Finding the optimal probabilty distribution

Question

I am trying to obtain the optimal probability distribution function to the following data.

My attempt:

data = Import["data_prob_new.dat", "Table"];
P00 = Histogram[Flatten[data], Automatic, "Probability", 
ChartStyle -> Gray, ChartBaseStyle -> EdgeForm[None], 
ImageSize -> 500]
fit = FindDistributionParameters[Flatten[data], 
LaplaceDistribution[a, b]]; 
a0 = a /. fit[[1]];
b0 = b /. fit[[2]];
lim = 1000;
t0 = Plot[PDF[LaplaceDistribution[a0, b0], x], {x, a0, lim}, 
PlotStyle -> {Blue, Thick}, PlotRange -> All];
P0 = Show[{P00, t0}, Frame -> True, Axes -> False, 
FrameStyle -> Thick, PlotRange -> {{0, All}, {-0.001, All}}, 
PlotRangePadding -> 0, PlotRangeClipping -> True]

As we can see, the Laplace probability distribution fails to smoothly fit the tail of the histogram. My question: How can we obtain the best fit (type of distribution) for this histogram?

Answer 1

Given that you just have data and an urge to fit a parametric probability distribution, the quality of the fit is in the eye of the beholder. Here's the fit with the gamma distribution (suggested by @mikado) (which I think you'll be hard-pressed to find a better fit):

P00 = Histogram[data, Automatic, "PDF", ChartStyle -> Gray, 
   ChartBaseStyle -> EdgeForm[None], ImageSize -> 500, 
   PlotRange -> All];
fit = FindDistributionParameters[data, 
   GammaDistribution[α, β, γ, μ]];

t0 = Plot[PDF[GammaDistribution[α, β, γ, μ] /. fit, x], {x, μ0, Max[data]},
   PlotStyle -> {Blue, Thick}, PlotRange -> All, PlotPoints -> 100];
P0 = Show[{P00, t0}, Frame -> True, Axes -> False, FrameStyle -> Thick,
  PlotRange -> {{0, All}, {0, All}}, PlotRangePadding -> 0, PlotRangeClipping -> True]