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Table 1 Some examples

From: Approximate moments of extremes

X  ∼  F ( x )

Support

i

α

\(\boldsymbol {b_{n} }\)

\(\boldsymbol {c_{n}}\)

exponential

(0,∞)

1

 

\(n_{1} \)

1

logistic

\(\mathbb{R} \)

1

 

\(n_{1} \)

1

normal

\(\mathbb{R} \)

1

 

\(B_{n}\)

\(( 2n_{1} )^{1/2}\)

log-normal

(0,∞)

1

 

\(B_{n}^{\prime}\)

\(( 2n_{1} )^{-1/2} B_{n}^{\prime}\)

1 − exp(1/x)

(−∞,1)

1

 

\(-n_{1}^{-1} \)

\(n_{1}^{-2}\)

Pareto \(1-(a/x)^{\alpha}\)

(a,∞)

2

α

0

\(an^{1/ \alpha}\)

Cauchy

\(\mathbb{R}\)

2

1

0

\(n^{-1}\)

uniform

(0,1)

3

1

1

\(n^{-1}\)

\(1 - (x_{0} - x )^{\alpha} \)

\((x_{0}-1, x_{0} )\)

3

α

\(x_{0}\)

\(n^{-1/\alpha}\)

\(\{ 1-\exp(-x) \} / \{ 1 - \exp (-x_{0} ) \}\)

\((0,x_{0} )\)

3

1

\(x_{0}\)

\(\{ \exp (x_{0} ) - 1 \} / n\)

\(G_{1}\)

\(\mathbb{R}\)

1

 

1

1

\(G_{2}\)

(0,∞)

2

α

0

\(n^{1/ \alpha}\)

\(G_{3}\)

(−∞,0)

3

α

0

\(n^{-1/\alpha}\)