Distribution fitting by using the mean of distances of empirical data

Described method was proposed by Yury S. Dodonov (2020)

On the one hand, for any given set of n values ${x}_{1}, {x}_{2}, ...,{x}_{n}$ with ${x}_{i}\in{}R$, we can calculate the mean of distances or mean square of distances from ${x}_{i}$ to the others point using, respectively,
(1) $ \Psi .1 = \frac{{\sum\limits_{j = 1}^n {\left| {{x_i} - {x_j}} \right|} }}{{{\rm{n}} - 1}} $

or

(2) $ \Psi .2 = \frac{{\sum\limits_{j = 1}^n {{{\left( {{x_i} - {x_j}} \right)}^2}} }}{{{\rm{n}} - 1}} $.

On the other hand, there is a functional dependency for theoretical distribution, which can be written (for the mean of distances) as

(3) $ \left\{ \begin{array}{l} \Phi .1\left( {x\,;\;\theta ,\;L,\;U} \right) = \frac{{\int\limits_0^{U - x} {\Omega \left( {t;\;x,\;\theta} \right)dt - \int\limits_{L - x}^0 {\Omega \left( {t;\;x,\;\theta} \right)dt} } }}{{CDF\left( {U;\;\theta } \right) - CDF\left( {L;\;\theta } \right)}}\\ L \le x \le U \end{array} \right. $

where CDF is a cumulative distribution function, L and U are the lower and upper of boundaries, respectively, and

(4) $ \Omega \left( {t;\;x,\;\theta } \right) = t \cdot PDF\left( {t;\;x,{\theta _{x\to \left( {t + x} \right)}}} \right) $

In dealing with a mean square of distances, the general functional dependency takes the following form:

(5) $ \left\{ \begin{array}{l} \Phi .2\left( {x\,;\;\theta ,\;L,\;U} \right) = \frac{{\int\limits_L^U {\left[ {{{\left( {x - t} \right)}^2} \cdot PDF\left( {t;\theta } \right)} \right]dt} }}{{CDF\left( {U;\;\theta } \right) - CDF\left( {L;\;\theta } \right)}}\\ L \le x \le U \end{array} \right. $

Thus, a general fitting algorithm contains two steps. The first is calculating the mean of distances or mean square of distances from ${x}_{i}$ to the other points, and then, by means of using (3) or (5), the distribution parameters are recovered by using a conventional regression method.

Example for normal distribution

Probability density and cumulative distribution functions for a Gaussian distribution are defined by:

(6) $PD{F_N} = \frac{1}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$

(7) $CD{F_N} = \frac{1}{2}\left[ {1 + erf\left( {\frac[[:Template:X - \mu]][[:Template:\sigma \sqrt 2]]} \right)} \right]$

For the mean of distances according to (6) and (4):

(8) ${\Omega _N}\left( {t;\;x,\;\mu ,\;\sigma } \right) = t \cdot PDF\left( {t;\;x,{\theta _{x \to \left( {t + x} \right)}}} \right) = \frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}$

according to (3), (7) and (8):

(9) $\left\{ \begin{array}{l} \Phi {.1_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \frac{{\int\limits_0^{U - x} {\frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dt - \int\limits_{L - x}^0 {\frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dt} } }}{{CD{F_N}\left( {U;\;\mu ,\;\sigma } \right) - CD{F_N}\left( {L;\;\mu ,\;\sigma } \right)}}\\ L \le x \le U \end{array} \right.$

and finally:

(10) $\left\{ \begin{array}{l} \Phi {.1_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \\ = \frac{{\left( {x - \mu } \right)\left( {2erf\left( {\frac[[:Template:X - \mu]][[:Template:\sigma \sqrt 2]]} \right) - erf\left( {\frac[[:Template:U - \mu]][[:Template:\sigma \sqrt 2]]} \right) - erf\left( {\frac[[:Template:L - \mu]][[:Template:\sigma \sqrt 2]]} \right)} \right) + \frac[[:Template:2\sigma]]{{\sqrt {2\pi } }}\left( {2{e^{\frac{{ - {{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}} - {e^{\frac{{ - {{\left( {U - \mu } \right)}^2}}}{{2{\sigma ^2}}}}} - {e^{\frac{{ - {{\left( {L - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} \right)}}{{erf\left( {\frac[[:Template:U - \mu]][[:Template:\sigma \sqrt 2]]} \right) - erf\left( {\frac[[:Template:L - \mu]][[:Template:\sigma \sqrt 2]]} \right)}}\\ L \le x \le U \end{array} \right.$

For the mean square of distances according to (5), (6) and (7):

(11) $\left\{ \begin{array}{l} \Phi {.2_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \frac{{\int\limits_L^U {\left[ {\frac{{{{\left( {x - t} \right)}^2}}}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {t - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} \right]dt} }}{{CD{F_N}(U;\mu ,\;\sigma ) - CD{F_N}(L;\mu ,\;\sigma )}}\\ L \le x \le U \end{array} \right.$

and finally:

(12) $\left\{ \begin{array}{l} \Phi {.2_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \\ = \frac{{\left( {{{\left( {\mu - x} \right)}^2} + {\sigma ^2}} \right)\left( {erf\left( {\frac[[:Template:U - \mu]][[:Template:\sigma \sqrt 2]]} \right) - erf\left( {\frac[[:Template:L - \mu]][[:Template:\sigma \sqrt 2]]} \right)} \right) + \frac[[:Template:2\sigma]]{{\sqrt {2\pi } }}\left( {{e^{\frac{{ - {{\left( {L - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}\left( {L + \mu - 2x} \right) - {e^{\frac{{ - {{\left( {U - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}\left( {\mu + U - 2x} \right)} \right)}}{{erf\left( {\frac[[:Template:U - \mu]][[:Template:\sigma \sqrt 2]]} \right) - erf\left( {\frac[[:Template:L - \mu]][[:Template:\sigma \sqrt 2]]} \right)}}\\ L \le x \le U \end{array} \right.$


Reference

Dodonov, Yury (2020): Distribution fitting by using the mean of distances of empirical data. figshare. Journal contribution. https://doi.org/10.6084/m9.figshare.13014410


Appendix. R code for fitting normal distribution