From Wikipedia, the free encyclopedia
The Moffat distribution , named after the physicist Anthony Moffat , is a continuous probability distribution based upon the Lorentzian distribution . Its particular importance in astrophysics is due to its ability to accurately reconstruct point spread functions , whose wings cannot be accurately portrayed by either a Gaussian or Lorentzian function.
Characterisation
Probability density function
The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (X ,Y ) centred at zero, and secondly as the distribution of the corresponding radii
R
=
X
2
+
Y
2
.
{\displaystyle R={\sqrt {X^{2}+Y^{2}}}.}
In terms of the random vector (X ,Y ), the distribution has the probability density function (pdf)
f
(
x
,
y
;
α
,
β
)
=
β
−
1
π
α
2
[
1
+
(
x
2
+
y
2
α
2
)
]
−
β
,
{\displaystyle f(x,y;\alpha ,\beta )={\frac {\beta -1}{\pi \alpha ^{2}}}\left[1+\left({\frac {x^{2}+y^{2}}{\alpha ^{2}}}\right)\right]^{-\beta },\,}
where
α
{\displaystyle \alpha }
and
β
{\displaystyle \beta }
are seeing dependent parameters. In this form, the distribution is a reparameterisation of a bivariate Student distribution with zero correlation.
In terms of the random variable R , the distribution has density
f
(
r
;
α
,
β
)
=
β
−
1
π
α
2
[
1
+
(
r
2
α
2
)
]
−
β
.
{\displaystyle f(r;\alpha ,\beta )={\frac {\beta -1}{\pi \alpha ^{2}}}\left[1+\left({\frac {r^{2}}{\alpha ^{2}}}\right)\right]^{-\beta }.\,}
Differential equation
The pdf of the Moffat distribution is a solution to the following differential equation :
{
(
r
3
+
α
2
r
)
f
′
(
r
)
+
f
(
r
)
(
−
α
2
+
2
β
r
2
−
r
2
)
=
0
,
f
(
1
)
=
2
(
β
−
1
)
(
1
α
2
+
1
)
−
β
α
2
}
{\displaystyle \left\{{\begin{array}{l}\left(r^{3}+\alpha ^{2}r\right)f'(r)+f(r)\left(-\alpha ^{2}+2\beta r^{2}-r^{2}\right)=0,\\f(1)={\frac {2(\beta -1)\left({\frac {1}{\alpha ^{2}}}+1\right)^{-\beta }}{\alpha ^{2}}}\end{array}}\right\}}
References
Discrete univariate
with finite support with infinite support
Continuous univariate
supported on a bounded interval supported on a semi-infinite interval supported on the whole real line with support whose type varies
Mixed univariate
Multivariate (joint) Directional Degenerate and singular Families