A simple and efficient test for the Pareto law
Supplementary material
Francisco J. Goerlich Gisbert
University of Valencia and Instituto Valenciano de Investigaciones Económicas
(Ivie)
Address: University of Valencia, Department of Economic Analysis, Campus
Tarongers, Av. Tarongers s/n, 46022-Valencia. Spain.
Phone: 00-34-96 382 82 53
Fax: 00-34-96 382 82 49
E-mail: Francisco.J.Goerlich@uv.es.
1
SUPPLEMENTARY MATERIAL
Useful results related to the Pareto law and the MLE estimation
We state some results related to the Pareto law used in the text. All of them follow
from standard results on the distribution of functions of random variables (Mood,
Graybill y Boes 1974, Chapter V).
x
Result 1: If x follows a Pareto law, then y1  log follows an exponential

1
distribution with parameter . Thus, E ( y1 )  .


Result 2: If x follows a Pareto law, then y2  follows a power function
x
distribution with parameter  (a special type of Pearson Type I


distribution). Thus, E ( y2 ) 
and E ( y22 ) 
.
 1
2

Result 3: If x follows a Pareto law with   1, then y2  follows a
x
uniform distribution on the interval [0,1].
Remark A.1. The MLE is very different from the OLS estimator of the
log-log linear relationship mentioned in the introduction (Nishiyama and
Osada 2004; Nishiyama, Osada and Sato 2008); although Aban and
Meerschaert (2004) show that the MLE coincides with the Generalized
Least Squares (GLS) estimator of this log-log linear relationship taking
into account the mean and covariance structure of the extreme order
statistics.
Remark A.2. Using well-known results and the fact that the exponential
distribution belongs to the class of gamma distributions, it is possible in
this case to obtain the small sample distribution of ̂ . Therefore, we know
the exact expectation and variance of the estimator (Johnson and Kotz
1970, 19.4.4).
Remark A.3. We have noticed in the text that  can be considered as a
measure of inequality, in the sense that higher values of  imply less
dispersion, and in fact lower values of standard relative inequality
measures, such as CV(X) or G(X).
Derivation of the LMp test statistic
In the sequel, we obtain the score and the information matrix of log LB and next
we derive the LM statistic for H0:   .
The gradient of (7) is given by
 log LB n n
 xi   
   log 1 


 i 1
 

n
 log LB
n
xi  
   (  1)


 xi   
i 1
2 1 

 

(A.1)
2
Note that, by equating these equations to zero we have a system of nonlinear equations that have to be solved by numerical methods. Therefore, the MLE
of the parameters for the Burr density does not have an explicit formula.
The Hessian is given by
 log 2 LB
n
 2
2


2
n
 log LB
xi  


 xi   
i 1
2 1 

 

n
 log 2 LB
n
( xi  )(2  xi  )


(


1)

2
2


2 (  xi  )2
i 1
(A.2)
Evaluating (A.1) at the restricted estimate of , ̂   , we have
 log LB


1 1 n
 xi  
 n    log   
  
  n i 1
(A.3a)
n(  1)  
1 n 


    1 n i 1 xi 
(A.3b)
and
 log LB



Remark A.4. Given that the restricted estimate of  is just the Pareto
MLE, ̂ (5), it is worth noting that evaluating (A.3a) at ̂ is just zero,
 log LB
 0.
 ˆ

 xi 
1 1 n
  log   , a function of , (A.3a)
 n i 1

measures a discrepancy between a population mean and a sample mean,
since given result 1 E ( z1 )  0 . This gives the MLE of  under the Pareto
law a nice moment interpretation.

1 n 
  , a function of , (A.3b)
Remark A.6. Defining z2 
  1 n i 1 xi
measures a discrepancy between a population mean and a sample mean,
since given result 2 E( z2 )  0 .
Remark A.5. Defining z1 
Evaluating (A.2) at the restricted estimate of , ̂   , we have
 log 2 LB



n 1 n 
1 
  n i 1 xi 
(A.4a)
3
and
 log 2 LB
2

n(  1)  
1 n  2 

   

2    1 n i 1  xi2  
(A.4b)
changing its sign and evaluating the expectation,
  log 2 LB
E 
 
  log 2 LB
E
2
 

n
n
 1 n  
   1  E      

(  1)
 n i 1 xi  

 
(A.5a)
 n(  1)  
 1 n  2  
n
(A.5b)

E


   2    2
2



1
n

(


2)
i 1  xi   



 


Thus, the Fisher sample information, evaluated at    , is given by

1

n 
(,   )  2
 
2

 (  1)
2 
(  1) 

3

 2 (  2) 

(A.6)
and its inverse is
 3
 2
1  2 (  2)(  1) 2   (  2)
1
 (,   )  .
 2
n


 (  1)
Now we can form the LM statistic. Noting that zˆ1 
2 
(  1) 

1 

(A.7)
 xi 
1 1 n
  log    0 ,
ˆ n i 1

the score vector evaluated at the restricted estimate is
0
 zˆ1 





q  n  (ˆ  1)   n  (ˆ  1) 
zˆ2
zˆ2
 

 

