13 Risk Measures in the Context of Robust and Reliability Based Optimization
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13.4.1 Estimation of Risk Functions Using ECDF
The Empirical Cumulative Distribution Function (ECDF) is the distribution function
associated with the empirical measure of a sample. Moreover, it can be seen as a
step function that jumps up by 1/n at each of the n data points. It takes as value the
fraction of observations of the variable that are less or equal to the specified value
[16].
Mathematically speaking, let X : Ω → R d a random variable, x i = (x i
1 , . . . , x i
d )
a random sample of X, μ a probability measure, and t = (t 1 , . . . , t d ) a generic
vector in R d . The ECDF is defined in Eq. 13.15 for n samples {x 1 , . . . , x n }.
ˆ
F
n
μ =
number of elements in the sample ≤ t
n
=
1
n
n
i=1
1{x
i
≤ t}
(13.15)
where 1{A} is the indicator of event A:
1 A (x) :=
1 if x ∈ A
0 if x /
∈ A
(13.16)
and x i ≤ t meaning x i
j ≤ t j , j = 1, . . . , d. The last relation defines a partial order
and if it is true, then x i is either dominated by t or equal to it.
The estimation of Value-at-Risk and Conditional Value-at-Risk by means of the
ECDF is explained in the following subsections.
13.4.1.1 Value-at-Risk (Quantile) Estimation Using ECDF
Value-at-Risk for a scalar random variable X at a given confidence level α can be
directly computed from Eq. 13.15. Hence, if X 1 , X 2 , . . . , X n are n independent and
identically distributed observations of the random variable X, then the estimation of
the α-VaR of X is given by
ˆ
ν
α;n
= X nα:n = ˆ
F
−1
n (α)
(13.17)
where X i:n is the i-th order statistic from the n observations, and
ˆ
F n (t) =
n
i=1
1{X i ≤ t}
(13.18)
is the empirical CDF constructed from the sequence ˜
X of x 1 , x 2 , . . . , x n . Note that
the hat symbol ( ˆ ) indicates estimated quantities.