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The multivariate-t distribution and the Simes inequality
Statistics and Probability Letters 83 (2013) 227–232
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
The multivariate-t distribution and the Simes inequality Henry W. Block a,∗ , Thomas H. Savits a , Jie Wang c , Sanat K. Sarkar b a
University of Pittsburgh, United States
b
Temple University, United States
c
Grand Valley State University, United States
article
info
Article history: Received 2 January 2012 Received in revised form 16 August 2012 Accepted 16 August 2012 Available online 23 August 2012 Keywords: Multivariate-t distribution Simes’ inequality Positive dependence MTP2
abstract Sarkar (1998) showed that certain positively dependent (MTP2 ) random variables satisfy the Simes inequality. The multivariate-t distribution does not satisfy this (MTP2 ) property, so other means are necessary. A corollary was given in Sarkar (1998) to handle this distribution, but there is an error. In this paper a direct proof is given to show the multivariate-t does satisfy the Simes inequality. © 2012 Published by Elsevier B.V.
1. Introduction This paper is motivated by the need to have a clarification of a result on Simes (1986) inequality in Sarkar (1998). While establishing this inequality for positively dependent multivariate random variables, as originally conjectured in Simes (1986), Sarkar (1998) assumed a specific type of positive dependence, e.g., the dependence that is characterized by the MTP2 property (see Theorem 3.1 of Sarkar (1998)). Having realized that the standard multivariate-t with the associated multivariate normal having non-negative correlation does not have this property, Sarkar (1998) made an attempt to prove the Simes inequality for this distribution under certain sign restrictions through a corollary (Corollary 3.1). Unfortunately the proof of this corollary does not appear to be correct, as we will point out in this paper. Thus, the claim that the multivariate-t distribution satisfies the Simes inequality does not follow from the argument of Remark 3.2 in Sarkar (1998). A careful study of Sarkar’s (1998) proof of the Simes inequality would reveal, and also it follows from the proofs of the FDR (false discovery rate) control of the method of Benjamini and Hochberg (1995) in the positive dependence case (Benjamini and Yekutieli, 2001; Sarkar, 2002), that the Simes inequality will continue to hold for distributions with a weaker version of dependence condition than the MTP2 condition, such as the positive dependent through stochastic ordering (PDS) condition of Block et al. (1985), which is essentially the same as the positive regression dependence on subset (PRDS) condition considered in Benjamini and Yekutieli (2001) and Sarkar (2002). However, as we will show in this paper, the standard multivariate-t with non-negative correlations does not satisfy the PDS or PRDS condition. Thus, the Simes inequality for the standard multivariate-t with non-negative correlations does not follow from the PDS or PRDS condition either. Nevertheless, the Simes inequality is indeed true for the standard multivariate-t with non-negative correlations under certain sign restrictions (Benjamini and Yekutieli, 2001; Sarkar, 2002, 2008). In this paper we provide a direct proof, one based on the paper of Sarkar (2002). Sarkar (2008) proves a general result, similar to the one in this paper, but under a stronger dependence condition.
∗
Corresponding author. Tel.: +1 4126248369; fax: +1 4126488814. E-mail address: [email protected] (H.W. Block).
0167-7152/$ – see front matter © 2012 Published by Elsevier B.V. doi:10.1016/j.spl.2012.08.013
228
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
The paper is organized as follows. In the next section, we make some comments on the results of Sarkar (1998) towards proving the Simes inequality for the multivariate-t distribution in Section 3. 2. Comments on Sarkar (1998) In this section, we make some improvements to Sarkar (1998) before giving proofs of the Simes inequality in the next section for the multivariate-t distribution. First, we present a more streamlined proof of his Lemma 2.1 using arguments similar to those used in more recent papers related to FDR methodologies; see also Sarkar (2008). Second, the Simes inequality is presented under a weaker form of positive dependence more suitable for multivariate t distributions. In addition, we give an example showing that the proof of Corollary 3.1 of Sarkar (1998) is not correct, and so his argument showing that the multivariate-t satisfies the Simes inequality does not apply. Lemma 2.1. Let X(1) ≤ · · · ≤ X(n) be the ordered components of a random vector X = (X1 , . . . , Xn )′ and a1 ≤ · · · ≤ an be real numbers. Then, P X(1) > a1 , . . . , X(n) > an
(−i)
where for each i = 1, . . . , n, X(1)
n n−1 I (Xi ≤ aj+1 ) I (Xi ≤ aj ) E − n i=1 j+1 j i=1 j=1 i) i) × I X((− > aj+1 , . . . , X((− , j) n−1) > an
= 1−
n 1
Fi (an ) +
(1)
i) ≤ · · · ≤ X((− n−1) denote the ordered components of the (n − 1)-dimensional random vector
X(−i) obtained by eliminating Xi from X.
