ARGUS distribution

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS, is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.

Definition
The probability density function (pdf) of the ARGUS distribution is:

f(x; \chi, c ) = \frac{\chi^3}{\sqrt{2\pi}\,\Psi(\chi)} \cdot \frac{x}{c^2} \sqrt{1-\frac{x^2}{c^2}} \exp\bigg\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\}, $$ for $$0 \leq x < c$$. Here $$\chi$$ and $$c$$ are parameters of the distribution and


 * $$\Psi(\chi) = \Phi(\chi)- \chi \phi( \chi ) - \tfrac{1}{2} ,$$

where $$ \Phi(x)$$ and $$\phi( x )$$ are the cumulative distribution and probability density functions of the standard normal distribution, respectively.

Cumulative distribution function
The cumulative distribution function (cdf) of the ARGUS distribution is
 * $$F(x) = 1 - \frac{\Psi\left(\chi\sqrt{1-x^2/c^2}\right)}{\Psi(\chi)}$$.

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, …, Xn 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
 * $$1 - \frac{3}{\chi^2} + \frac{\chi\phi(\chi)}{\Psi(\chi)} = \frac{1}{n}\sum_{i=1}^n \frac{x_i^2}{c^2}$$.

The solution exists and is unique, provided that the right-hand side is greater than 0.4; the resulting estimator is consistent and asymptotically normal.

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

f(x) = \frac{2^{-p}\chi^{2(p+1)}}{\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)} \cdot \frac{x}{c^2} \left( 1 - \frac{x^2}{c^2} \right)^p \exp\left\{ -\frac12 \chi^2\left(1-\frac{x^2}{c^2}\right) \right\}, \qquad 0 \leq x \leq c, \qquad c>0,\,\chi>0,\,p>-1 $$

F(x) = \frac{\Gamma\left(p+1,\,\tfrac{1}{2}\chi^2\left( 1 - \frac{x^2}{c^2} \right)\right)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)}{\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)}, \qquad 0 \leq x \leq c, \qquad 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:
 * $$\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2p-1)+\sqrt{\chi^2(\chi^2-4p+2)+(1+2p)^2}}$$

The mean is:
 * $$\mu=c \,p \, \sqrt{\pi}\frac{\Gamma(p)}{\Gamma(\tfrac{5}{2}+p)}\frac{\chi^{2p+2}}{2^{p+2}}\frac{M\left(p+1,\tfrac{5}{2}+p,-\tfrac{\chi^2}{2}\right)}{\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)}$$

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

The variance is:
 * $$\sigma^2=c^2 \frac{\left(\frac{\chi}{2} \right)^{p+1}\chi^{p+3}e^{-\tfrac{\chi^2}{2}}+\left(\chi^2-2(p+1)\right)\left\{\Gamma(p+2)-\Gamma(p+2,\,\tfrac{1}{2}\chi^2)\right\}}

{\chi^2(p+1)\left(\Gamma(p+1)-\Gamma(p+1,\,\tfrac{1}{2}\chi^2)\right)}-\mu^2 $$

p = 0.5 gives a regular ARGUS, listed above.