21
(A.8)
ˆ
1 n 
  . Therefore, only the (2,2) element of (A.8) matters in the
ˆ  1 n i 1 xi
1  2 (  2)(  1) 2
LM statistic, .
. The feasible LM statistic is then given by
n

with zˆ2 
LM P  n
(ˆ  2)(ˆ  1)4 2 asy 2
.zˆ2 ~  (1) on H0 :   
ˆ
(A.9)
as stated in the text.
4
Power tables behind the figures 2, 3 and 4
Table A.1 Power of the LMP test against the Burr distribution. Percentage of
rejections, n = 1000 and asymptotic critical values
H1:  = .(1 + )
Significance
level


10%
5%
1%
1
2
3
5
10
-0.5
100.0%
99.8%
97.5%
80.9%
42.9%
-0.4
99.8%
96.7%
87.4%
63.7%
32.2%
-0.3
93.7%
80.9%
65.7%
43.3%
23.0%
-0.2
64.3%
49.5%
38.0%
25.6%
15.7%
-0.1
25.0%
20.5%
17.2%
14.2%
11.4%
0.0
10.0%
10.0%
9.9%
10.0%
9.7%
0.1
22.5%
18.2%
15.3%
12.5%
10.3%
0.2
51.3%
39.5%
30.2%
20.7%
12.9%
0.3
77.9%
63.8%
50.4%
32.7%
17.5%
0.4
92.8%
82.9%
69.6%
47.4%
23.5%
0.5
98.4%
93.6%
84.1%
62.0%
31.1%
-0.5
100.0%
99.6%
95.4%
73.4%
33.2%
-0.4
99.4%
94.2%
81.1%
53.4%
23.2%
-0.3
89.2%
72.5%
54.9%
33.4%
15.5%
-0.2
52.9%
38.1%
27.9%
17.4%
9.6%
-0.1
16.1%
12.8%
10.6%
8.3%
6.3%
0.0
5.0%
5.1%
4.9%
5.0%
4.8%
0.1
13.8%
10.5%
8.3%
6.3%
5.0%
0.2
38.2%
27.0%
19.2%
11.8%
6.5%
0.3
67.0%
50.4%
36.6%
20.9%
9.5%
0.4
87.1%
72.9%
56.3%
33.8%
13.6%
0.5
96.4%
88.0%
74.1%
47.9%
19.3%
-0.5
100.0%
98.2%
88.2%
55.7%
18.1%
-0.4
97.6%
85.2%
64.6%
34.0%
11.0%
-0.3
75.1%
52.5%
34.4%
17.2%
6.4%
-0.2
30.5%
19.6%
13.1%
7.2%
3.3%
-0.1
5.7%
4.3%
3.5%
2.6%
1.7%
0.0
1.0%
1.0%
1.0%
1.0%
1.0%
0.1
4.1%
2.7%
1.8%
1.2%
0.9%
0.2
17.0%
9.8%
5.9%
2.9%
1.2%
0.3
41.6%
25.4%
14.9%
6.2%
2.0%
0.4
68.3%
47.0%
29.3%
12.7%
3.3%
0.5
86.9%
68.7%
47.8%
21.7%
5.4%
Source: Own Monte Carlo simulations using the direct inversion method, 100,000 replications and  = 1.
5
Table A.2 Power against the exponential distribution. Percentage of rejections.
(a) LMP

Significance
level
1
2
3
5
10
10%
100.0%
100.0%
100.0%
99.9%
83.9%
5%
100.0%
100.0%
100.0%
99.7%
72.1%
1%
100.0%
100.0%
100.0%
97.7%
40.8%
(b) Bootstraped K-S test of Clauset et al (2009)

Significance
level
1
2
3
5
10
10%
100.0%
100.0%
100.0%
96.9%
59.9%
5%
100.0%
100.0%
99.9%
93.0%
45.9%
1%
100.0%
100.0%
99.3%
76.2%
22.1%
Source: Own Monte Carlo simulations using the direct inversion method and 100,000 replications for the LMP test. Matlab
script plpva.m of Aaron Clauset and 10,000 replications for the K-S test.
Table A.3 Power against the lognormal distribution. Percentage of the tail
observations included in the test. Percentage of rejections
(a) LMP
Significance
level
10%
5%
2.5%
1%
0.5%
10%
100.0%
99.2%
96.4%
88.4%
81.1%
5%
99.9%
98.0%
92.2%
79.5%
69.5%
1%
98.7%
90.3%
75.0%
53.5%
40.6%
(b) Bootstraped K-S test of Clauset et al (2009)
Significance
level
10%
5%
2.5%
1%
0.5%
10%
98.3%
91.9%
81.8%
67.7%
58.0%
5%
95.9%
84.6%
70.1%
54.6%
44.4%
1%
84.6%
62.9%
43.3%
28.8%
20.4%
Source: Own Monte Carlo simulations using the direct inversion method and 100,000 replications for the LMP test. Matlab
script plpva.m of Aaron Clauset and 10,000 replications for the K-S test.
References
Aban IB Meerschaert MM (2004) Generalized least-squares estimators for the thickness of heavy
tails. J Stat Plann Infer 119:341-352.
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Mood AM Graybill FA Boes DC (1974) Introduction to the Theory of Statistics. 3rd edn.
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distribution. Discussion Paper 009 (January), 21COE, Interfaces for Advanced Economic
Analysis, Kyoto University. http://www.kier.kyoto-u.ac.jp/coe21/dp/01-10/DP009nishiyama%26oasada.pdf. Accessed September 2012
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