ARGUS distribution

ARGUS

Parameters	$c > 0$ cut-off (real) $\chi > 0$ curvature (real)
Support	$x \in (0, c)\!$
PDF	see text
CDF	see text
Mean	$\mu = c\sqrt{\pi/8}\;\frac{\chi e^{-\frac{\chi^2}{4}} I_1(\tfrac{\chi^2}{4})}{ \Psi(\chi) }$ where I1 is the Modified Bessel function of the first kind of order 1, and $\Psi(x)$ is given in the text.
Mode	$\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2)+\sqrt{\chi^4+4}}$
Variance	$c^2\!\left(1 - \frac{3}{\chi^2} + \frac{\chi\phi(\chi)}{\Psi(\chi)}\right) - \mu^2$

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 ≤ x < c. Here χ, and c are parameters of the distribution and

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

and Φ(·), ϕ(·) are the cumulative distribution and probability density functions of the standard normal distribution, respectively.

Differential equation

The pdf of the ARGUS distribution is a solution of the following differential equation:

$\left\{\begin{array}{l} c^2 x (c-x) (c+x) f'(x)+f(x) \left(-c^4-c^2 \left(\chi ^2-2\right) x^2+\chi ^2 x^4\right)=0, \\[10pt] f(1)=-\frac{\sqrt{2-\frac{2}{c^2}} \chi ^3 e^{\frac{\chi ^2}{2 c^2}}}{c^2 \left(\sqrt{2} \chi -\sqrt{\pi} e^{\frac{\chi ^2}{2}} \operatorname{erf}\left(\frac{\chi }{\sqrt{2}}\right)\right)} \end{array}\right\}$

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 speed of light), 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

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

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

Generalized ARGUS distribution

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

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

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:

$\frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2p-1)+\sqrt{\chi^2(\chi^2-4p+2)+(1+2p)^2}}$

p = 0.5 gives a regular ARGUS, listed above.