Estimating Confidence Interval Variance covariance matrix from weibull distribution

Question

In The following Example 50 obs . I estimated distribution parameters (by point) but *i want to estimate parameters by confidence interval and calculate covariance matrix i need help *

 `W = {1.1387057619769245, 0.9413936042888995, 3.4223017890169585, 
  4.7312263532673775, 2.667241278799745, 3.3983134686460916, 
  2.636138174853885, 0.9705263776287205, 3.6649383055213263, 
  2.2143855476032965, 2.446558570934328, 4.58500104927277, 
  1.2265125056342852, 4.941351320784421, 1.8537116849991357, 
  1.0307354130400692, 2.9330720895216333, 2.181876850133328, 
  1.9478680535068564, 1.5797371732389491, 3.6973649466255814, 
  1.9142930470727724, 2.9830638549889015, 4.035435643539325, 
  2.531189805912063, 2.7105005421868675, 3.1664423157344235, 
  3.961740252540448, 0.7741223547449858, 5.586933987488866, 
  1.1120311064331565, 2.5478799049118757, 4.099380042175206, 
  2.866064958359309, 2.267794578506713, 0.8004793762222566, 
  0.9552216662568689, 0.9493583653491414, 4.05651509710675, 
  1.344905272000637, 2.1643243303245954, 2.760896995257153, 
  3.75913686974915, 1.1082504794305363, 0.7222438008616832, 
  2.2536543275392185, 2.8622705077499395, 2.31755560051214, 
  5.211296471564994, 1.745559742802774}`

FindDistributionParameters[W, WeibullDistribution[a, b]]
{a -> 2.1449, b -> 2.89501}

Answer 1

FindDistributionParameters and EstimatedDistribution do not provide information to construct confidence intervals conveniently. A possible approach is to use NonlinearModelFit using the empirical cumulative distribution of the data as input and the CDF of WeibullDistribution as the model to be estimated.

edistdata = Table[{x, CDF[EmpiricalDistribution[W], x]}, {x, W}];
cdf[a_, b_, x_] := Simplify[CDF[WeibullDistribution[a, b], x], x > 0];
cdf[a, b, x]

$1-e^{-\left(\frac{x}{b}\right)^a}$

nlm = NonlinearModelFit[edistdata, cdf[a, b, x], {a, b}, x];
nlm["BestFit"]

$1-e^{-0.126786 x^{1.96278}}$

nlm["BestFitParameters"]
(* {a -> 1.96278,b -> 2.86397} *)

nlm["ParameterConfidenceIntervalTable"]

nlm["CovarianceMatrix"] // MatrixForm