How we fit our own generated probability distribution on real data

Question

I want to fit derived distribution on real data and I need MLE parameters, standard errors of the estimated parameters, AIC, Loglikelihood, Goodness of fit test (Chi-square, Anderson etc), PDF, CDF, Plots and QQplots. The data,CDF and PDF are

data = {3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 
   4.42, 2.41, 3.19, 3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 3.75, 
   2.43, 2.95, 2.97, 3.39, 2.96, 2.53, 2.67, 2.93, 3.22, 3.39, 2.81, 
   4.20, 3.33, 2.55, 3.31, 3.31, 2.85, 2.56, 3.56, 3.15, 2.35, 2.55, 
   2.59, 2.38, 2.81, 2.77, 2.17, 2.83, 1.92, 1.41, 3.68, 2.97, 1.36, 
   0.98, 2.76, 4.91, 3.68, 1.84, 1.59, 3.19, 1.57, 0.81, 5.56, 1.73, 
   1.59, 2.00, 1.22, 1.12, 1.71, 2.17, 1.17, 5.08, 2.48, 1.18, 3.51, 
   2.17, 1.69, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 
   4.70, 2.03, 1.80, 1.57, 1.08, 2.03, 1.61, 2.12, 1.89, 2.88, 2.82, 
   2.05, 3.65};
cdf = (1 - (1 + ((1 - (1 + x^ξ)^-ψ)^λ/(1 - (1 - (1 + x^ξ)^-ψ)^λ))^γ)^-α)^(β)
pdf = D[cdf, x];
\[ScriptCapitalD] = ProbabilityDistribution[pdf, {x, 0, Infinity}, 
   Assumptions -> α > 0 && β > 0 && γ > 0 && λ > 0 && ψ > 0 && ξ > 0];

Answer 1

This is an extended comment rather than an answer.

I think that there are two issues: (1) the amount of data available is inadequate to estimate all 6 parameters, and (2) there is a potential for a severe amount of numeric instability.

A straightforward approach to estimate the parameters is to use FindDistributionParameters:

mle = FindDistributionParameters[data, \[ScriptCapitalD]]
(* {α -> 10.6474, β -> 73.7566, γ -> 67.2255, λ -> 73.7566, ξ -> 67.2255, ψ -> 10.6474} *)

But then try to obtain a plot of the pdf:

Plot[pdf /. mle, {x, Min[data], Max[data]}]

Generating just 10 bootstrap estimates of the parameters results in the following:

SeedRandom[12345]
boot = RandomChoice[data, {10, Length[data]}];
mle = {α, β, γ, λ, ξ, ψ} /. FindDistributionParameters[#, \[ScriptCapitalD]] & /@ boot;
TableForm[mle, TableHeadings -> {Range[10], {"α", "β", "γ", "λ", "ξ", "ψ"}}]

We see only two different values for $\alpha$ and $\psi$ and pairs of estimates are identical: (1) $\alpha$ and $\psi$, (2) $\beta$ and $\lambda$, and (3) $\gamma$ and $\xi$.

So I think your data is woefully inadequate to estimate all of those parameters. You should set some values for the parameters, take a large random sample, and see if the estimates get closer to the true values.