Proof. Let Rn = max r | X(r ) ≤ ar , so that
{Rn = r } = X(r ) ≤ ar , X(r +1) > ar +1 , . . . , X(n) > an ,
for r = 1, . . . , n,
and
{Rn = 0} = X(1) > a1 , . . . , X(n) > an . Then I {Rn > 0} =
n r r =1
=
I (Rn = r ) =
r
r =1
n n 1
r
i=1 r =1 n
=
r i =1
I (Xi ≤ ar ) I (Rn = r )
I (Xi ≤ ar ) I X(r ) ≤ ar , X(r +1) > ar +1 , . . . , X(n) > an
n
1
(−i)
I (Xi ≤ ar ) I Rn−1 = r − 1 r
i=1 r =1
i) = max r | X((− r −1) ≤ ar , defined analogously to Rn , i.e., (−i) (−i) (−i) (−i) Rn−1 = r − 1 = X(r −1) ≤ ar , X(r ) > ar +1 , . . . , X(n−1) > an
where
(−i) Rn−1
n n 1
for r = 2, . . . , n,
and
(−i)
(−i)
(−i)
Rn−1 = 0 = X(1) > a2 , . . . , X(n−1) > an . (−i)
(−i)
(−i)
(−i)
(−i)
Since I {Rn−1 = r − 1} = I {Rn−1 ≤ r − 1} − I {Rn−1 ≤ r − 2}, for r = 1, . . . , n, with I (Rn−1 < 0) = 0 and I {Rn−1 ≤ n − 1} = 1, it follows that I {Rn > 0} =
n n−1
(−i)
I Rn−1 ≤ r − 1
I (X ≤ a ) i r r
i=1 r =1
−
n n
(−i)
n 1
n i=1
I ( X i ≤ an ) +
i=1
n
I (X ≤ a ) i r
I Rn−1 ≤ r − 2
r
i=1 r =2
=
n I (Xi ≤ an ) i) + I R(− n −1 ≤ n − 1
n n−1
(−i)
I (X ≤ a ) i r
I Rn−1 ≤ r − 1
i=1 r =1
r
−
I (Xi ≤ ar +1 ) r +1
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
=
n 1
n i =1
I (Xi ≤ an ) +
n n −1
(−i)
(−i)
I X(r ) > ar +1 , . . . , X(n−1) > an ×
i=1 r =1
229
I ( X i ≤ ar ) r
−
I (Xi ≤ ar +1 )
r +1
Using I (X(1) > a1 , . . . , X(n) > an ) = 1 − I (Rn > 0), and taking expectations gives the required result.
Lemma 2.2 (Simes’ Inequality.). Let X be a random vector. Then, (i) we have P X(1) > a1 , . . . , X(n) > an ≥ 1 −
n 1
n i=1
P { X i ≤ an } ,
for any fixed a1 ≤ · · · ≤ an such that j−1 P X1 ≤ aj is nondecreasing in j = 1, . . . , n under the assumption that
(−i)
P X(j)
i) > aj+1 , . . . , X((− n−1) > an | Xi ≤ aj ↑ in aj ,
for every j = 1, . . . , n − 1 and all i = 1, . . . , n; and (ii) we have P X(1) < b1 , . . . , X(n) < bn ≥
n 1
n i=1
P { X i < b1 } ,
for any fixed b1 ≤ · · · ≤ bn such that j−1 P X1 > bn−j+1 is nondecreasing in j = 1, . . . , n under the assumption that
(−i)
(−i)
P X(1) < b1 , . . . , X(j)
< bj | Xi ≥ bi ↓ in bi ,
for every j = 1, . . . , n − 1 and all i = 1, . . . , n. Proof. The first part of the lemma follows from the following inequalities:
I Xi ≤ aj+1 I Xi ≤ aj − E I > > an j+1 j i) i) = P X((− > aj+1 , . . . , X((− (j + 1) j) n−1) > an , Xi ≤ aj+1 i) i) −P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj P {Xi ≤ aj }/j i) (−i) ≥ P X((− > a , . . . , X > a , X ≤ a (j + 1) j + 1 n i j + 1 j) (n−1) i) i) − P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj+1 P {Xi ≤ aj }/j i) (−i) ≥ P X((− > a , . . . , X > a , X ≤ a (j + 1) j +1 n i j +1 j) (n−1) i) i) − P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj+1 P {Xi ≤ aj+1 }/(j + 1)
(−i) X(j)
(−i) aj+1 , . . . , X(n−1)
= 0, for any fixed pair of i and j. Then apply Lemma 2.1. The second part follows from the first part. Example 2.1. The following example shows that the proof of Corollary 3.1 of Sarkar (1998) is not correct. Let g (x, y) = 1/y exp(−x/y) for all x and y > 0. Also let q(z ) = λ exp(−λz ) for all z > 0 where λ > 0. It is easy to show that g ∗ (x, y) =
∞
zg (zx, y)q(z ) dz = 0
λy (λy + x)2
.
