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E. M. Tirado and D. Quagliarella
13.4.1.2 Cumulative Value-at-Risk (Superquantile) Estimation Using
ECDF
Regarding the estimation of the superquantile, according to [15], c α can also be
written as a stochastic program:
c
α
= inf
t∈R
t +
1
1 − α
E[X − t]
+
(13.19)
with [a] + = max{0, a}. The set of optimal solutions to the stochastic program is
T = [ν α , u α ] with u α = sup t : F (t) ≤ α. In particular, ν α ∈ T , so
c
α
= ν
α
+
1
1 − α
E[X − ν
α
]
+
(13.20)
When X has a positive density in the neighbourhood of ν α , then ν α = u α . Under
these conditions, the above formula can be also directly derived from Eq. 13.13.
So, in the case of finite number of samples, with X 1 , X 2 , . . . , X n independent and
identically distributed (i.i.d.) observations of the random variable X, the estimation
of c α is given by:
ˆ
c
α;n
= ˆ
ν
α;n
+
1
n(1 − α)
n
i=1
[X i − ˆ
ν
α;n
]
+
(13.21)
13.4.2 Estimation of Risk Functions Using WECDF
It was above stated that the ECDF is a step function that jumps up a fixed quantity,
1/n, for each data point belonging to the sorted set of samples. Conversely, the
WECDF can be considered as a step function that has a variable size jump, w i :
ˆ
F
n
μ;w (t) =
n
i=1
w i 1{x
i
≤ t}
(13.22)
with the related constraint
n
i=1
w i = 1
(13.23)
The formula for VaR estimation starting from a WECDF is a generalization of
Eq. 13.18, and requires two steps. Firstly, the k α index of the sorted sample set
has to be chosen according to the following inequalities: