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Asymptotic approximation for the probability density function of an arbitrary sequence of random variables
Statistics and Probability Letters 90 (2014) 100–107
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Asymptotic approximation for the probability density function of an arbitrary sequence of random variables Cyrille Joutard ∗ Université Montpellier 3 and Institut de Mathématiques et de Modélisation de Montpellier (I3M), Route de Mende, 34199 Montpellier, France
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Article history: Received 30 September 2013 Received in revised form 20 March 2014 Accepted 20 March 2014 Available online 26 March 2014
abstract We establish a large deviation approximation for the density function of an arbitrary sequence of random variables. The results are analogous to those obtained by Chaganty and Sethuraman (1985). We apply our theorems to the sample variance and the Mann–Whitney two-sample statistic. © 2014 Elsevier B.V. All rights reserved.
Keywords: Density function Large deviations Sample variance Mann–Whitney statistic
1. Introduction In this paper, we obtain large deviation results analogous to Chaganty and Sethuraman (1985) who derived large deviation local limit theorems for the probability density function of an arbitrary sequence of random variables n−1 Tn . More precisely, we generalize their results by establishing asymptotic approximations of large deviation type for the probability 1 density function (or the probability mass function) of an arbitrary sequence of random variables Zn = b− n Tn , where the rate bn is a sequence of real positive numbers going to infinity as n → ∞. These approximations are obtained, in particular, by assuming an asymptotic expansion for the normalized cumulant generating function (c.g.f.) of bn Zn . As in Chaganty and Sethuraman (1985), we distinguish the case where Zn is lattice-valued from the case where Zn is absolutely continuous. Unlike the results of Chaganty and Sethuraman (1985), the asymptotic expansion of the normalized c.g.f. allows us to obtain, in the absolutely continuous case, an explicit asymptotic expression for the density function of Zn that depends on n only through bn . In particular, the real τa , used to make an exponential change of measure (see Section 2), does not depend on n. Our results can also be useful when the (normalized) c.g.f. of bn Zn is intractable and we have to use its asymptotic function instead to make the exponential change of measure (see, e.g., Bercu et al., 2000). The proofs of the results follow the same lines as in Chaganty and Sethuraman (1985). The main difference lies in the use of an approximation for the c.g.f. 1 of bn Zn , so that basically ϕn is replaced with its approximation ϕ + b− n H (see Section 2). Once again, this allows us to deal with the case of intractable c.g.f. We illustrate our theorems with two statistical applications, namely the sample variance for the absolutely continuous case and the Mann–Whitney two-sample statistic for the lattice case. Note that the techniques used in the proofs of the corollaries of Section 3 are different from the ones of Chaganty and Sethuraman (1985), in particular when we want to show, in the Mann–Whitney case, that the conditions of Chaganty and Sethuraman (1985, Lemma 2.7) are satisfied. We eventually
∗
Tel.: +33 672356596. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.spl.2014.03.018 0167-7152/© 2014 Elsevier B.V. All rights reserved.
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
101
carry out some numerical comparisons for the Mann–Whitney statistic Mm,n (see Mann and Whitney, 1947 and Wilcoxon, 1945). More precisely, we compare the (exact) probabilities P(Mm,n = l) with the large deviation approximation and a saddle point approximation derived by Jin and Robinson (2003). The results of Chaganty and Sethuraman (1985) can be seen as an extension of Richter (1957) who obtained an asymptotic approximation for the probability density function of the sample mean. Around the same time, Blackwell and Hodges (1959) established large deviation approximations for the probability mass function of the sample mean, for lattice-valued random variables. In the particular case of the Mann–Whitney statistic, Jin and Robinson (1999, 2003) obtained saddle point approximations for the probability mass function of this statistic. Regarding the approximations for the tail probabilities of the Mann–Whitney statistic, one can refer to the works of Bean et al. (2004), Buckle et al. (1969), Jin and Robinson (2003) and Joutard (2013c). The paper is organized as follows. In Section 2, we introduce the assumptions before presenting the main results. Section 3 deals with the two statistical applications and Section 4 is devoted to the numerical comparisons. 