230
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
which is clearly TP2 in x and y. For n = 2, a1 = 1, a2 = 3 and λ = 1, after some calculations it can be shown that P {T1 ≤ 1} −
1 2
3−y
P {T2 ≤ 3} =
2(y + 1)(y + 3)
h(y) dy,
and P {T(1) ≥ 1, T(2) ≥ 3} −
P {X(∗1)
≥
1, X(∗2)
y(y2 − 3y − 24)
≥ 3} = −3
(y + 1)(y + 3)2 (y + 4)(y + 6)
h(y) dy,
where h is as in Corollary 3.1 of Sarkar. The integrand in the first expression is negative for any probability density h(y) whose support is contained in (3, ∞) and so satisfies the condition
P {T1 ≤aj }
√
j
nondecreasing in j = 1, 2. Moreover, since the
roots of y − 3y − 24 = 0 are = −3.62348, 6.62348, the integrand in the second expression is negative if the support of h(y) is contained in (6.62348, ∞). Thus (3.8) of Sarkar (1998) does not hold for this h. However, we have not been able to show whether the statement of the corollary is correct or not. 3± 105 2
2
3. Multivariate-t distribution In this section we show that the standard multivariate-t, with the associated normal having nonnegative correlations, satisfies the Simes inequality under certain sign restrictions. The standard multivariate-t distribution is given in Tong (1990, Chapter 9) as follows. Let R = (ρij ) be an n × n symmetric matrix which is either positive definite or positive semidefinite with ρii = 1 for i = 1, 2, . . . , n. In the following we will always assume positive definiteness. Let Z = (Z1 , . . . , Zn )′ have a multivariate normal distribution with mean vector 0 and correlation matrix R and let the univariate random variable S be such that (i) S is independent of Z and (ii) ν S 2 has a chi-square distribution with ν degrees of freedom. Then the multivariate T is defined by T = (T1 , . . . , Tn ) = ′
Z1 S
,...,
Zn
′
S
.
This is called ‘The Multivariate-t with Common Denominator’ by Johnson and Kotz (1972) (see various other multivariatet distributions defined there) and is also the basic multivariate-t given by (1.1) of Kotz and Nagarajah (2004). In Sarkar (1998) this is called the multivariate-t of Dunnett and Sobel (1954) type, but Johnson and Kotz (1972) seem to also attribute this distribution to Cornish and Laurent (see (Johnson and Kotz, 1972). It is not hard to show that for ρij ≥ 0, i ̸= j, the multivariate-t has several positive dependence properties. See Section 9.2.1 of Tong (1990), where various lower and upper orthant inequalities are derived and it is also shown that the distribution is associated. However it does not possess higher order dependence properties such as MTP2 , as shown in Sampson (1983), and also is not positive dependent through stochastic ordering (Block et al., 1985), which we will show. This latter property is essentially the same as PRDS of Benjamini and Yekutieli (2001). We now prove the Simes inequality for the multivariate-t distribution under certain sign restrictions (i.e., the ai are negative and the bj are positive) by establishing the following theorem. Theorem 3.1. Let T be the multivariate-t distribution as given above. Then, (i) for fixed a1 ≤ · · · ≤ an < 0, we have P T(1) > a1 , . . . , T(n) > an ≥ 1 −
n 1
n i =1
P {Xi ≤ an } ,
(2)
if j−1 P T1 ≤ aj is nondecreasing in j = 1, . . . , n; and (ii) for fixed 0 < b1 ≤ · · · ≤ bn , we have
P T(1) ≤ b1 , . . . , T(n) ≤ bn ≥
n 1
n i=1
P { T i ≤ b1 } ,
if j−1 P T1 > bn−j+1 is nondecreasing in j = 1, . . . , n.
Proof. Part (i) of the theorem will follow from Lemma 2.2 if we can show that
(−i)
P T(j)
i) > aj+1 , . . . , T((− n−1) > an | Ti ≤ aj ↑ in aj < 0
for every j = 1, . . . , n − 1 and all i = 1, . . . , n. Since Ti = Zi /S with Z independent of S, we can write
(−i)
P T(j)
i) > aj+1 , . . . , T((− n−1) > an ; Ti ≤ aj = E [ψ(Zi , S ); Ti ≤ aj ]
(3)
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
231
by conditioning on (Zi , S ), where i) i) ψ(zi , s) = P {Z((− > saj+1 , . . . , Z((− j) n−1) > san |Zi = zi }
since S is independent of Zi . Since Z is PDS (see Block et al. (1985)), ψ(zi , s) ↑ zi for each s; also ψ(zi , s) ↑ s for each zi because the aj ’s are negative. Now set Si∗ = (Zi2 + ν S 2 )1/2 and Ui = Zi /Si∗ . Then Ui and Si∗ are independent. Noting that Ti ≤ aj is equivalent to Ui ≤ aj /(ν + aj 2 )1/2 , we write
i) > aj+1 , . . . , T((− n−1) > an | Ti ≤ aj 1/2 ∗ ∗ 1 − Ui Si 2 1/2 = E ψ Ui Si , ; Ui ≤ aj /(ν + aj ) P (Ui ≤ aj /(ν + a2j )1/2 ) √ ν
(−i)
P T(j)
= E [φ(Ui ); Ui ≤ aj /(ν + a2j )1/2 ]/P (Ui ≤ aj /(ν + a2j )1/2 ). Here
2 1/2 ∗ Si ∗ ( 1 − ui ) , φ(ui ) = E ψ ui Si , √ ν which is nondecreasing for ui ∈ (−1, 0). Set
ˆ ui ) = φ(ui ) φ( φ(0)
for − 1 < ui < 0 for 0 ≤ ui < 1.
ˆ ui ) is nondecreasing on (−1, 1) and Then φ( E φ(Ui ); Ui ≤
aj
(ν + a2j )1/2
ˆ Ui ); Ui ≤ = E φ(
aj
(ν + a2j )1/2
.
Using the following Lemma 3.1, we conclude that the right hand side of the above equation is nondecreasing in aj /(ν + a2j )1/2 ∈ (−1, 0). Since aj /(ν + a2j )1/2 is nodecreasing in aj , it follows that the left hand side of the equation is nondecreasing in aj . This establishes part (i) of the Theorem. Part (ii) of the Theorem follows from part (i). Lemma 3.1. Let X be a continuous random variable with support on an interval (c , d) and g (x) a continuous nondecreasing function there. Then E {g (X )|X ≤ ξ } is nondecreasing for ξ ϵ(c , d). The above proof of Theorem 3.1 is a slight variation of that based on an idea briefly outlined in Sarkar (2002) and elaborated in Sarkar (2008). Example. This example shows that the bivariate-t distribution for ρ ≥ 0 does not satisfy the positive dependence property of stochastic increasing. Let (T1 , T2 ) have a bivariate-t distribution with ρ = 0. Then it can be shown that d dt2
P {T1 > −1|T2 = t2 } =
−2t2 2 + t22 π (3 + t22 )2
.
This is negative for t2 > 0 and hence P {T1 > −1|T2 = t2 } is not increasing there; i.e., (T1 , T2 ) is not PDS. Acknowledgments H.W. Block and T.H. Savits were supported by NSA Grant H 98230-07-1-0018. S.K. Sarkar was supported by NSF Grant DMS-0603868. References Benjamini, Y., Hochberg, Y., 1995. Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of Royal Statistical Society Series B 57, 289–300. Benjamini, Y., Yekutieli, D., 2001. The control of the false discovery rate in multiple testing under dependency. Annals of Statistics 29, 1165–1188. Block, H.W., Savits, T.H., Shaked, M., 1985. A concept of negative dependence using stochastic ordering. Statistics and Probability Letters 3, 81–86. Dunnett, C.W., Sobel, M., 1954. A bivariate generalization of Student’s t distribution with tables for certain special cases. Biometrika 41, 153–169. Johnson, N.L., Kotz, S., 1972. Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York. Kotz, S., Nagarajah, S., 2004. Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge. Sampson, A.R., 1983. Positive dependence properties of elliptically symmetric distributions. Journal of Multivariate Analysis 13, 375–381.
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Sarkar, S.K., 1998. Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Annals of Statistics 26, 494–504. Sarkar, S.K., 2002. Some results on false discovery rate in stepwise multiple testing procedures. Annals of Statistics 30, 239–257. Sarkar, S.K., 2008. On the Simes inequality and its generalization. IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen 1, 231–242. Simes, R.J., 1986. An improved Bonferroni procedure for multiple tests of significance. Biometrika 73, 751–754. Tong, Y.L., 1990. The Multivariate Normal Distribution. Springer-Verlag, New York.
Contents lists available at SciVerse ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
The multivariate-t distribution and the Simes inequality Henry W. Block a,∗ , Thomas H. Savits a , Jie Wang c , Sanat K. Sarkar b a
University of Pittsburgh, United States
b
Temple University, United States
c
Grand Valley State University, United States
article
info
Article history: Received 2 January 2012 Received in revised form 16 August 2012 Accepted 16 August 2012 Available online 23 August 2012 Keywords: Multivariate-t distribution Simes’ inequality Positive dependence MTP2
abstract Sarkar (1998) showed that certain positively dependent (MTP2 ) random variables satisfy the Simes inequality. The multivariate-t distribution does not satisfy this (MTP2 ) property, so other means are necessary. A corollary was given in Sarkar (1998) to handle this distribution, but there is an error. In this paper a direct proof is given to show the multivariate-t does satisfy the Simes inequality. © 2012 Published by Elsevier B.V.
1. Introduction This paper is motivated by the need to have a clarification of a result on Simes (1986) inequality in Sarkar (1998). While establishing this inequality for positively dependent multivariate random variables, as originally conjectured in Simes (1986), Sarkar (1998) assumed a specific type of positive dependence, e.g., the dependence that is characterized by the MTP2 property (see Theorem 3.1 of Sarkar (1998)). Having realized that the standard multivariate-t with the associated multivariate normal having non-negative correlation does not have this property, Sarkar (1998) made an attempt to prove the Simes inequality for this distribution under certain sign restrictions through a corollary (Corollary 3.1). Unfortunately the proof of this corollary does not appear to be correct, as we will point out in this paper. Thus, the claim that the multivariate-t distribution satisfies the Simes inequality does not follow from the argument of Remark 3.2 in Sarkar (1998). A careful study of Sarkar’s (1998) proof of the Simes inequality would reveal, and also it follows from the proofs of the FDR (false discovery rate) control of the method of Benjamini and Hochberg (1995) in the positive dependence case (Benjamini and Yekutieli, 2001; Sarkar, 2002), that the Simes inequality will continue to hold for distributions with a weaker version of dependence condition than the MTP2 condition, such as the positive dependent through stochastic ordering (PDS) condition of Block et al. (1985), which is essentially the same as the positive regression dependence on subset (PRDS) condition considered in Benjamini and Yekutieli (2001) and Sarkar (2002). However, as we will show in this paper, the standard multivariate-t with non-negative correlations does not satisfy the PDS or PRDS condition. Thus, the Simes inequality for the standard multivariate-t with non-negative correlations does not follow from the PDS or PRDS condition either. Nevertheless, the Simes inequality is indeed true for the standard multivariate-t with non-negative correlations under certain sign restrictions (Benjamini and Yekutieli, 2001; Sarkar, 2002, 2008). In this paper we provide a direct proof, one based on the paper of Sarkar (2002). Sarkar (2008) proves a general result, similar to the one in this paper, but under a stronger dependence condition.
∗
Corresponding author. Tel.: +1 4126248369; fax: +1 4126488814. E-mail address: [email protected] (H.W. Block).
0167-7152/$ – see front matter © 2012 Published by Elsevier B.V. doi:10.1016/j.spl.2012.08.013
228
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
The paper is organized as follows. In the next section, we make some comments on the results of Sarkar (1998) towards proving the Simes inequality for the multivariate-t distribution in Section 3. 2. Comments on Sarkar (1998) In this section, we make some improvements to Sarkar (1998) before giving proofs of the Simes inequality in the next section for the multivariate-t distribution. First, we present a more streamlined proof of his Lemma 2.1 using arguments similar to those used in more recent papers related to FDR methodologies; see also Sarkar (2008). Second, the Simes inequality is presented under a weaker form of positive dependence more suitable for multivariate t distributions. In addition, we give an example showing that the proof of Corollary 3.1 of Sarkar (1998) is not correct, and so his argument showing that the multivariate-t satisfies the Simes inequality does not apply. Lemma 2.1. Let X(1) ≤ · · · ≤ X(n) be the ordered components of a random vector X = (X1 , . . . , Xn )′ and a1 ≤ · · · ≤ an be real numbers. Then, P X(1) > a1 , . . . , X(n) > an
(−i)
where for each i = 1, . . . , n, X(1)
n n−1 I (Xi ≤ aj+1 ) I (Xi ≤ aj ) E − n i=1 j+1 j i=1 j=1 i) i) × I X((− > aj+1 , . . . , X((− , j) n−1) > an
= 1−
n 1
Fi (an ) +
(1)
i) ≤ · · · ≤ X((− n−1) denote the ordered components of the (n − 1)-dimensional random vector
X(−i) obtained by eliminating Xi from X.
