ARGUS distribution

Cumulative distribution function

The cumulative distribution function (cdf) of the ARGUS distribution is

${\displaystyle F(x)=1-\frac{\mathrm{\Psi }\left(\chi \sqrt{1-x^2/c^2}\right)}{\mathrm{\Psi }(\chi )}}$.

Parameter estimation

Parameter c is assumed to be known (the kinematic limit of the invariant mass distribution), whereas χ can be estimated from the sample X₁, …, X_n using the maximum likelihood approach. The estimator is a function of sample second moment, and is given as a solution to the non-linear equation

${\displaystyle 1-\frac{3}{{\chi }^2}+\frac{\chi \varphi (\chi )}{\mathrm{\Psi }(\chi )}=\frac{1}{n}\sum _{i=1}^n\frac{x_i^2}{c^2}}$.

The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator ${\displaystyle \hat{\chi }}$ is consistent and asymptotically normal.

Generalized ARGUS distribution

Sometimes a more general form is used to describe a more peaking-like distribution:

${\displaystyle f(x)=\frac{2^{-p}{\chi }^{2(p+1)}}{\mathrm{\Gamma }(p+1)-\mathrm{\Gamma }(p+1,\frac{1}{2}{\chi }^2)}\cdot \frac{x}{c^2}{\left(1-\frac{x^2}{c^2}\right)}^p\exp \left\{-\frac{1}{2}{\chi }^2\left(1-\frac{x^2}{c^2}\right)\right\},0\le x\le c,c>0,\chi >0,p>-1}$

${\displaystyle F(x)=\frac{\mathrm{\Gamma }\left(p+1,\frac{1}{2}{\chi }^2\left(1-\frac{x^2}{c^2}\right)\right)-\mathrm{\Gamma }(p+1,\frac{1}{2}{\chi }^2)}{\mathrm{\Gamma }(p+1)-\mathrm{\Gamma }(p+1,\frac{1}{2}{\chi }^2)},0\le x\le c,c>0,\chi >0,p>-1}$

where Γ(·) is the gamma function, and Γ(·,·) is the upper incomplete gamma function.

Here parameters c, χ, p represent the cutoff, curvature, and power respectively.

The mode is:

${\displaystyle \frac{c}{\sqrt{2}\chi }\sqrt{({\chi }^2-2p-1)+\sqrt{{\chi }^2({\chi }^2-4p+2)+(1+2p{)}^2}}}$

The mean is:

${\displaystyle \mu =cp\sqrt{\pi }\frac{\mathrm{\Gamma }(p)}{\mathrm{\Gamma }(\frac{5}{2}+p)}\frac{{\chi }^{2p+2}}{2^{p+2}}\frac{M\left(p+1,\frac{5}{2}+p,-\frac{{\chi }^2}{2}\right)}{\mathrm{\Gamma }(p+1)-\mathrm{\Gamma }(p+1,\frac{1}{2}{\chi }^2)}}$

where M(·,·,·) is the Kummer's confluent hypergeometric function.

The variance is:

${\displaystyle {\sigma }^2=c^2\frac{{\left(\frac{\chi }{2}\right)}^{p+1}{\chi }^{p+3}{e}^{-\frac{{\chi }^2}{2}}+\left({\chi }^2-2(p+1)\right)\left\{\mathrm{\Gamma }(p+2)-\mathrm{\Gamma }(p+2,\frac{1}{2}{\chi }^2)\right\}}{{\chi }^2(p+1)\left(\mathrm{\Gamma }(p+1)-\mathrm{\Gamma }(p+1,\frac{1}{2}{\chi }^2)\right)}-{\mu }^2}$

p = 0.5 gives a regular ARGUS, listed above.