2. Theorems In this section, we give asymptotic approximations of large deviation type for the density function (or the mass function) of an arbitrary sequence of random variables. The context is the following. Let (Zn ) be a sequence of real random variables and let (bn ) be a sequence of real positive numbers such that limn→∞ bn = ∞. Let φn be the moment generating function (m.g.f.) 1 of bn Zn , that is, φn (t ) = E{exp(tbn Zn )}, and let ϕn be the normalized c.g.f. of bn Zn , namely ϕn (t ) = b− n log E{exp(tbn Zn )}. Assume that there exists a differentiable function ϕ defined on an interval (−α, α), α > 0, such that limn→∞ ϕn (t ) = ϕ(t ) for all t ∈ (−α, α). Let a be real such that there exists τa ∈ (−α, α) satisfying ϕ ′ (τa ) = a. Furthermore, we assume the following assumptions: (A.1) The c.g.f. ϕn is analytic in DC := {z ∈ C : |z | < α}, and there exists M > 0 such that |ϕn (z )| < M for all z ∈ DC and n large enough. (A.2) There exists H, a continuous function at τa , such that for n large enough,
−2 1 ϕn (τa ) = ϕ(τa ) + b− , n H (τa ) + O bn
(1)
1 ϕn′ (τa ) = ϕ ′ (τa ) + O(b− n ),
(2)
ϕn (τa ) = ϕ (τa ) + O(bn ), ′′
′′
−1
(3)
where the function ϕ is twice differentiable at τa , and ϕ (τa ) > 0. (A.3) There exist γ , δ1 > 0 such that for 0 < δ < δ1 , ′′
1 φn (τa ) inf b− n log |t |≥δ φn (τa + it )
≥ Ln (δ),
(4) γ
where Ln is a function satisfying bn Ln (b−λ n ) = O(bn ) for some 1/3 < λ < 1/2. (A.4) There exist p, l > 0 such that
φn (τa + it ) l/bn p φ (τ ) dt = O(bn ). n a −∞ ∞
Below is the theorem dealing with absolutely continuous random variables. Theorem 1. Let assumptions (A.1)–(A.4) hold. Then for n large enough, we have the following asymptotic approximation for the probability density function kn of Zn , kn (a) =
bn
1/2
2π ϕ ′′ (τa )
1 exp(−bn ϕ ∗ (a) + H (τa )) 1 + O(b− n )
(5)
where τa is such that ϕ ′ (τa ) = a, and ϕ ∗ (a) = τa a − ϕ(τa ). Proof. The proof follows the same lines as in Chaganty and Sethuraman (1985). First, set Gn (t ) = −ϕn (τa + it ) + ϕn (τa ) + ita.
(6)
Then, by (2) and noting that ϕ (τa ) = a, one can find η0 > 0 small enough such that for |t | < η0 , ′
Gn (t ) =
t 2 ′′ it 3 (3) 1 ϕn (τa ) + ϕn (τa ) − Rn (τa + it ) − itO(b− n ), 2 3!
(7)
and by Assumption (A.1), there exists a constant M0 > 0 such that the remainder term Rn satisfies
|Rn (τa + it )| ≤ M0 t 4 .
(8)
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C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
Using an exponential change of measure, let us define dHn (y) = exp(τa y − bn ϕn (τa ))dKn (y), where Kn is the distribution function of bn Zn . By the inversion formula, the density function kn of Zn is then given by kn (y) =
∞
bn
φn (τa + it )e−bn (τa +it )y dt .
2π
−∞
bn
Next, by (3), kn (a) =
1/2
2π ϕ ′′ (τa )
1 exp(−bn ϕ ∗ (a) + H (τa ))In [1 + O(b− n )]
where ϕ ∗ (a) = sups∈(−α,α) {sa − ϕ(s)} = τa a − ϕ(τa ) and
In =
bn ϕn′′ (τa )
1/2
2π
=
e−H (τa )
∞
exp(bn [ϕ ∗ (a) − (τa + it )a])φn (τa + it )dt −∞
bn ϕn′′ (τa )
1/2
−H (τa )
∞
exp(bn [ϕn (τa + it ) − ϕn (τa ) − ita
e
2π
−∞
1 −2 −1 + b− n H (τa ) + O(bn )])dt = (In1 + In2 )[1 + O(bn )] ′′ 1/2 ′′ 1/2 b ϕ (τ ) b ϕ (τ ) e−bn Gn (t ) dt , In2 = n 2nπ a e−bn Gn (t ) dt . Note that we have used (1) and λ is chosen with In1 = n 2nπ a |t |≥b−λ |t |
goes exponentially fast to zero. Now, as for the second term In2 , by (7) we have
In2 =
bn ϕn′′ (τa )
1/2
2π
1/2−λ
|s|
is
bn
s
+ √ O(1) + bn Rn τa + i √
s2 ′′ ϕn (τa ) 2
is3 1 − √ ϕn(3) (τa ) 6 bn
exp −
bn
+ Ln (s) ds
(9)
is where Ln (s) = ezn − 1 − zn , with zn = − 6√ ϕn(3) (τa ) + √isb O(1) + bn Rn (τa + i √sb ). On the right-hand side of (9), the first b 3
n
1/2−λ
n
√
n
