Stable Distributions



1.5 Estimation of parameters

11


Sample of size N=10^4

-4           -2           0            2            4

log(x)


Figure 1.4: A double logarithmic plot of the right tail of an empirical symmetric
1
.9-stable distribution function for a sample of size N = 104 (left
panel
) and N = 106 (right panel). Thick red lines represent the
linear regression fit. The tail index estimate (<^ = 3
.7320) obtained
for the smaller sample is close to the initial power-law like decay of
the larger sample (<^ = 3
.7881). The far tail estimate α = 1.9309 is
close to the true value of
α.

θ STFstab04.xpl


i.e. the negative tail) observations, after a crossover from a temporary power-
like decay (which surprisingly indicates
α ≈ 3.7). Moreover, the obtained
estimates still have a slight positive bias, which suggests that perhaps even
larger samples than 10
6 observations should be used. In Figure 1.4 we used
only the upper 0.15% of the records to estimate the true tail exponent. In
general, the choice of the observations used in the regression is subjective and
can yield large estimation errors, a fact which is often neglected in the literature.

A well known method for estimating the tail index that does not assume a
parametric form for the entire distribution function, but focuses only on the
tail behavior was proposed by Hill (1975). The Hill estimator is used to estimate
the tail index
α, when the upper (or lower) tail of the distribution is of the
form: 1
- F (x) = Cx, see Figure 1.5. Like the log-log regression method, the
Hill estimator tends to overestimate the tail exponent of the stable distribution



More intriguing information

1. Publication of Foreign Exchange Statistics by the Central Bank of Chile
2. The name is absent
3. Imperfect competition and congestion in the City
4. The name is absent
5. Does Competition Increase Economic Efficiency in Swedish County Councils?
6. WP 36 - Women's Preferences or Delineated Policies? The development or part-time work in the Netherlands, Germany and the United Kingdom
7. Conditions for learning: partnerships for engaging secondary pupils with contemporary art.
8. Cross border cooperation –promoter of tourism development
9. Understanding the (relative) fall and rise of construction wages
10. A Regional Core, Adjacent, Periphery Model for National Economic Geography Analysis