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Formatted code. To OP: Please check to see if I did so correctly.
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JimB
JimB
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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^\[Xi]x^ξ)^-\[Psi] ψ)^\[Lambda]^λ/(1 - (1 - (1 + x^\[Xi]x^ξ)^-\[Psi] ψ)^\[Lambda]^λ))^[Gamma]^γ)^-\[Alpha]α)^(\\[Beta]β)
\[ScriptCapitalD]pdf = ProbabilityDistribution[-((\[Alpha] \[Beta] \[Gamma] \[Lambda] \\[Xi] \[Psi] x^(-1 + \[Xi]) (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^-\[Alpha])^\\[Beta])/((1 + x^\[Xi]) (-1 + (1 + x^\[Xi])^\[Psi]) (-1 + (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma]) (-1 + (1 + \(-1 + 1/(1 - (1 -D[cdf, (1x];
\[ScriptCapitalD] += x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^\[Alpha])))\ProbabilityDistribution[pdf, {x, 0, Infinity} , 
   Assumptions -> \[Alpha]α > 0 && \[Beta]β > 0 && \[Gamma]γ > 0 && \[Lambda]λ > 0 && \[Psi]ψ > 0 && \[Xi]ξ > 0  ];0];

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^\[Xi])^-\[Psi] )^\[Lambda]/(1 - (1 - (1 + x^\[Xi])^-\[Psi] )^\[Lambda]))^[Gamma])^-\[Alpha])^(\\[Beta])
\[ScriptCapitalD] = ProbabilityDistribution[-((\[Alpha] \[Beta] \[Gamma] \[Lambda] \\[Xi] \[Psi] x^(-1 + \[Xi]) (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^-\[Alpha])^\\[Beta])/((1 + x^\[Xi]) (-1 + (1 + x^\[Xi])^\[Psi]) (-1 + (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma]) (-1 + (1 + \(-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^\[Alpha])))\, {x, 0, Infinity} , Assumptions -> \[Alpha] > 0 && \[Beta] > 0 && \[Gamma] > 0 && \[Lambda] > 0 && \[Psi] > 0 && \[Xi] > 0  ];

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];
added 50 characters in body
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b.gates.you.know.what
b.gates.you.know.what
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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^[Xi])^-[Psi] )^[Lambda]/(1 - (1 - (1 + x^[Xi])^-[Psi] )^[Lambda]))^[Gamma])^-[Alpha])^(\[Beta]) [ScriptCapitalD] = ProbabilityDistribution[-(([Alpha] [Beta] [Gamma] [Lambda] \[Xi] [Psi] x^(-1 + [Xi]) (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^-[Alpha])^\[Beta])/((1 + x^[Xi]) (-1 + (1 + x^[Xi])^[Psi]) (-1 + (1 - (1 + x^[Xi])^-[Psi])^[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma]) (-1 + (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^[Alpha]))), {x, 0, Infinity} , Assumptions -> [Alpha] > 0 && [Beta] > 0 && [Gamma] > 0 && [Lambda] > 0 && [Psi] > 0 && [Xi] > 0 ];

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^\[Xi])^-\[Psi] )^\[Lambda]/(1 - (1 - (1 + x^\[Xi])^-\[Psi] )^\[Lambda]))^[Gamma])^-\[Alpha])^(\\[Beta])
\[ScriptCapitalD] = ProbabilityDistribution[-((\[Alpha] \[Beta] \[Gamma] \[Lambda] \\[Xi] \[Psi] x^(-1 + \[Xi]) (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^-\[Alpha])^\\[Beta])/((1 + x^\[Xi]) (-1 + (1 + x^\[Xi])^\[Psi]) (-1 + (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma]) (-1 + (1 + \(-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^\[Alpha])))\, {x, 0, Infinity} , Assumptions -> \[Alpha] > 0 && \[Beta] > 0 && \[Gamma] > 0 && \[Lambda] > 0 && \[Psi] > 0 && \[Xi] > 0  ];

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^[Xi])^-[Psi] )^[Lambda]/(1 - (1 - (1 + x^[Xi])^-[Psi] )^[Lambda]))^[Gamma])^-[Alpha])^(\[Beta]) [ScriptCapitalD] = ProbabilityDistribution[-(([Alpha] [Beta] [Gamma] [Lambda] \[Xi] [Psi] x^(-1 + [Xi]) (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^-[Alpha])^\[Beta])/((1 + x^[Xi]) (-1 + (1 + x^[Xi])^[Psi]) (-1 + (1 - (1 + x^[Xi])^-[Psi])^[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma]) (-1 + (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^[Alpha]))), {x, 0, Infinity} , Assumptions -> [Alpha] > 0 && [Beta] > 0 && [Gamma] > 0 && [Lambda] > 0 && [Psi] > 0 && [Xi] > 0 ];

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^\[Xi])^-\[Psi] )^\[Lambda]/(1 - (1 - (1 + x^\[Xi])^-\[Psi] )^\[Lambda]))^[Gamma])^-\[Alpha])^(\\[Beta])
\[ScriptCapitalD] = ProbabilityDistribution[-((\[Alpha] \[Beta] \[Gamma] \[Lambda] \\[Xi] \[Psi] x^(-1 + \[Xi]) (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^-\[Alpha])^\\[Beta])/((1 + x^\[Xi]) (-1 + (1 + x^\[Xi])^\[Psi]) (-1 + (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma]) (-1 + (1 + \(-1 + 1/(1 - (1 - (1 + x^\[Xi])^-\[Psi])^\[Lambda]))^\[Gamma])^\[Alpha])))\, {x, 0, Infinity} , Assumptions -> \[Alpha] > 0 && \[Beta] > 0 && \[Gamma] > 0 && \[Lambda] > 0 && \[Psi] > 0 && \[Xi] > 0  ];
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Amin
Amin
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How we fit our own generated probability distribution on real data

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^[Xi])^-[Psi] )^[Lambda]/(1 - (1 - (1 + x^[Xi])^-[Psi] )^[Lambda]))^[Gamma])^-[Alpha])^(\[Beta]) [ScriptCapitalD] = ProbabilityDistribution[-(([Alpha] [Beta] [Gamma] [Lambda] \[Xi] [Psi] x^(-1 + [Xi]) (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma] (1 - (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^-[Alpha])^\[Beta])/((1 + x^[Xi]) (-1 + (1 + x^[Xi])^[Psi]) (-1 + (1 - (1 + x^[Xi])^-[Psi])^[Lambda]) (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma]) (-1 + (1 + (-1 + 1/(1 - (1 - (1 + x^[Xi])^-[Psi])^[Lambda]))^[Gamma])^[Alpha]))), {x, 0, Infinity} , Assumptions -> [Alpha] > 0 && [Beta] > 0 && [Gamma] > 0 && [Lambda] > 0 && [Psi] > 0 && [Xi] > 0 ];