Journal of Statistical Planning and Inference 137 (2007) 359 – 361
www.elsevier.com/locate/jspi
Letter to the Editor
Proof of a conjecture on Spearman’s and Kendall’s for sample
minimum and maximum
Xiaohu Li
∗
, Zhouping Li
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Received 28 July 2004; received in revised form 18 August 2005; accepted 18 August 2005
As two nonparametric measures of association between two random variables X and Y , Kendall’s and Spearman’s
(see Schweizer and Wolff, 1981, and the references therein for more details) are known to be closely related to the
copula C of X and Y. Specifically, by Nelsen (1999, 1992), it holds that
(X, Y ) = 4
(0,1)
2
C(u, v) dC(u, v) - 1 = 4E
C
[C(U,V)]- 1,
where U and V are two U(0, 1) random variables with copula C and E
C
denotes the expectation with respect to the
Lebesgue–Stieltjes measure P
C
induced by C. Additionally,
(X, Y ) = 12
(0,1)
2
uv dC(u, v) - 3 = 12E[C(U,V)]- 3,
here U and V are two i.i.d. U(0, 1) random variables.
Recently, Schmitz (2004) investigates the dependence structure of the minimum X
(1)
= min{X
1
,...,X
n
} and
maximum X
(n)
= max{X
1
,...,X
n
} of an i.i.d. sample X
1
,...,X
n
through determining their copula C(u, v), the
closed form expression of Kendall’s
n
= (X
(1)
,X
(n)
) and Spearman’s
n
= (X
(1)
,X
(n)
) are given by the following
expressions:
(X
(1)
,X
(n)
) =
1
2n - 1
,
(X
(1)
,X
(n)
) = 3 -
12n
2n
n
n
k=0
(-1)
k
2n - k
2n
n + k
+ 12
(n!)
3
(3n)!
(-1)
n
.
Based upon some empirical evidence, Schmitz (2004) further conjectures that
n
n
→
3
2
as n →∞. (1)
Now, we address a mathematical proof of (1), which validates the conjecture. As a result, for a large n, Spearman’s
n
can be evaluated as 3/(4n - 2).
∗
Corresponding author.
E-mail address: xhli@lzu.edu.cn (X. Li).
0378-3758/$ - see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2005.08.048