in PROBABILITY
TIGHTNESS OF THE STUDENT T-STATISTIC
PHILIP S. GRIFFIN
Department of Mathematics, Syracuse University, Syracuse, NY 13244-1150, U.S.A.
email: [email protected]
submitted March 19, 2002Final version accepted September 27, 2002 AMS 2000 Subject classification: 60F05
Tightness, t-statistic, self-normalized sum
Abstract
LetX, X1, X2, . . . be a sequence of nondegenerate, independent and identically distributed ran-dom variables and set Sn =X1+· · ·+Xn, Vn2 =X12+· · ·+Xn2. We answer a question of
G¨otze, Gin´e and Mason by providing a simple necessary and sufficient condition for tightness of Sn/Vn.
1
Introduction
Let X, X1, X2, . . . be a sequence of nondegenerate, independent and identically distributed random variables and set
Sn=X1+· · ·+Xn, Vn2=X12+· · ·+Xn2 (1.1)
and
tn= q Sn
n n−1
Pn
1(Xi−X)2
, (1.2)
where X = n−1Sn. Then tn is the classical Student t-statistic which may be expressed equivalently as
tn= Sn
Vn
s
n−1
n−(Sn
Vn)
2. (1.3)
In a beautiful paper G¨otze, Gin´e and Mason (1997) solved a long standing conjecture of Logan, Mallows, Rice and Shepp (1973), by proving thattn, or equivalently the self-normalized sum
Sn/Vn (see Proposition 1), is asymptotically standard normal if and only ifXis in the domain of attraction of the normal law andEX = 0. A key step in their proof was to show that if
Sn
Vn is tight (1.4)
then it is uniformly subgaussian in the sense that
sup
n Eexp
µ
tSn Vn
¶
≤2 exp(ct2) (1.5)
for allt∈Rand somec >0. This is clearly an important property of the self-normalized sum
Sn/Vn which is not shared by scalar normalized sums,i.e. sums of the form (Sn−an)b−1
n for
scalar sequences an andbn. Thus Gin´e, G¨otze and Mason asked for precise conditions under which (1.4) holds. In a subsequent paper, Gin´e and Mason (1998) gave such a characterization for distributions which are in the Feller class. Here we will solve the problem in general. To describe the result we need to introduce a little notation. Forr >0 set
G(r) =P(|X|> r), K(r) =r−2E(X2;|X| ≤r), M(r) =r−1E(X;|X| ≤r), (1.6)
and
Q(r) =G(r) +K(r). (1.7)
Each of these functions is right continuous with left limits and tends to 0 as r approaches infinity. We can now give an analytic characterization of the two classes of random variables mentioned above. X is in the domain of attraction of the normal law andEX = 0 if and only if
lim sup
r→∞
G(r) +|M(r)|
K(r) = 0, (1.8)
whileX is in the Feller class if and only if
lim sup
r→∞
G(r)
K(r) <∞. (1.9)
The result of Gin´e and Mason is that ifX is in the Feller class, then (1.4) holds if and only if
lim sup
r→∞
|M(r)|
K(r) <∞. (1.10)
The main result of this paper is
Theorem 1 The following are equivalent:
tn is tight, (1.11)
Sn
Vn is tight, (1.12)
lim sup
r→∞
|M(r)|
Examples of distributions satisfying (1.13) but not (1.9) and (1.10) are easily found, for ex-ample any symmetric distribution for which the tail functionGis slowly varying.
In the course of the proof of Theorem 1 we also answer the question of when does there exist a centering sequenceαn for which
Sn−αn
Vn is tight. (1.14)
In the case of scalar normalization this reduces to centering at the median ofSnsince for any scalar sequencebn, if (Sn−αn)b−1
n is tight for someαn, then it is tight withαn=median(Sn).
For self-normalization this is not the case. We illustrate this by giving an example for which (1.12) holds but (1.14) fails whenαn=median(Sn).
In concluding the introduction we would like to mention that there have been several other interesting lines of investigation into the Student t-statistic. These include large deviation results (Shao (1997)), law of the iterated logarithm results (Griffin and Kuelbs (1991), Gin´e and Mason (1998)) and Berry-Esseen bounds (Bentkus and G¨otze (1994)). In addition Chistyakov and G¨otze (2001) have recently confirmed a second conjecture of Logan, Mallows, Rice and Shepp that the Student t-statistic has a non-trivial limiting distribution if and only ifX is in the domain of attraction of a stable law. This last paper contains further references to the literature on self-normalized sums.
2
Preliminaries
We begin by showing that for tightness, and indeed for many asymptotic properties, the behavior oftn and Sn/Vn are equivalent. In order that Sn/Vn always make sense, we define
Sn/Vn = 0 ifVn = 0.
