Don't bin if you don't have to bin. But if that's all you have....
If you have binned data with $n$ bins, boundaries $x_ 1< x_ 2< \cdots < x_ {n +
1}$, and counts $c_ 1, c_ 2, \ldots, c_n$ for a proposed
distribution with cumulative distribution function (CDF) $F$,
then maximum likelihood estimators are the values of the parameters
that maximum the likelihood. Usually the log of the likelihood is maximized as that can be
more numerically stable when iteration is needed and sometimes
results in simple closed-form estimators. We have
$$log (L) = \sum_ {i = 1}^n c_i \log (F (x_ {i + 1}) - F (x_ {i})) $$
Here is some code when the distribution is normal with unknown mean and variance:
(* Random sample from a known distribution *)
SeedRandom[12345];
n = 10000;
data = RandomVariate[NormalDistribution[5, 3], n];
(* Create a histogam *)
nBins = 20;
h = HistogramList[data, nBins];
(* Bin boundaries *)
x = h[[1]]
(* {-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18} *)
(* Frequency counts *)
c = h[[2]]
(* {4,7,27,65,136,244,443,656,949,1234,1299,1292,1148,932,690,420,250,122,53,17,9,0,2,1} *)
(* Find the log of the likelihood for the binned data *)
logL = Total[Table[c[[i]] Log[CDF[NormalDistribution[μ, σ], x[[i + 1]]] -
CDF[NormalDistribution[μ, σ], x[[i]]]], {i, nBins}]];
(* Find values of μ and σ that maximize the log of the likelihood *)
(* Initial values *)
(μ0 = Sum[c[[i]] (x[[i + 1]] + x[[i]])/2, {i, nBins}]/Total[c]) // N
(* 4.9439 *)
(σ0 = (Sum[c[[i]] ((x[[i + 1]] + x[[i]])/2 - μ0)^2, {i, nBins}]/Total[c])^(1/2)) // N
(* 2.9738228281705013 *)
(* Maximim likelihood estimates *)
mle = FindMaximum[{logL, σ > 0}, {{μ, μ0}, {σ, σ0}}]
(* {-25063.7, {μ -> 4.94984, σ -> 2.96156}} *)
(* Now get estimates of the associated standard errors *)
(covMat = -Inverse[D[logL, {{μ, σ}, 2}] /. mle[[2]]]) // MatrixForm
seμ = covMat[[1, 1]]^0.5
(* 0.029773837258604677 *)
seσ = covMat[[2, 2]]^0.5
(* 0.021152624920503942 *)
(* Display histogram and estimated density *)
Show[Histogram[data, nBins, "PDF"],
Plot[PDF[NormalDistribution[μ, σ] /. mle[[2]], z], {z, x[[1]], x[[nBins + 1]]}]]
Your comment
For example data that may appear Gaussian distributed, may have a
slight exponential tilt only revealed when binning is fine enough.
is true but has nothing to do with fitting a specific distribution. The fit is conditional on assuming the form of the distribution (i.e., the form known but not necessarily all of the parameters). If you suspect deviations from a particular distribution then you need to try different forms of distributions or fit a nonparametric density estimate (using SmoothHistogram or SmoothKernelDistribution) but that requires un-binned data.
WeightedData(and hopefully with an associated bootstrap to get estimates of standard errors) or use a maximum likelihood approach which uses theCDFfunction which gives you the probability of getting a particular count in a bin which in turn goes into the log likelihood function that you'll choose the parameters that maximize the log likelihood. You can also get estimates of standard errors. (If someone doesn't write that up, I will after I get some pizza and beer for dinner.) $\endgroup$