New results for ratios of extremes from distributions with a regularly varying tail are presented. Deriving from independence results for certain functions of order statistics, 'consecutive' ratios of extremes are shown to be independent as well as non-distribution specific. They have tractable distributions related to beta distributions. The minimum variance unbiased maximum likelihood estimator has the form of Hill's estimator. It achieves the Cramer-Rao minimum variance bound and is a function of a sufficient statistic. For small sample sizes the form of the moment generating function of the estimator shows it has a gamma distribution.Tail-index, Minimum variance unbiased, Maximum likelihood, Asymptotically normal

Roger Gay

English

Research Papers in Economics

 
 
Australia 
 
 
 
 
 
Department of Econometrics 
and Business Statistics 
 
http://www.buseco.monash.edu.au/depts/ebs/pubs/wpapers/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Minimum Variance Unbiased Maximum Likelihood 
Estimation of the Extreme Value Index 
 
Roger Gay  
 
 
 
 
 
 
 
 
 
April 2005 
 
 
 
 
Working Paper 8/05 
 
  1 
MINIMUM VARIANCE UNBIASED  
MAXIMUM LIKELIHOOD ESTIMATION  
OF THE EXTREME VALUE INDEX 
 
By ROGER GAY 
Dept. of Accounting and Finance, Monash University 
Clayton, Australia, 3800 
roger.gay@buseco.monash.edu.au 
 
 
SUMMARY 
 
New results for ratios of extremes from distributions with a regularly varying tail at 
infinity are presented. If the appropriately normalized order statistic X(n-j+1) from 
distribution with tail exponent  δ, (δ > 0) converges weakly to X
*
(j) then: 
 
(i)  For ρ = 1/δ, the ratio X
*
(j)/X
*
(j+k) is distributed like {U(j,k)}
-ρ  where  U(j,k) has 
density                                       
                                   B(u:j,k) = B
-1(j,k)u
j-1(1-u)
k-1,  0 ≤u ≤1  
 
where  B(a,b) = Γ(a)Γ(b)/Γ(a+b), a > 0, b > 0, Γ(ּ) denoting the gamma function, 
 
(ii)  Consecutive ratios of extremes X
*
(1)/X
*
(2), X
*
(2)/X
*
(3)     ּ ּ  X
*
(m/X
*
(m+1)   ּ ּ ּ  are 
independently distributed. 
 
(iii)  The maximum likelihood estimator (mle) based on the first  k ratios is 
 
                                     ρ ˆ  = k
-1 ∑
=
k
m
m
1
log{X
*
(m)/X
*
(m+1)} 
 
(iv) The mle ρ ˆ  is unbiased, has variance  ρ
2/k (the Cramer-Rao minimum variance 
bound) and is asymptotically normally distributed.  
 
(v)  It has moments of all orders and moment generating function  
 
                                   Mρ(θ) = (1-ρθ/k)
-k, θ>0. 
 
 
KEYWORDS: Tail-index, Minimum variance unbiased, Maximum likelihood, 
Asymptotically normal 
 
JEL CLASSIFICATION:  C13 
 
  2 
 
1.  INTRODUCTION AND MOTIVATION 
 
Considerable research effort has been devoted over the last thirty years to estimation of 
the exponent of regular variation, alternatively described depending on context, as the 
tail-index or the extreme value index. 
Popular estimators based on independent random samples such as those due to Hill 
(1975), Pickands (1975) and maximum likelihood estimators of Smith (1987) and Drees 
et al (2004) are consistent, asymptotically normal (i.e. biased) estimators. They are beset 
by an inherent dilemma; large variability when the number of extremes is too low, large 
bias when it is too high. A survey of bias problems is contained in Beirlant et al. (1999). 
Bias reduction of tail-index estimators, including maximum likelihood estimators has 
generally been approached by applying second order properties of regularly varying 
functions to the tail-quantile function of the distribution. Invariably the bias depends on 
the unknown exponent, and is distribution-specific (e.g. Drees et al. (2004), Teugels and 
Vanroelen (2004)). Another approach focuses on optimal threshold selection, trading off 
bias reduction against variability (e.g. Matthys, (2001)). 
 
