During testing my code for calculate parameters The following exception is thrown:

FindMaximum::eit: The algorithm does not converge to the tolerance of _4.806217383937354`*^-6_ in _500_ iterations
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This is an extended comment about your pdf not integrating to 1.

I've typed in the code from the image you posted and I think it looks exactly like the image:

dist = ProbabilityDistribution[(1 - Exp[λ*(1 - Exp[1])])^(-1) *λ*β*(γ/α)*(x/α)^(γ - 1)*
  (1 - Exp[-(x/α)^γ])^(β - 1)*Exp[λ*(1 - Exp[-(x/α)^γ])^β - (x/α)^γ + (1 - Exp[-(x/α)^γ])], {x, 1, ∞}, 
  Assumptions -> {λ > 0, β > 0, α > 0, γ > 0}]

My result as an image: My distribution

Your image of the resulting distribution:

Your image of the resulting distribution

But if a set of parameters is tried, the pdf does not integrate to 1:

parms = {λ -> 2, α -> 1, β -> 1, γ -> 1};
Integrate[PDF[dist /. parms, x], {x, 1, ∞}] // N
(* 9.2468 *)

There is an option for ProbabilityDistribution to normalize the pdf (Method->"Normalize") but the proposed pdf seems too complicated for that to work on the symbolic distribution. (In other words, doing so does not return an answer in a reasonable or unreasonable amount of time.)


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