Proof. Let Rn = max r | X(r ) ≤ ar , so that
{Rn = r } = X(r ) ≤ ar , X(r +1) > ar +1 , . . . , X(n) > an ,
for r = 1, . . . , n,
and
{Rn = 0} = X(1) > a1 , . . . , X(n) > an . Then I {Rn > 0} =
n r r =1
=
I (Rn = r ) =
r
r =1
n n 1
r
i=1 r =1 n
=
r i =1
I (Xi ≤ ar ) I (Rn = r )
I (Xi ≤ ar ) I X(r ) ≤ ar , X(r +1) > ar +1 , . . . , X(n) > an
n
1
(−i)
I (Xi ≤ ar ) I Rn−1 = r − 1 r
i=1 r =1
i) = max r | X((− r −1) ≤ ar , defined analogously to Rn , i.e., (−i) (−i) (−i) (−i) Rn−1 = r − 1 = X(r −1) ≤ ar , X(r ) > ar +1 , . . . , X(n−1) > an
where
(−i) Rn−1
n n 1
for r = 2, . . . , n,
and
(−i)
(−i)
(−i)
Rn−1 = 0 = X(1) > a2 , . . . , X(n−1) > an . (−i)
(−i)
(−i)
(−i)
(−i)
Since I {Rn−1 = r − 1} = I {Rn−1 ≤ r − 1} − I {Rn−1 ≤ r − 2}, for r = 1, . . . , n, with I (Rn−1 < 0) = 0 and I {Rn−1 ≤ n − 1} = 1, it follows that I {Rn > 0} =
n n−1
(−i)
I Rn−1 ≤ r − 1
I (X ≤ a ) i r r
i=1 r =1
−
n n
(−i)
n 1
n i=1
I ( X i ≤ an ) +
i=1
n
I (X ≤ a ) i r
I Rn−1 ≤ r − 2
r
i=1 r =2
=
n I (Xi ≤ an ) i) + I R(− n −1 ≤ n − 1
n n−1
(−i)
I (X ≤ a ) i r
I Rn−1 ≤ r − 1
i=1 r =1
r
−
I (Xi ≤ ar +1 ) r +1
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
=
n 1
n i =1
I (Xi ≤ an ) +
n n −1
(−i)
(−i)
I X(r ) > ar +1 , . . . , X(n−1) > an ×
i=1 r =1
229
I ( X i ≤ ar ) r
−
I (Xi ≤ ar +1 )
r +1
Using I (X(1) > a1 , . . . , X(n) > an ) = 1 − I (Rn > 0), and taking expectations gives the required result.
Lemma 2.2 (Simes’ Inequality.). Let X be a random vector. Then, (i) we have P X(1) > a1 , . . . , X(n) > an ≥ 1 −
n 1
n i=1
P { X i ≤ an } ,
for any fixed a1 ≤ · · · ≤ an such that j−1 P X1 ≤ aj is nondecreasing in j = 1, . . . , n under the assumption that
(−i)
P X(j)
i) > aj+1 , . . . , X((− n−1) > an | Xi ≤ aj ↑ in aj ,
for every j = 1, . . . , n − 1 and all i = 1, . . . , n; and (ii) we have P X(1) < b1 , . . . , X(n) < bn ≥
n 1
n i=1
P { X i < b1 } ,
for any fixed b1 ≤ · · · ≤ bn such that j−1 P X1 > bn−j+1 is nondecreasing in j = 1, . . . , n under the assumption that
(−i)
(−i)
P X(1) < b1 , . . . , X(j)
< bj | Xi ≥ bi ↓ in bi ,
for every j = 1, . . . , n − 1 and all i = 1, . . . , n. Proof. The first part of the lemma follows from the following inequalities:
I Xi ≤ aj+1 I Xi ≤ aj − E I > > an j+1 j i) i) = P X((− > aj+1 , . . . , X((− (j + 1) j) n−1) > an , Xi ≤ aj+1 i) i) −P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj P {Xi ≤ aj }/j i) (−i) ≥ P X((− > a , . . . , X > a , X ≤ a (j + 1) j + 1 n i j + 1 j) (n−1) i) i) − P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj+1 P {Xi ≤ aj }/j i) (−i) ≥ P X((− > a , . . . , X > a , X ≤ a (j + 1) j +1 n i j +1 j) (n−1) i) i) − P X((− > aj+1 , . . . , X((− j) n−1) > an |Xi ≤ aj+1 P {Xi ≤ aj+1 }/(j + 1)
(−i) X(j)
(−i) aj+1 , . . . , X(n−1)
= 0, for any fixed pair of i and j. Then apply Lemma 2.1. The second part follows from the first part. Example 2.1. The following example shows that the proof of Corollary 3.1 of Sarkar (1998) is not correct. Let g (x, y) = 1/y exp(−x/y) for all x and y > 0. Also let q(z ) = λ exp(−λz ) for all z > 0 where λ > 0. It is easy to show that g ∗ (x, y) =
∞
zg (zx, y)q(z ) dz = 0
λy (λy + x)2
.
