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C Local exponent
In this Appendix, we come back to the notion of a local Pareto exponent discussed above in section 4.3.1,
with figure 7 and expression (33). We call the local exponent β.
Generally speaking, any positive smooth function g(x) can be represented in the form
g(x) = 1/x β(x) , x 1, (72)
by defining β(x) = −ln(g(x))/lnx. Thus, the local index β(x) of a distribution F(x) will be defined as
β(x) = −
95
ln(1 − F(x))
. (73)
ln(x)
For example, the log-normal distribution can be mistaken for a power law over several decades with very
slowly (logarithmically) varying exponents if its variance is large (see figures 4.2 and 4.3 in section 4.1.3
of (Sornette 2000)). When we approximate a sample distribution by some parametric family with moving
index β(x), it is important that β(x) should have as small a variation as possible, i.e., that the representation
(72) be parsimonious. Given a sample x1,x2,··· ,xN, drawn from a distribution function F(x), x ≥ 1, the tail
index β(x) is consistently estimated by
ˆβ(x) = lnN − ln
∑i 1 {xi>x}
, (74)
ln(x)
where 1 {·} is the indicator function which equals one if its argument is true and zero otherwhise. The
asymptotic distribution of the estimator is easily derived and reads:
with
N 1/2 ·
e ln(x)·ˆ β(x) − e ln(x)·β(x) d
−→ N (0,σ 2 ), (75)
σ 2 =
1
. (76)
F(x) · (1 − F(x))
As an example, let us illustrate the properties of the local index β(x) for regularly varying distributions. The
general representation of any regularly varying distribution is given by ¯F(x) = L(x) · x −α , where L(·) is a
slowly varying function. In such a case, the local power index can be written as
β(x) = α − lnL(x)
, (77)
ln(x)
which goes to α as x goes to infinity, as expected from the build-in slow variation of L(x). The upper panel
of figure 10 shows the local index β(x) for a simulated Pareto distribution with power index b = 1.2 (X > 1).
The estimate β(x) oscillates very closely to the true value b = 1.2.
Let us now assume that we observe a regularly varying local exponent
β(x) = L(x) · x c , with c > 0, (78)
it would be clearly the charaterization of a Stretched-Exponential distribution
¯F(x) = exp −L ′ (x) · x c , (79)
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