fit#, *args, **kwds)[source]#

Return estimates of shape (if applicable), location, and scale parameters from data. The default estimation method is Maximum Likelihood Estimation (MLE), but Method of Moments (MM) is also available.

Starting estimates for the fit are given by input arguments; for any arguments not provided with starting estimates, self._fitstart(data) is called to generate such.

One can hold some parameters fixed to specific values by passing in keyword arguments f0, f1, …, fn (for shape parameters) and floc and fscale (for location and scale parameters, respectively).

dataarray_like or CensoredData instance

Data to use in estimating the distribution parameters.

arg1, arg2, arg3,…floats, optional

Starting value(s) for any shape-characterizing arguments (those not provided will be determined by a call to _fitstart(data)). No default value.

**kwdsfloats, optional
  • loc: initial guess of the distribution’s location parameter.

  • scale: initial guess of the distribution’s scale parameter.

Special keyword arguments are recognized as holding certain parameters fixed:

  • f0…fn : hold respective shape parameters fixed. Alternatively, shape parameters to fix can be specified by name. For example, if self.shapes == "a, b", fa and fix_a are equivalent to f0, and fb and fix_b are equivalent to f1.

  • floc : hold location parameter fixed to specified value.

  • fscale : hold scale parameter fixed to specified value.

  • optimizer : The optimizer to use. The optimizer must take func and starting position as the first two arguments, plus args (for extra arguments to pass to the function to be optimized) and disp. The fit method calls the optimizer with disp=0 to suppress output. The optimizer must return the estimated parameters.

  • method : The method to use. The default is “MLE” (Maximum Likelihood Estimate); “MM” (Method of Moments) is also available.

parameter_tupletuple of floats

Estimates for any shape parameters (if applicable), followed by those for location and scale. For most random variables, shape statistics will be returned, but there are exceptions (e.g. norm).

TypeError, ValueError

If an input is invalid


If fitting fails or the fit produced would be invalid


With method="MLE" (default), the fit is computed by minimizing the negative log-likelihood function. A large, finite penalty (rather than infinite negative log-likelihood) is applied for observations beyond the support of the distribution.

With method="MM", the fit is computed by minimizing the L2 norm of the relative errors between the first k raw (about zero) data moments and the corresponding distribution moments, where k is the number of non-fixed parameters. More precisely, the objective function is:

(((data_moments - dist_moments)
  / np.maximum(np.abs(data_moments), 1e-8))**2).sum()

where the constant 1e-8 avoids division by zero in case of vanishing data moments. Typically, this error norm can be reduced to zero. Note that the standard method of moments can produce parameters for which some data are outside the support of the fitted distribution; this implementation does nothing to prevent this.

For either method, the returned answer is not guaranteed to be globally optimal; it may only be locally optimal, or the optimization may fail altogether. If the data contain any of np.nan, np.inf, or -np.inf, the fit method will raise a RuntimeError.

When passing a CensoredData instance to data, the log-likelihood function is defined as:

\[\begin{split}l(\pmb{\theta}; k) & = \sum \log(f(k_u; \pmb{\theta})) + \sum \log(F(k_l; \pmb{\theta})) \\ & + \sum \log(1 - F(k_r; \pmb{\theta})) \\ & + \sum \log(F(k_{\text{high}, i}; \pmb{\theta}) - F(k_{\text{low}, i}; \pmb{\theta}))\end{split}\]

where \(f\) and \(F\) are the pdf and cdf, respectively, of the function being fitted, \(\pmb{\theta}\) is the parameter vector, \(u\) are the indices of uncensored observations, \(l\) are the indices of left-censored observations, \(r\) are the indices of right-censored observations, subscripts “low”/”high” denote endpoints of interval-censored observations, and \(i\) are the indices of interval-censored observations.


Generate some data to fit: draw random variates from the beta distribution

>>> import numpy as np
>>> from scipy.stats import beta
>>> a, b = 1., 2.
>>> rng = np.random.default_rng()
>>> x = beta.rvs(a, b, size=1000, random_state=rng)

Now we can fit all four parameters (a, b, loc and scale):

>>> a1, b1, loc1, scale1 =
>>> a1, b1, loc1, scale1
(1.0198945204435628, 1.9484708982737828, 4.372241314917588e-05, 0.9979078845964814)

The fit can be done also using a custom optimizer:

>>> from scipy.optimize import minimize
>>> def custom_optimizer(func, x0, args=(), disp=0):
...     res = minimize(func, x0, args, method="slsqp", options={"disp": disp})
...     if res.success:
...         return res.x
...     raise RuntimeError('optimization routine failed')
>>> a1, b1, loc1, scale1 =, method="MLE", optimizer=custom_optimizer)
>>> a1, b1, loc1, scale1
(1.0198821087258905, 1.948484145914738, 4.3705304486881485e-05, 0.9979104663953395)

We can also use some prior knowledge about the dataset: let’s keep loc and scale fixed:

>>> a1, b1, loc1, scale1 =, floc=0, fscale=1)
>>> loc1, scale1
(0, 1)

We can also keep shape parameters fixed by using f-keywords. To keep the zero-th shape parameter a equal 1, use f0=1 or, equivalently, fa=1:

>>> a1, b1, loc1, scale1 =, fa=1, floc=0, fscale=1)
>>> a1

Not all distributions return estimates for the shape parameters. norm for example just returns estimates for location and scale:

>>> from scipy.stats import norm
>>> x = norm.rvs(a, b, size=1000, random_state=123)
>>> loc1, scale1 =
>>> loc1, scale1
(0.92087172783841631, 2.0015750750324668)