Extrasolar Moons as Gravitational Microlenses Christine Liebig

CHAPTER 3. METHOD 25
Now, we can look up the probability density function of the χ2-distribution with n
degrees of freedom and find
⎧
⎪⎨ x
fn(x) =
⎪⎩
n
2 −1 x
− e 2
2 n
2 Γ x > 0
n
,
2
0 x ≤ 0
where Γ(r) is the gamma function. Γ
n can be evaluated via the formulae
2
1
Γ =
2
√ π , Γ(1) = 1
Γ(r + 1) = r · Γ(r) with r ∈ R + .
The probability P (χ 2 ≥ Q 2 µ) that any random set of n data points compared with
the correct(!) theory µ would yield a value of χ 2 as large as or larger than Q 2 µ is
P (χ 2 ≥ Q 2 µ) =
∞
Q 2 µ
fn(x)dx = 1 − Fn(Q 2 µ),
with Fn(z) = z
f(x)dx as the cumulative distribution function. If it is very
−∞
small, Q2 µ is outside the expected range for χ2-distributed random variables and we
conclude that our theory is probably wrong.
For the standard χ2-test one compares Q2 µ with the expectation value of χ2 ,
which is equal to the number of degrees of freedom.
Q 2 µ = 〈χ 2 〉 ± √ 2n = n ± √ 2n
is a rough criterion for an acceptable goodness-of-fit, where too small a value of
Q 2 µ is indicative of possible hidden correlations between the data points or an overestimation
of the standard errors, whereas too high a value can be caused by an
underestimation of the standard errors or, in fact, by using a wrong model.
3.3.2 Using the χ 2 -distribution as a measure for significance
of deviation between two theoretical curves
We are faced with the task of comparing two simulated light curves, one of a ternary
lens system, the other of a related binary lens system. One common approach is
to random-generate artificial data around one of them and then consecutively fit
the two theoretical curves (theoretical meaning the non-random-scattered simulated
curve) to the data with some free parameters. Two χ2-values for the goodness-of-fit
will result. With these the so-called ∆χ2 = χ2 1 − χ2 2 is calculated. A threshold
value ∆χ2 thresh is, more or less arbitrarily, chosen. By keeping it rather large one
can try to account for systematic errors. ∆χ2 > ∆χ2 thresh then is the condition
for reported non-negligible deviation and, thus, detectability of the deviation. A