View source for Distribution fitting by using the mean of distances of empirical data

Described method was proposed by '''''Yury S. Dodonov''''' (2020)<br><br>
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,<br>

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

$$or$$

$$ \tag{2}
\Psi .2 = \frac{{\sum\limits_{j = 1}^n {{{\left( {{x_i} - {x_j}} \right)}^2}}
}}{{{\rm{n}} - 1}}.
$$
<br><br>
On the other hand, there is a functional dependency for theoretical distribution, which can be written (for the mean of distances) as
<br><br>

$$ \tag{3}
\left\{ \begin{array}{l}
\Phi .1\left( {x\,;\;\theta ,\;L,\;U} \right) = \frac{{\int\limits_0^{U - x}
{\Omega \left( {t;\;x,\;\theta} \right)dt - \int\limits_{L - x}^0 {\Omega \left(
{t;\;x,\;\theta} \right)dt} } }}{{CDF\left( {U;\;\theta } \right) - CDF\left(
{L;\;\theta } \right)}}\\
L \le x \le U
\end{array} \right.
$$
<br><br>
where CDF is a cumulative distribution function, L and U are the lower and upper of boundaries, respectively, and
<br><br>

$$ \tag{4}
\Omega \left( {t;\;x,\;\theta } \right) = t \cdot PDF\left( {t;\;x,{\theta _{x\to \left( {t + x} \right)}}} \right)
$$
<br><br>
In dealing with a mean square of distances, the general functional dependency takes the following form:
<br><br>

$$ \tag{5}
\left\{ \begin{array}{l}
\Phi .2\left( {x\,;\;\theta ,\;L,\;U} \right) = \frac{{\int\limits_L^U {\left[
{{{\left( {x - t} \right)}^2} \cdot PDF\left( {t;\theta } \right)} \right]dt}
}}{{CDF\left( {U;\;\theta } \right) - CDF\left(
{L;\;\theta } \right)}}\\
L \le x \le U
\end{array} \right.
$$
<br><br>
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'''
<br><br>
Probability density and cumulative distribution functions for a Gaussian distribution are defined by:
<br><br>
$$ \tag{6}
PD{F_N} = \frac{1}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {x - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}
$$
<br><br>
$$ \tag{7}
CD {F_N}  = \frac{1}{2}\left[  {1 + erf\left(  {\frac{ {x - \mu } }{ {\sigma \sqrt 2 } }}  \right)}  \right]
$$
<br><br>
For the mean of distances according to (6) and (4):
<br><br>
$$ \tag{8}
{\Omega _N}\left( {t;\;x,\;\mu ,\;\sigma } \right) = t \cdot PDF\left( {t;\;x,{\theta _{x \to \left( {t + x} \right)}}} \right) = \frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}
$$
<br><br>
according to (3),  (7) and (8):
<br><br>
$$ \tag{9}
\left\{ \begin{array}{l}
\Phi {.1_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \frac{{\int\limits_0^{U - x} {\frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dt - \int\limits_{L - x}^0 {\frac{t}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {\left( {t + x} \right) - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}dt} } }}{{CD{F_N}\left( {U;\;\mu ,\;\sigma } \right) - CD{F_N}\left( {L;\;\mu ,\;\sigma } \right)}}\\
L \le x \le U
\end{array} \right.
$$
<br><br>
and finally:
<br><br>
$$ \tag{10}
\left\{ \begin{array}{l}\Phi  {.1_N} \left(  {x\,;\;\mu ,\;\sigma ,\;L,\;U}  \right) = \\ = \frac{ {\left(  {x - \mu }  \right)\left(  {2erf\left(  {\frac{ {x - \mu } }{ {\sigma \sqrt 2 } }}  \right) - erf\left(  {\frac{ {U - \mu } }{ {\sigma \sqrt 2 } }}  \right) - erf\left(  {\frac{ {L - \mu } }{ {\sigma \sqrt 2 } }}  \right)}  \right) + \frac{ {2\sigma } }{ {\sqrt  {2\pi }  } }\left(  {2 {e^ {\frac{ { -  { {\left(  {x - \mu }  \right)} ^2} } }{ {2 {\sigma ^2} } }} }  -  {e^ {\frac{ { -  { {\left(  {U - \mu }  \right)} ^2} } }{ {2 {\sigma ^2} } }} }  -  {e^ {\frac{ { -  { {\left(  {L - \mu }  \right)} ^2} } }{ {2 {\sigma ^2} } }} } }  \right)} }{ {erf\left(  {\frac{ {U - \mu } }{ {\sigma \sqrt 2 } }}  \right) - erf\left(  {\frac{ {L - \mu } }{ {\sigma \sqrt 2 } }}  \right)} }\\L \le x \le U\end{array} \right.
$$
<br><br>
For the mean square of distances according to (5), (6) and (7):
<br><br>
$$ \tag{11}
\left\{ \begin{array}{l}
\Phi {.2_N}\left( {x\,;\;\mu ,\;\sigma ,\;L,\;U} \right) = \frac{{\int\limits_L^U {\left[ {\frac{{{{\left( {x - t} \right)}^2}}}{{\sigma \sqrt {2\pi } }}{e^{\frac{{ - {{\left( {t - \mu } \right)}^2}}}{{2{\sigma ^2}}}}}} \right]dt} }}{{CD{F_N}(U;\mu ,\;\sigma ) - CD{F_N}(L;\mu ,\;\sigma )}}\\
L \le x \le U
\end{array} \right.
$$
<br><br>
and finally:
<br><br>
$$ \tag{12}
\left\{ \begin{array}{l}\Phi  {.2_N} \left(  {x\,;\;\mu ,\;\sigma ,\;L,\;U}  \right) = \\ = \frac{ {\left(  { { {\left(  {\mu  - x}  \right)} ^2}  +  {\sigma ^2} }  \right)\left(  {erf\left(  {\frac{ {U - \mu } }{ {\sigma \sqrt 2 } }}  \right) - erf\left(  {\frac{ {L - \mu } }{ {\sigma \sqrt 2 } }}  \right)}  \right) + \frac{ {2\sigma } }{ {\sqrt  {2\pi }  } }\left(  { {e^ {\frac{ { -  { {\left(  {L - \mu }  \right)} ^2} } }{ {2 {\sigma ^2} } }} } \left(  {L + \mu  - 2x}  \right) -  {e^ {\frac{ { -  { {\left(  {U - \mu }  \right)} ^2} } }{ {2 {\sigma ^2} } }} } \left(  {\mu  + U - 2x}  \right)}  \right)} }{ {erf\left(  {\frac{ {U - \mu } }{ {\sigma \sqrt 2 } }}  \right) - erf\left(  {\frac{ {L - \mu } }{ {\sigma \sqrt 2 } }}  \right)} }\\L \le x \le U\end{array} \right.
$$
<br><br><br>
== 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 ==
Available at https://doi.org/10.6084/m9.figshare.13042745
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