Proof. IfEX2 < ∞and EX 6= 0 then (2.1) follows immediately from (1.3) and the strong law. To prove (2.2) it suffices to show
lim
Hence
Thus again by the strong law
lim sup
The functions defined in (1.6) and (1.7) are defined forr >0. It will be convenient to extend them to r= 0 by continuity. Thus set
G(0) =P(|X|>0), K(0) =M(0) = 0. (2.7)
We will be particularly interested in the functionQof (1.7) which is in fact continuous. This is most easily seen by observing that
Q(r) =r−2E(X2∧r2) =r−2Z
r
0
2sG(s)ds. (2.8)
Thus by a reverse Chebyshev inequality, see Durrett (1996) Exercise 3.8 on page 16, for any
Proof. This follows immediately from Lemma 1 sinceV2
n ≥Un(λ) forn >(λQ(0))−1. ✷
Proof. SinceV2
Proof. Observe that by the Cauchy-Schwartz inequality
Hence by Chebyshev’s inequality, (2.15) and (2.19)
P(|Sn−αn(λ)|> LVn)≤na
Theorem 2 Fix a centering sequenceαn. Then the following are equivalent:
On the other hand by Chebyshev’s inequality
P(|Tn(λ)−αn(λ)|> L2an(λ))≤ na2n(λ)K(an(λ))
That (3.2) implies (3.3) is trivial. Finally assume (3.3) holds. Fixλ >0 sufficiently small that
c(λ) =: sup
Hence by Corollary 2, (3.7) and (3.8), for allλsufficiently small
Theorem 3 The following are equivalent:
Sn
Vn is tight, (3.9)
lim sup
r→∞
|M(r)|
Q(r) <∞. (3.10)
Proof. Assume (3.10). Then with αn = 0 we have forn >(λQ(0))−1
|αn−nan(λ)M(an(λ))
an(λ) |= 1
λ
|M(an(λ))|
Q(an(λ)) ,
so (3.9) holds by Theorem 2. Conversely assume (3.10) fails, so
|M(rk)|
Q(rk) → ∞ (3.11)
for some rk → ∞. Set
nk= max{n:nQ(rk)≤1}. (3.12)
Then fornk >(Q(0))−1 we have that
rk=ank(λk) (3.13)
where 1≤λk≤1 +n−k1. Now for suchk
|nkank(λk)M(ank(λk))−nkank(1)M(ank(1))| ≤ank(λk)nkG(ank(1))
≤ank(λk),
while by (3.11)
nk|M(ank(λk))|=
|M(ank(λk))|
λkQ(ank(λk)) =
|M(rk)|
λkQ(rk) → ∞. Consequently
nkank(1)|M(ank(1))|
ank(λk)
→ ∞.
Sinceank(1)≤ank(λk) it then follows thatnk|M(ank(1))| → ∞. Thus we conclude that (3.2) fails with αn = 0 andλ= 1.Hence by Theorem 2, (3.9) fails. ✷
Theorem 1 follows immediately from Proposition 1 and Theorem 3. We conclude by giving an example showing that it is possible for (1.12) to hold, but for (1.14) to fail with αn =
median(Sn).
Example 1 Let X >0have distribution given by
G(r) = 1
Since Gis slowly varying, it follows from Darling (1952) that
→1 and in particular (1.12) holds. Set
bn(λ) =eλn. (3.16)
Hence the medianmn of Sn satisfies lim sup
by (3.15) and (3.17). Thus by (3.17), (3.19) and (3.20), mn/Vn is not tight. Since, as we have already observed, (1.12)holds, it then follows that (1.14) must fail when αn=mn.
References
Bentkus, V., and G¨otze, F., (1994). The Berry-Esseen bound for student’s statistic. Ann. Probab.24,491–503.
Chistyakov, G.P., and G¨otze, F., (2001). Limit distibutions of studentized means. (Preprint)
Darling, D.A., (1952). The influence of the maximum term in the addition of independent random variables.Trans. Amer. Math. Soc.73, 95–107.
Durrett, R., (1996). Probability:Theory and Examples,Second Edition,(Duxbury Press). Gin´e, E., G¨otze, F., and Mason, D.M., (1997). When is the student t-statistic asymptotically
normal? Ann. Probab.25, 1514–1531.
Griffin, P.S., and Kuelbs, J., (1991). Some extensions of the LIL via self-normalizations.Ann. Probab. 19, 380–395.
Logan, B.F., Mallows, C.L., Rice, S.O., and Shepp, L.A., (1973). Limit distributions of self-normalized sums.Ann. Probab.1, 788–809.
