ARGUS distribution

ARGUS
Probability density function
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Cumulative distribution function
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Parameters c > 0 cut-off (real)
χ > 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)%2B\sqrt{\chi^4%2B4}}
Variance c^2\!\left(1 - \frac{3}{\chi^2} %2B \frac{\chi\phi(\chi)}{\Psi(\chi)}\right) - \mu^2

In physics, the ARGUS distribution, named after the particle physics experiment ARGUS[1], is the probability distribution of the reconstructed invariant mass of a decayed particle candidate in continuum background.

Contents

Definition

The probability density function 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.

Cumulative distribution function

The cdf of the ARGUS distribution is


    F(x) = 1 - \frac{\Psi\Big(\chi\sqrt{1-x^2/c^2}\,\Big)}{\Psi(\chi)}.

Parameter estimation

Parameter c is assumed to be known (the speed of light), 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} %2B \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%2B1)}}{\Gamma(p%2B1)-\Gamma(p%2B1,\,\tfrac{1}{2}\chi^2)}\ \cdot\ 
           \frac{x}{c^2} \bigg( 1 - \frac{x^2}{c^2} \bigg)^p
           \exp\bigg\{ -\frac12 \chi^2\Big(1-\frac{x^2}{c^2}\Big) \bigg\},
           \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.

mode = \frac{c}{\sqrt2\chi}\sqrt{(\chi^2-2p-1)%2B\sqrt{\chi^2(\chi^2-4p%2B2)%2B(1%2B2p)^2}}

p = 0.5 gives a regular ARGUS, listed above.

References

  1. ^ Albrecht, H. (1990). "Search for hadronic b→u decays". Physics Letters B 241 (2): 278–282. doi:10.1016/0370-2693(90)91293-K.  edit (More formally by the ARGUS Collaboration, H. Albrecht et al.) In this paper, the function has been defined with parameter c representing the beam energy and parameter p set to 0.5. The normalization and the parameter χ have been obtained from data.

Further reading