statistique, théorie et gestion de portefeuille - Docs at ISFA

Equation (46) depends on c only and must be solved numerically. Then, the resulting value of c can be
reinjected in (47) to get d. The maximum of the log-likelihood function is
1
T LSE T (ĉ, d) ˆ = ln ĉ ĉ − 1
+
dˆ ĉ T
T
∑
i=1
91
lnxi − 1. (48)
Since c > 0, the vector √ N(ĉ − c, ˆ
d − d) is asymptotically normal, with a covariance matrix whose
expression is given in appendix D by the inverse of K11.
It should be noted that maximum likelihood equations (46-47) admit a solution with positive c not for
all possible samples (x1,··· ,xN). Indeed, the function
h(c) = 1
c −
1
T ∑T xi c xi
i=1 u ln
u
c +
− 1 1 T
ln
T
xi
, (49)
u
1
T ∑T xi
i=1 u
which is the total derivative of LSE T (c, d(c)), ˆ is a decreasing function of c. It means, as one can expect,
that the likelihood function is concave. Thus, a necessary and sufficient condition for equation (46) to
admit a solution is that h(0) is positive. After some calculations, we find
which is positive if and only if
h(0) = 2 1
T
xi ∑ln u
2 xi
T ∑ln u
∑
i=1
2 1 xi − T ∑ln2 u
, (50)
2 1 xi
2
T
∑ln −
u
1 2 xi
T
∑ln > 0. (51)
u
However, the probability of occurring a sample providing a negative maximum-likelyhood estimate of
c tends to zero (under Hypothesis of SE with a positive c) as
Φ − c√
N σ
√ e
σ 2π Nc − c2N 2σ2 , (52)
i.e. exponentially with respect to N. Here σ2 is the variance of the limit Gaussian distribution of
maximum-likelihood c-estimator that can be derived explicitly. If h(0) is negative, LSE T reaches its
maximum at c = 0 and in such a case
1
T LSE
1 xi
T (c = 0) = −ln
T
∑ln −
u
1
T ∑lnxi − 1. (53)
Now, if we apply maximum likelihood estimation based on SE assumption to samples distributed
differently from SE, then we can get negative c-estimate with some positive probability not tending
to zero with N → ∞. If sample is distributed according to Pareto distribution, for instance, then
maximum-likelihood c-estimate converges in probability to a Gaussian random variable with zero
mean, and thus the probability of negative c-estimates converges to 0.5.
A.3 The Exponential distribution
The Exponential distribution function is given by equation (24), and its density is
fu(x|d) = exp
u
d exp −
d
x
,
d
x ≥ u. (54)
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