‘Nowadays, . . . applications of extreme value theory can be seen in a large variety of 
fields such as hydrology, engineering, economics, astronomy and finance. The estimation 
of the extreme value index is the first and main statistical challenge’ (Teugels and 
Vanroelen, (2004). 
 
2.  MAIN RESULTS 
 
The class F of distributions F(ּ) have a regularly varying tail with index  δ (equivalently 
belong to the maximum domain of attraction of the Frechet distribution, (e.g. Embrechts 
et al. (1997), p.131)) if  1-F(x) = L(x)x
-δ, x > 0, δ > 0. 
The function  L(x) is slowly varying at infinity (see for instance Feller, (1971), p.276). 
 
Theorem 1  (distribution of a ratio of extremes) 
 
Denote by X(1), X(2), ⋅  ⋅  ⋅ X(n) ascending order statistics of common parent F ε F. 
The variables X
*
(1), X
*
(2),  ⋅ ⋅ ⋅ are descending Frechet extremes, i.e  X
*
(j) is the weak limit 
of  normalized order statistic  X(n-j+1)/vn  from the parent F ε F, the normalizing sequence 
{vn}obtained from the  parent tail-quantile function, satisfying n[1-F(vn)]=1, 
(see for instance, David and Nagaraja, (1993), Chapter 10). 
 
Then for any F ε F,  
                                              








+
*
) (
*
) (
k j
j
X
X
 = {U
*(j,k)}
-ρ  
 
where  U
*(j,k) has  beta density B(u:j,k) = B
-1(j,k)u
j-1(1-u)
k-1, (0 ≤ u ≤1) and 
where B(a,b) = Γ(a)Γ(b)/Γ(a+b), (a >0, b>0), Γ(·) representing the gamma function. 
  3 
Proof  (see Appendix, Note 1) 
 
Theorem 2  (independence of ‘consecutive’ ratios of extremes) 
 
Using the same notation as in Theorem 1 
 
 
For any F ε F,  
                                







+
*
) 1 (
*
) 1 (
j X
X
 = 








*
) 2 (
*
) 1 (
X
X
×








*
) 3 (
*
) 2 (
X
X
× ⋅ ⋅ ⋅ 








+
*
) 1 (
*
) (
j
j
X
X
 
 
where the right side is a product of independent {U(m,1)}
-ρ  variables, the U(m,1) having 
beta density B(u:m,1) = mu
m-1,(0 ≤ u ≤1). 
 
Proof  (see Appendix, Note 1) 
 
 
Theorem 3 (minimum variance maximum likelihood estimation based on ratios of 
extremes) 
 
(i)  The maximum likelihood estimator ρ ˆ  based on the k observed ratios  
Ym = X
*
(m)/X
*
(m+1)  m = 1,2,  ּ ּ ּ  k  is given by 
 
                                            ρ ˆ  = k
-1 ∑
=
k
m
m
1
log{X
*
(m)/X
*
(m+1)} 
 
(ii)  It is unbiased and with variance ρ
2/k  and is asymptotically normally distributed. 
The variance ρ
2/k is the Cramer-Rao minimum variance bound for ρ. 
 
(iii)  It has moment generating function  Mρ(θ) = (1-ρθ/k)
-k, θ > 0. 
 
Proof 
 
(i)   Note that L =    while ℓ = lnL  ) (
1
m
k
m
m y f ∏
=
 
                                                                = const. +klnδ –   lny ∑
=
+
m
m
m
1
) 1 ( δ m 
since X
*
(m)/X
*
(m) has distribution of  {U(m,1)}
-ρ where U(m,1) has density 
B(ym:m,1); i.e. fm(ym) = mδ(ym)
-mδ-1 
 
Thus 
  4 
                           
δ ∂
∂l
 = kρ -  lny ∑
=
m
m
m
1
m  
 
i.e.                                                    ρ ˆ  = k
-1 ∑
=
k
m
m
1
ln{X
*
(m)/X
*
(m+1)} 
Note that E[-
2
2
δ ∂
∂ l ] = kρ
2 
 