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which is clearly TP2 in x and y. For n = 2, a1 = 1, a2 = 3 and λ = 1, after some calculations it can be shown that P {T1 ≤ 1} −
1 2
3−y
P {T2 ≤ 3} =
2(y + 1)(y + 3)
h(y) dy,
and P {T(1) ≥ 1, T(2) ≥ 3} −
P {X(∗1)
≥
1, X(∗2)
y(y2 − 3y − 24)
≥ 3} = −3
(y + 1)(y + 3)2 (y + 4)(y + 6)
h(y) dy,
where h is as in Corollary 3.1 of Sarkar. The integrand in the first expression is negative for any probability density h(y) whose support is contained in (3, ∞) and so satisfies the condition
P {T1 ≤aj }
√
j
nondecreasing in j = 1, 2. Moreover, since the
roots of y − 3y − 24 = 0 are = −3.62348, 6.62348, the integrand in the second expression is negative if the support of h(y) is contained in (6.62348, ∞). Thus (3.8) of Sarkar (1998) does not hold for this h. However, we have not been able to show whether the statement of the corollary is correct or not. 3± 105 2
2
3. Multivariate-t distribution In this section we show that the standard multivariate-t, with the associated normal having nonnegative correlations, satisfies the Simes inequality under certain sign restrictions. The standard multivariate-t distribution is given in Tong (1990, Chapter 9) as follows. Let R = (ρij ) be an n × n symmetric matrix which is either positive definite or positive semidefinite with ρii = 1 for i = 1, 2, . . . , n. In the following we will always assume positive definiteness. Let Z = (Z1 , . . . , Zn )′ have a multivariate normal distribution with mean vector 0 and correlation matrix R and let the univariate random variable S be such that (i) S is independent of Z and (ii) ν S 2 has a chi-square distribution with ν degrees of freedom. Then the multivariate T is defined by T = (T1 , . . . , Tn ) = ′
Z1 S
,...,
Zn
′
S
.
This is called ‘The Multivariate-t with Common Denominator’ by Johnson and Kotz (1972) (see various other multivariatet distributions defined there) and is also the basic multivariate-t given by (1.1) of Kotz and Nagarajah (2004). In Sarkar (1998) this is called the multivariate-t of Dunnett and Sobel (1954) type, but Johnson and Kotz (1972) seem to also attribute this distribution to Cornish and Laurent (see (Johnson and Kotz, 1972). It is not hard to show that for ρij ≥ 0, i ̸= j, the multivariate-t has several positive dependence properties. See Section 9.2.1 of Tong (1990), where various lower and upper orthant inequalities are derived and it is also shown that the distribution is associated. However it does not possess higher order dependence properties such as MTP2 , as shown in Sampson (1983), and also is not positive dependent through stochastic ordering (Block et al., 1985), which we will show. This latter property is essentially the same as PRDS of Benjamini and Yekutieli (2001). We now prove the Simes inequality for the multivariate-t distribution under certain sign restrictions (i.e., the ai are negative and the bj are positive) by establishing the following theorem. Theorem 3.1. Let T be the multivariate-t distribution as given above. Then, (i) for fixed a1 ≤ · · · ≤ an < 0, we have P T(1) > a1 , . . . , T(n) > an ≥ 1 −
n 1
n i =1
P {Xi ≤ an } ,
(2)
if j−1 P T1 ≤ aj is nondecreasing in j = 1, . . . , n; and (ii) for fixed 0 < b1 ≤ · · · ≤ bn , we have
P T(1) ≤ b1 , . . . , T(n) ≤ bn ≥
n 1
n i=1
P { T i ≤ b1 } ,
if j−1 P T1 > bn−j+1 is nondecreasing in j = 1, . . . , n.
