How to fit a Weibull distribution function to histogram with unequal bins?

Question

Whenever the function "FindDistributionParameter" is used in Wolfram Mathematica, it needs two inputs: raw data and distribution function.

I have a problem where I have have data and I need to use Weibull distribution function, but bins in my histogram are unevenly spaced.

How do I add that information into "FindDistributionParameters" function in order to get a proper fit?

Answer 1

One can't use FindDistributionParameters once the data becomes binned. However, one can construct the log of the likelihood function and then use FindMaximum as long as you still have the counts. (The relative frequencies won't do.)

(* Generate histogram data with unequal bin widths *)
SeedRandom[12345]
x = RandomVariate[WeibullDistribution[2, 3], 1000];
{borders, counts} = HistogramList[Log[x]];
borders = Exp[borders] // N
(* {0.090718, 0.110803, 0.135335, 0.165299, 0.201897, 0.246597, 0.301194, 0.367879,
    0.449329, 0.548812, 0.67032, 0.818731, 1., 1.2214, 1.49182, 1.82212, 2.22554, 
    2.71828, 3.32012, 4.0552, 4.95303, 6.04965, 7.38906, 9.02501} *)
counts
(* {1, 1, 1, 1, 3, 3, 4, 7, 6, 16, 17, 26, 58, 59, 94, 117, 148, 176, 112, 78, 58, 13, 1} *)

(* Log of the likelihood *)
logL = Sum[counts[[i]] Log[CDF[WeibullDistribution[a, b], borders[[i + 1]]] - 
  CDF[WeibullDistribution[a, b], borders[[i]]]], {i, Length[counts]}];

(* Find maximum likelihood estimates using the default starting values of 1 for both parameters *)
mle = FindMaximum[{logL, a > 0 && b > 0}, {{a, 1}, {b, 1}}]
(* {-2460.29, {a -> 2.0714, b -> 2.9929}} *)

(* Estimates of standard errors *)
cov = -Inverse[(D[logL, {{a, b}, 2}]) /. mle[[2]]];
ase = cov[[1, 1]]^0.5
(* 0.0516358 *)
bse = cov[[2, 2]]^0.5
(* 0.0485239 *)