 
(ii)  Since  Ym =  X
*
(m)/X
*
(m+1) is distributed like {U(m,1)}
-ρ  where U(m,1) has density 
 
B(u:m,1) = mu
m-1, (integer m ≥ 1),  
 
                                   E[{mln(X
*
(m)/X
*
(m+1))}
j] = Γ(j+1)ρ
j, (integer j ≥ 1) 
 
 
Note that   Ij = E[{mln(X
*
(m)/X
*
(m+1))}
j] 
 
                     = (-ρ)
jE[{mlnU(m,1)}
j]  since X
*
(m)/X
*
(m+1)  is distributed like {U(m,1)}
-ρ 
 
                     = (-ρ)
j(-j)E[{mlnU(m,1)}
j-1] 
 
                     = jρ(-ρ)
j-1 E[{mlnU(m,1)}
j-1] 
 
                     = jρIj-1  while I0 = 1 
 
leading to E[{mln(X
*
(m)/X
*
(m+1))}
j] = Γ(j+1)ρ
j, (integer j ≥ 1) 
 
 
In particular, E[{mln(X
*
(m)/X
*
(m+1))}] = ρ 
 
                     E[{mln(X
*
(m)/X
*
(m+1))}
2] = 2ρ
2  (so that Var[mln(X
*
(m)/X
*
(m+1))] = ρ
2 
 
                     E[{mln(X
*
(m)/X
*
(m+1))}
3] = 6ρ
3 
 
From these results, 
 
                     E[ρ ˆ ] = k
-1E[ {X ∑
=
k
m
m
1
ln
*
(m)/X
*
(m+1)}] = ρ 
 
                  Var[ρ ˆ ] = k
-2 Var[ {X ∑
=
k
m
m
1
ln
*
(m)/X
*
(m+1)}] = ρ
2/k 
 
  5Since E[{mln(X
*
(m)/X
*
(m+1))}
3] = 6ρ
3 < ∞,  asymptotic normality follows for example 
from Liapunov’s Central Limit Theorem, (Rao, (1973), p.127). 
 
Since E[-
2
2
δ ∂
∂ l ] =kρ
2, the Cramer-Rao minimum variance bound for g(δ) = δ
-1 is given by 
 
Var[ρ ˆ ] ≥ {g´(δ)}/ E[-
2
2
δ ∂
∂ l ] = ρ
4/kρ
2 = ρ
2/k, 
 
(iii)  Recalling the independence of the ratios {X
*
(m)/X
*
(m+1)},  m = 1,2,    ּ ּ ּ k,  the 
moment generating function of ρ ˆ  is  
 
                              E[exp(θρ ˆ )] = E[ ]  k m
m
k
m
m X X / *
) 1 (
1
*
) ( } / { ϑ
+
=
∏
 
                                                  =  (1-ρθ/k)
-k   
 
Since E[{{X
*
(m)/X
*
(m+1)}
mθ/k] = E[{U(m,1)}
-ρmθ/k]  where U(m,1) has density  
B(u:m,1) = mu
m-1, (0 ≤u ≤ 1), its expectation is (1-ρθ/k)
-1. 
 
This completes the proof. 
 
3. SUMMARY AND CONCLUSIONS 
 
Fundamental new results for heavy-tailed extremes are formulated in terms of ratios of 
extremes. Such ratios are non-distribution specific, being independent of normalizing 
constants. 
Individual ratios in the form  {X
*
(j)/X
*
(j+k)} are distributed like powers of beta variates. 
‘Consecutive’ ratios  X
*
(1)/X
*
(2), X
*
(2)/X
*
(3),  ּ ּ ּ   are independently distributed. 
These results affect considerable simplification in dealing with samples of extremes. 
One consequence is that unbiased estimators of the tail-index are available. 
The maximum likelihood estimator ρ ˆ  = k
-1 ∑
=
k
m
m
1
ln{X
*
(m)/X
*
(m+1)}based on  k  observed 
ratios is itself unbiased. It achieves minimum variance ρ
2/k among unbiased estimators, 
and is asymptotically normal. Moreover ρ ˆ  has moments of all orders, with moment 
generating function Mρ(θ) = (1-ρθ/k)
-k. 
 
