ARGUS distribution - Alchetron, The Free Social Encyclopedia

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Parameters c > 0 {\displaystyle c>0} cut-off (real) χ > 0 {\displaystyle \chi >0} curvature (real) Support x ∈ ( 0 , c ) {\displaystyle x\in (0,c)\!} Mean μ = c π / 8 χ e − χ 2 4 I 1 ( χ 2 4 ) Ψ ( χ ) {\displaystyle \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 Ψ ( x ) {\displaystyle \Psi (x)} is given in the text. Mode c 2 χ ( χ 2 − 2 ) + χ 4 + 4 {\displaystyle {\frac {c}{{\sqrt {2}}\chi }}{\sqrt {(\chi ^{2}-2)+{\sqrt {\chi ^{4}+4}}}}}

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 ; χ , c ) = χ 3 2 π Ψ ( χ ) ⋅ x c 2 1 − x 2 c 2 exp ⁡ { − 1 2 χ 2 ( 1 − x 2 c 2 ) } ,

for 0 ≤ x < c. Here χ, and c are parameters of the distribution and

Ψ ( χ ) = Φ ( χ ) − χ ϕ ( χ ) − 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:

{ c 2 x ( c − x ) ( c + x ) f ′ ( x ) + f ( x ) ( − c 4 − c 2 ( χ 2 − 2 ) x 2 + χ 2 x 4 ) = 0 , f ( 1 ) = − 2 − 2 c 2 χ 3 e χ 2 2 c 2 c 2 ( 2 χ − π e χ 2 2 erf ⁡ ( χ 2 ) ) }

Cumulative distribution function

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

F ( x ) = 1 − Ψ ( χ 1 − x 2 / c 2 ) Ψ ( χ ) .

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

1 − 3 χ 2 + χ ϕ ( χ ) Ψ ( χ ) = 1 n ∑ i = 1 n 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 χ ^ is consistent and asymptotically normal.

Generalized ARGUS distribution

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

f ( x ) = 2 − p χ 2 ( p + 1 ) Γ ( p + 1 ) − Γ ( p + 1 , 1 2 χ 2 ) ⋅ x c 2 ( 1 − x 2 c 2 ) p exp ⁡ { − 1 2 χ 2 ( 1 − x 2 c 2 ) } , 0 ≤ x ≤ 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:

c 2 χ ( χ 2 − 2 p − 1 ) + χ 2 ( χ 2 − 4 p + 2 ) + ( 1 + 2 p ) 2

p = 0.5 gives a regular ARGUS, listed above.

References

ARGUS distribution Wikipedia

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