According to this Wikipedia article:
https://en.wikipedia.org/wiki/Exponentially_modified_Gaussian_distribution
One way to estimate the exp_gaussian parameteres given a data sample is with the method of moments. For example, to estimate the exponential decay term, $τ$ :
$ τ = s \cdot ( \frac{γ_{1}}{2} ) ^ {1/3}$
Where $γ_{1}$ is the skewness and $s$ the standard deviation.
If we take a random sample (too small for estimation, but still) $[957,1,5,271,12,85, 12,658,4719]$:
$ m = 746.66 $
$ s = 1528.118 $
$ γ_{1} = 2.254 $
$ τ = 1590.351 $
I would like to know several things. Are this results correct (particularly for the exponential term)? What does the τ value mean graphically (what happens when it's negative, positive, etc)?
The probability density function of an exponentially modified gaussian is given by this equation:
$f(x;μ,σ,λ) = \frac{λ}{2}e^{\frac{λ}{2}(2μ+λσ^2-2x)} erfc(\frac{μ+λσ^2-x}{\sqrt{2}σ}) $
Where erfc is the complementary error function or:
$erfc(x) = 1 - erf(x)$
The python library SciPy has implemented erf as
scipy.special.erf(). We can use it with Matplotlib to represent how the function changes by varying λ (which has the same value as $\frac{1}{τ}$):EMG for different λ
My interpretation of this is that, as τ increases, a higher percentage of the distribution is accumulated in the right end of the distribution. τ is exponent relaxation time, meaning a lower τ would make the distribution reach it's horizontal asymptote much faster than a higher τ.
To confirm this practically, I created 2 lists that contained 20000 groups of numbers. In the first list I added the group [-2,2] while in the second list I added the group [4,-1,-1,-1,-1]. After calculating the parameters of this 2 lists, I get:
$m_1 = m_2 = 0$
$s_1 = s_2 = 2$
$τ_1 = 0.027$
$τ_2 = 4229.471$
Given that $m_2 = 0$ and $s_2 = 2$, 4 is 2 standard deviations above the mean. By adding [4,-1,-1,-1,-1] n-times we have that 20% of the data is in the far right of the distribution (in a normal distribution only ≈ 2% would be there). Because of this, τ is higher (feel free to comment).