 
 
APPENDIX   
 
Note 1: Theorem 1  (The distribution of a ratio of extremes) 
 
Denote by X(1), X(2), ⋅  ⋅  ⋅ X(n) ascending order statistics of common parent F ε F. 
  6The variables X
*
(1), X
*
(2),  ⋅ ⋅ ⋅ are descending Frechet extremes, i.e  X
*
(j) is the weak limit 
of  normalized order statistic  X(n-j+1)/vn  from the parent F ε F, the normalizing sequence 
{vn}obtained from the  parent tail-quantile function, satisfying n[1-F(vn)]=1, 
(see for instance, David and Nagaraja, (1993), Chapter 10). 
 
Then for any F ε F,  
                                              








+
*
) (
*
) (
k j
j
X
X
 = {U
*(j,k)}
-ρ  
 
where  U
*(j,k) has  beta density B(u:j,k) = B
-1(j,k)u
j-1(1-u)
k-1, (0 < u < 1) and 
where B(a,b) = Γ(a)Γ(b)/Γ(a+b), (a >0, b>0), Γ(·) representing the gamma function. 
 
Proof:   
 
For the order statistics from common distribution F ε F and corresponding density f(⋅) the 
joint density of  (X(n-j-k+1), X(n-j+1)), (j < k) denoted by f
#(x(n-j-k+1),x(n-j+1)) is 
 
 B(u:j,k) × B(v:j+k,n-j-k+1) = B
-1(j, k)u
j-1(1-u)
k-1 ×B
-1(j+k,n-j-k+1)
 v
j+k-1(1-v)
n-j-k  
 
i.e. that of independent random variables U and V deriving from transformations 
(see for instance, Arnold et al. (1993), Chapter 2). 
 
(i)             u(x(n-j-k+1), x(n-j+1))  = {1-F(x(n-j+1))}/{1-F(x(n-j-k+1))} u ε [0,1] 
 
(ii)            v(x(n-j-k+1), x(n-j+1))   = x(n-j-k+1)   
 
Note that |
) , (
) , (
) 1 ( ) 1 ( + − + − − ∂
∂
j n k j n x x
v u
| = {f(x(n-j+1))/(1-F(x(n-j-k+1))}×f(x(n-j-k+1)) 
 
where ‘|·|’ represents absolute value. 
 
The implication of the transformation: 
       
                     u(x(n-j-k+1), x(n-j+1)) = {1-F(x(n-j+1))}/{1-F(x(n-j-k+1))}  
 
for F ε F, i.e. 1-F(x) = L(x)x
-δ when n is large, can be determined as follows: 
 
Put:  (i)   x(n-j+1) = vnx
*
(j) 
 
        (ii)  x(n-j-k+1) = vnx
*
(j+k) 
 
Note that  for 1-F(x) = L(x)x
-δ,     vn
δ = nL(vn) 
 
and                                                  u = {1-F(x(n-j+1))}/{1-F(x(n-j-k+1))} 
  7 
  i.e.                                                 u = L(vnx
*
(j))vn
-δ(x
*
(j))
-δ/ L(vnx
*
(j+k))vn
-δ(x
*
(j+k))
-δ 
 
                                                           = (x
*
(j))
-δ/ (x
*
(j+k))
-δ 
 
using for example L(vnx
*
(j))/ L(vn) → 1 as n → ∞. 
 
Thus the u-transformation implies for large  n  that 
 
(A.1)                               {U
*(j, k)}
-ρ = 







+
*
) (
*
) (
k j
j
X
X
  
 
where  U
*(j,k) has density  B(u:j, k), independent of X
*
(j+k).  
 
 
Check: The independence can be checked by showing equivalence of the moments of 
 
{X
*
(j)}
φ  and those of {X
*
(j)/X
*
(j+k)}
φ×{X
*
(j+k)}
φ on any dense φ-set interior to (0, j-ρφ), j-
ρφ>0. 
 
