}]

$$ \begin{array}{l|ll} \text{} & \text{Statistic} & \text{P-Value} \\ \hline \text{Anderson-Darling} & 18.343 & \text{8.698112014204895$\grave{ }$*${}^{\wedge}$-6} \\ \end{array} $$

Notes:

I've modified the domain of the given CDF to $[1,\infty)$ so that it stays within the required range $[0,1]$.

The empirical distribution is EmpiricalDistribution[data]:

datacdf[x_] = CDF[datadist][x];
Plot[datacdf[x], {x, 1, 4}, PlotPoints -> 100]