TERStatistics of Extremes.pdf

1. Basics and Recalled Results

1

Basics and Recalled Results

1.1

Order Statistics

Let X1 , . . . , Xn be a sample, i.e. a family of non-degenerate independent and identically
distributed random variables with a common distribution F . We define the ordered
sample as the family Xn,n ≤ · · · ≤ X1,n so that Xn,n = min(X1 , . . . , Xn ) and X1,n =
Mn = max(X1 , . . . , Xn ).
The random variable Xk,n is called the k-order statistic. Xn,n and X1,n = Mn are
called smallest order statistic and larger order statistic, respectively. The difference
X1,n − Xn,n is called the sample range.

1.2

Cumulative Distribution Function

The cumulative distribution function of a real-valued random variable X is the function
given by F (x) = P(X ≤ x) and the survival function is defined by F = 1 − F (x).
We also define the right endpoint xF of the distribution function F as:
xF = sup{x : F (x) < 1}.
Let X1 , . . . , Xn be independent, identically distributed real random variables with the
common cumulative distribution F . Then we define the empirical distribution function
as:
n
1X
Fˆn (x) =
1{Xi ≤x}
n i=1

1.3

Strong Law of Large Numbers

Let X1 , . . . , Xn be a sequence of independent and identically distributed random variables with expected value E[X1 ] = E[X2 ] = µ < ∞.
The sample mean X = n1 (X1 + · · · + Xn ) → µ when n → ∞.

1.4

Central Limit Theorem

Let X1 , . . . , Xn be a sequence of independent and identically distributed random variables with mean µ and finite, positive variance σ 2 . Then, defining:
Sn =

X 1 + · · · + Xn
,
n

√ (Sn − µ) d
n
→
− Z
σ
as n → ∞ where Z ∼ N (0, 1).

5

(1.1)