Note that E[{X
*
(j)}
φ] = Γ(j-ρφ)/Γ(j)  
 
This is the same as E{X
*
(j)/X
*
(j+k)}
φ×E{X
*
(j+k)}
φ = E[{U(j,k)}
-ρφ]× E{X
*
(j+k)}
φ] 
 
                                                                             = {B(j-ρφ,k)/B(j,k)}×Γ(j+k-ρφ)/Γ(j+j) 
 
                                                                             = Γ(j-ρφ)/Γ(j) 
 
 
 
This completes the proof of Theorem 1 
 
Note 2:  Theorem 2  (independence of consecutive ratios of extremes) 
 
Using the same notation as in Theorem 1 
 
 
For any F ε F,  
(A.2)                           








+
*
) 1 (
*
) 1 (
j X
X
= 








*
) 2 (
*
) 1 (
X
X
×








*
) 3 (
*
) 2 (
X
X
× ⋅ ⋅ ⋅ 








+
*
) 1 (
*
) (
j
j
X
X
 
 
  8where the right side is a product of independent {U
*(m,1)}
-ρ  variables, the U
*(m,1) being  
beta (m,1) random variables, for m = 1,2, ⋅  ⋅  ⋅  j. 
 
Proof 
 
We re-write product (A.2) as 
 
(A.3)               
δ −
+ 







*
) 1 (
*
) 1 (
j X
X
= 
δ −








*
) 2 (
*
) 1 (
X
X
×
δ −








*
) 3 (
*
) 2 (
X
X
× ⋅ ⋅ ⋅  ×
δ −
+ 







*
) 1 (
*
) (
j
j
X
X
 
 
Each of the ratios on the right side is a B(m,1) variable, m = 1, 2, ⋅ ⋅ ⋅ j,  using Theorem 1. 
Their independence then follows using Kendall and Stuart (1969), Exercise 11.8, and 
noting that the left side cannot be beta if the betas on the right side are not independent 
An alternative proof by equivalence of moments on a dense set (as outlined in the proof 
of Theorem 1) is also available using the uniqueness of moments for distributions (like 
beta) confined to closed intervals (see for instance Feller (1971) p.227). 
 
In fact Theorem 2 also succumbs to a proof based on extending the proof of Theorem 1 
as follows: 
 
 
Consider the joint distribution of  X(n-i-j-k+1), X(n-i-j+1) and  X(n-i+1) jointly distributed order 
statistics with common parent  F(·) ε F with density denoted by f
#, i.e. 
 
 
f
#(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1))  
 
                 = Γ(n+1)/{Γ(n-i-j-k+1)Γ(k)Γ(j)Γ(i)×F(x(n-i-j-k+1))
n-i-j-k×f(x(n-i-j-k+1)) × 
 
                     {F(x(n-i-j+1))-F(x(n-i-j-k+1))}
k-1× f(x(n-i-j+1))× 
 
                     {F(x(n-i+1))-F(x(n-i-j+1))}
j-1× f(x(n-i+1))× 
 
                     {1-F(x(n-i+1))}
i-1  
 
 
This can be re-written as 
 
f
#(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) =  
 
  Γ(n+1)/{Γ(i+j+k)Γ(n-i-j-k+1)}×F(x(n-i-j-k+1))
n-i-j-k×{1-F(x(n-i-j-k+1))
i+j+k-1×f(x(n-i-j-k+1))× 
 
  Γ(i+j+k)/{Γ(k)Γ(i+j)}×[1-{(1-F(x(n-i-j+1))/(1-F(x(n-i-j-k+1))}
k-1]× 
{(1-F(x(n-i-j+1))/(1-F(x(n-i-j-k+1))}
i+j-1×f(x(n-i-j+1)/(1-F(x(n-i-j-k+1)))× 
  9 
 Γ(i+j)/{Γ(j)Γ(i)}[1-{(1-F(x(n-i+1))/(1-F(x(n-i-j+1))
j-1]× 
{(1-F(x(n-i+1))/(1-F(x(n-i-j+1))}
i-1×f(x(n-i+1))/(1-F(x(n-i-j+1))) 
 
