Fit distribution using moments

There is a `fitdist` function to find distribution based on the given data.

Is there a function to find a distribution given moments?

Thank you.

Answer 1

[John D'Errico]

This is something commonly done using either the Pearson or Johnson family of distributions. (It has been many years since I looked at this, so there may be other choices by now too.) The MATLAB stats TB offers the Pearson family (which I think I recall is the better choice anyway.)

lookfor pearson

pearscdf                       - Pearson cumulative distribution function
pearspdf                       - Pearson probability density function
pearsrnd                       - Pearson system random numbers

For example, given the first 4 moments, we can use those tools.

help pearscdf

 PEARSCDF Cumulative distribution function (CDF) for Pearson system of distributions
 
    F = PEARSCDF(X, MU, SIGMA, SKEW, KURT) returns an array F of the
    Pearson system's cumulative distribution function evaluated at
    double/single array X. The evaluated Pearson distribution has mean MU,
    standard deviation SIGMA, skewness SKEW, and kurtosis KURT. MU and SIGMA
    can be double/single scalars or arrays. If they are arrays, each must
    have the same size as X. SKEW, and KURT must be double/single 
    scalars.
 
    The kurtosis KURT must be greater than the square of the skewness SKEW
    plus 1.  The kurtosis of the normal distribution 3.
 
    F = PEARSCDF(__, 'upper') returns the complement of the CDF. For all
    Pearson types except 4, this syntax uses an algorithm that more
    accurately computes the extreme upper-tail probabilities.
 
    [F, TYPE] = PEARSCDF(__) returns the type TYPE of the specified 
    distribution within the Pearson system.  Type is a scalar integer from 
    0 to 7. The eight distribution types in the Pearson system correspond 
    to the following distributions:
 
       Type 0: Normal distribution
       Type 1: Four-parameter beta
       Type 2: Symmetric four-parameter beta
       Type 3: Three-parameter gamma
       Type 4: Not related to any standard distribution. Density proportional
               to (1+((x-mu)/sigma)^2)^(-a) * exp(-b*arctan((x-mu)/sigma))
               where a and b are quantities related to the differential
               equation that defines the Pearson system.
       Type 5: Inverse gamma location-scale
       Type 6: F location-scale
       Type 7: Student's t location-scale
 
    [F,TYPE,COEFS] = PEARSCDF(__) returns the coefficients of the quadratic
    polynomial that defines the distribution via the following differential 
    equation:
    d(log(p(x)))/dx = -(a + x) / (COEFS(1) + COEFS(2)*x + COEFS(3)*x^2)
 
    Example: Evaluate the Pearson distribution CDF at parameters that
             correspond to the standard Normal:
 
    % mu=0, sigma=1, skew=0, kurt=3 for the Pearson distribtion
    % corresponds to the standard Normal distribution
    % Evaluate this CDF from -1 to 1 in increments of 0.1
    [cdf, type] = pearscdf(-1:0.1:1, 1, 2, 0, 3)
 
    See also PEARSPDF, PEARSRND, CDF

    Documentation for pearscdf
       doc pearscdf

How would you use them? Simple enough. For example, I know the normal distribution has skewness zero, and kurtosis 3. So a normal distribution, with mean 1, and standard deviation 1/2 would have the cdf...

[~,T] = pearscdf([],1,1/2,0,3)

T = 0

A Pearson type 0 is a normal distribution. Whoopsie do! I got it right!

And we can plot the cdf as:

fplot(@(x) pearscdf(x,1,1/2,0,3))

You can use pearsrnd to generate random numbers from that distribution, etc.