Proof. Part (i) of the theorem will follow from Lemma 2.2 if we can show that
(−i)
P T(j)
i) > aj+1 , . . . , T((− n−1) > an | Ti ≤ aj ↑ in aj < 0
for every j = 1, . . . , n − 1 and all i = 1, . . . , n. Since Ti = Zi /S with Z independent of S, we can write
(−i)
P T(j)
i) > aj+1 , . . . , T((− n−1) > an ; Ti ≤ aj = E [ψ(Zi , S ); Ti ≤ aj ]
(3)
H.W. Block et al. / Statistics and Probability Letters 83 (2013) 227–232
231
by conditioning on (Zi , S ), where i) i) ψ(zi , s) = P {Z((− > saj+1 , . . . , Z((− j) n−1) > san |Zi = zi }
since S is independent of Zi . Since Z is PDS (see Block et al. (1985)), ψ(zi , s) ↑ zi for each s; also ψ(zi , s) ↑ s for each zi because the aj ’s are negative. Now set Si∗ = (Zi2 + ν S 2 )1/2 and Ui = Zi /Si∗ . Then Ui and Si∗ are independent. Noting that Ti ≤ aj is equivalent to Ui ≤ aj /(ν + aj 2 )1/2 , we write
i) > aj+1 , . . . , T((− n−1) > an | Ti ≤ aj 1/2 ∗ ∗ 1 − Ui Si 2 1/2 = E ψ Ui Si , ; Ui ≤ aj /(ν + aj ) P (Ui ≤ aj /(ν + a2j )1/2 ) √ ν
(−i)
P T(j)
= E [φ(Ui ); Ui ≤ aj /(ν + a2j )1/2 ]/P (Ui ≤ aj /(ν + a2j )1/2 ). Here
2 1/2 ∗ Si ∗ ( 1 − ui ) , φ(ui ) = E ψ ui Si , √ ν which is nondecreasing for ui ∈ (−1, 0). Set
ˆ ui ) = φ(ui ) φ( φ(0)
for − 1 < ui < 0 for 0 ≤ ui < 1.
ˆ ui ) is nondecreasing on (−1, 1) and Then φ( E φ(Ui ); Ui ≤
aj
(ν + a2j )1/2
ˆ Ui ); Ui ≤ = E φ(
aj
(ν + a2j )1/2
.
Using the following Lemma 3.1, we conclude that the right hand side of the above equation is nondecreasing in aj /(ν + a2j )1/2 ∈ (−1, 0). Since aj /(ν + a2j )1/2 is nodecreasing in aj , it follows that the left hand side of the equation is nondecreasing in aj . This establishes part (i) of the Theorem. Part (ii) of the Theorem follows from part (i). Lemma 3.1. Let X be a continuous random variable with support on an interval (c , d) and g (x) a continuous nondecreasing function there. Then E {g (X )|X ≤ ξ } is nondecreasing for ξ ϵ(c , d). The above proof of Theorem 3.1 is a slight variation of that based on an idea briefly outlined in Sarkar (2002) and elaborated in Sarkar (2008). Example. This example shows that the bivariate-t distribution for ρ ≥ 0 does not satisfy the positive dependence property of stochastic increasing. Let (T1 , T2 ) have a bivariate-t distribution with ρ = 0. Then it can be shown that d dt2
P {T1 > −1|T2 = t2 } =
−2t2 2 + t22 π (3 + t22 )2
.
This is negative for t2 > 0 and hence P {T1 > −1|T2 = t2 } is not increasing there; i.e., (T1 , T2 ) is not PDS. Acknowledgments H.W. Block and T.H. Savits were supported by NSA Grant H 98230-07-1-0018. S.K. Sarkar was supported by NSF Grant DMS-0603868. References Benjamini, Y., Hochberg, Y., 1995. Controlling the false discovery rate: A practical and powerful approach to multiple testing. Journal of Royal Statistical Society Series B 57, 289–300. Benjamini, Y., Yekutieli, D., 2001. The control of the false discovery rate in multiple testing under dependency. Annals of Statistics 29, 1165–1188. Block, H.W., Savits, T.H., Shaked, M., 1985. A concept of negative dependence using stochastic ordering. Statistics and Probability Letters 3, 81–86. Dunnett, C.W., Sobel, M., 1954. A bivariate generalization of Student’s t distribution with tables for certain special cases. Biometrika 41, 153–169. Johnson, N.L., Kotz, S., 1972. Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York. Kotz, S., Nagarajah, S., 2004. Multivariate t Distributions and Their Applications. Cambridge University Press, Cambridge. Sampson, A.R., 1983. Positive dependence properties of elliptically symmetric distributions. Journal of Multivariate Analysis 13, 375–381.
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Sarkar, S.K., 1998. Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Annals of Statistics 26, 494–504. Sarkar, S.K., 2002. Some results on false discovery rate in stepwise multiple testing procedures. Annals of Statistics 30, 239–257. Sarkar, S.K., 2008. On the Simes inequality and its generalization. IMS Collections Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen 1, 231–242. Simes, R.J., 1986. An improved Bonferroni procedure for multiple tests of significance. Biometrika 73, 751–754. Tong, Y.L., 1990. The Multivariate Normal Distribution. Springer-Verlag, New York.