 
Now make the substitutions: 
 
(i)   u(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) = (1-F(x(n-i+1))/(1-F(x(n-i-j+1)) 
 
 
(ii)   v(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) = (1-F(x(n-i-j+1))/(1-F(x(n-i-j-k+1)) 
 
(iii)  v(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) = x(n-i-j-k+1) 
 
 
Note that  
 
 
) , , (
) , , (
) 1 ( ) 1 ( ) 1 ( + − + − − + − − − ∂
∂
i n j i n k j i n x x x
w v u
 =  
 
                              {f(x(n-i+1)/(1-F(x(n-i-j+1))}× {f(x(n-i-j+1)/(1-F(x(n-i-j-k+1))} 
 
 
So f
#(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1))  
 
= B
-1(n-i-j-k,i+j+k)w
n-i-j-k(1-w)
i+j+k-1× 
 
   B
-1(i+j,k)v
i+j-1(1-v)
k-1× 
 
   B
-1(i,j)u
i-1(1-u)
j-1×
) , , (
) , , (
) 1 ( ) 1 ( ) 1 ( + − + − − + − − − ∂
∂
i n j i n k j i n x x x
w v u
 
 
showing U,V, and W to be independently distributed. 
 
As in the proof of Theorem 1, consider the implications as n → ∞ when  
1-F(x) = L(x)x
-δ, and 
 
(i')  x(n-i-j-k+1) = vnx
*
(i+j+k) 
 
(ii')   x(n-i-j+1) = vnx
*
(i+j) 
 
(iii')  x(n-i-j+1) = vnx
*
(i) 
 
As outlined in the proof of Theorem 1, under these changes the transformation 
  10 
 
u(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) = (1-F(x(n-i+1))/(1-F(x(n-i-j+1)) 
 
becomes 
 
u(x
*
(i+j+k), x
*
(i+j), x
*
(i)) = {x
*
(i)/x
*
(i+j)}
-δ 
 
while the transformation  
 
v(x(n-i-j-k+1), x(n-i-j+1), x(n-i+1)) = (1-F(x(n-i-j+1))/(1-F(x(n-i-j-k+1)) 
 
becomes 
 
u(x
*
(i+j+k), x
*
(i+j), x
*
(i)) = {x
*
(i+j)/x
*
(i+j+k)}
-δ 
 
These transformations imply that  
 
{X
*
(i)/X
*
(i+j)} has the distribution of {U(i,j)}
-ρ  where U(i,j) has beta density B(u:i,j) 
 
and is distributed independently of  
 
{X
*
(i+j)/X
*
(i+j+k)} which has the distribution of {V(i+j,k)}
-ρ  where V(i+j,k) has beta 
density B(v:i+j,k). 
 
Thus the dependent set of extremes  {X
*
(i), X
*
(i+j), X
*
(i+j+k)} is transformed to the 
independent random variables{(X
*
(i)/X
*
(i+j))
-δ,  (X
*
(i+j)/X
*
(i+j+k))
-δ, X
*
(i+j+k)} the first two 
random variables having densities  B(u:i,j) and B(v:i+j,k) respectively. 
 
A special case of Theorem 2 then follows on putting i = j = k = 1; 
 
i.e. (X
*
(1)/X
*
(2)), (X
*
(2)/X
*
(3)) are independently distributed with distributions like 
{U(1,1)}
-ρ, {U(2,1)}
-ρ  where U(m,1) has beta density B(u:m,1), m = 1, 2. 
 
Evidently the mechanism extends to k ratios and Theorem 2 follows. 
 
 
 
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reduction and optimal threshold selection. Doctoral Thesis, Katholieke Universiteit 
Leuven. 
 
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119-131 
 
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Minimum Variance Unbiased Maximum Likelihood Estimation of the Extreme Value Index

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Minimum Variance Unbiased Maximum Likelihood Estimation of the Extreme Value Index

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