2 integral is 1 + O(b− ϕ ′′ (τa )) (Φ is the standard normal distribution) goes exponentially fast to n ) since the term Φ (−bn zero. The second and third integrals are zero since the integrands are odd functions. As in Chaganty and Sethuraman (1985), 1 −λ √is using (8), one can easily see that the fourth integral is O(b− n ). Finally, noting that | b O(1)| ≤ O(bn ) → 0, as n → ∞, for n
1/2−λ
| s| ≤ bn , we can show that |zn | < 1/2 and prove as in Chaganty and Sethuraman (1985) that the fifth integral is also 1 −1 O(b− n ). Therefore, In2 = 1 + O(bn ). This completes the proof of Theorem 1. Now we assume that Zn is lattice-valued. We recall that a random variable is said to be lattice-valued if its range lies in a subset of the lattice set {d0 + ks, k ∈ Z}, where the real d0 is called the displacement and the positive real s is the span. Denote by dn and sn the displacement and the span of bn Zn . We also assume that the real an is such that bn an belongs to the range of bn Zn , and there exists τan ∈ (−α1 , α1 ), with 0 < α1 < α , such that ϕ ′ (τan ) = an . The versions of assumptions (A.2)–(A.4) for lattice-valued random variables are given below. (A′ .2) There exists H, a continuous function in (−α1 , α1 ), such that for n large enough, (1)–(3) hold (with a replaced by an ) wherein the function ϕ is twice differentiable in (−α1 , α1 ) and there exists β > 0 such that ϕ ′′ (τan ) ≥ β . (A′ .3) There exist γ , δ1 > 0 such that for 0 < δ < δ1 , inf
−1
δ≤|t |≤π/sn
bn
φn (τan ) ≥ Ln (δ), log φn (τan + it ) γ
where Ln is a function satisfying bn Ln (b−λ n ) = O(bn ) for some 1/3 < λ < 1/2.
(A .4) There exist p, l > 0 such that ′
sup
τ ∈(−α,α)
φn (τ + it ) l/bn p φ (τ ) dt = O(bn ). n −π/sn π/sn
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
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The theorem dealing with lattice-valued random variables is as follows. Theorem 2. Let assumptions (A.1) and (A′ .2)–(A′ .4) hold. Then for a real an such that bn an belongs to the range of the latticevalued random variable bn Zn and for n large enough,
P(Zn = an ) = |sn |
1/2
1 2π bn ϕ ′′ (τan )
1 exp(−bn ϕ ∗ (an ) + H (τan )) 1 + O(b− n )
(10)
where τan is such that ϕ ′ (τan ) = an , and ϕ ∗ (an ) = τan an − ϕ(τan ). Proof. The proof is similar to that of Theorem 1 and we shall not consider it here.
3. Statistical applications In this section, we give an application of Theorem 1 to illustrate the case of absolutely continuous random variables and an application of Theorem 2 to illustrate the case of lattice-valued random variables. 3.1. Example 1. The sample variance We consider the sample variance Zn =
n (Xi − X )2 ,
1
n − 1 i=1 where the random variables X1 , X2 , . . . are independent and identically distributed (i.i.d.). Assuming that the Xi ’s have a n normal distribution N (µ; σ 2 ), σ 2 > 0, we know that σ −2 i=1 (Xi − X )2 follows a chi-square distribution with n − 1 degrees of freedom. Large deviation approximations were obtained in Joutard (2013a,b) for the tail probabilities of Zn . Here, by applying Theorem 1 to Zn with bn = n, we obtain a large deviation approximation for the density function of Zn . Corollary 1. Let Zn be defined as above. Then for a real a > 0 and n large enough, the following asymptotic approximation holds for the probability density function kn of Zn , kn (a) =
where ISV (a) =
1 2
n 1
π 2a a
σ2
exp(−(n − 1)ISV (a)) 1 + O(n−1 ) ,
− log
a
σ2
(11)
−1 .
Remark 1. Note that in this particular example assuming a normality assumption, the exact distribution of the statistic is known, which allows to obtain an explicit expression for the c.g.f. of nZn . It would be more interesting to drop the normality assumption and assume, for example, a sub-Gaussian distribution for X1 . Unfortunately in this case, one cannot derive an expression for the c.g.f. ϕn that satisfies (1). This therefore remains an open problem and this example can be considered merely as an illustration of Theorem 1. Proof. By the proof of Joutard (2013a, Corollary 2.2), Assumption (A.1) is satisfied and (1) holds with bn = n, ϕ(t ) =
log(1 − 2σ 2 t ) + 1−σ2σt 2 t . Besides, one can easily verify that (2)–(3) hold with ϕ ′ (t ) = 1−σ2σ 2 t 4 −σ 2 and ϕ ′′ (t ) = (1−22σσ 2 t )2 . From Joutard (2013a), we have τa = a2a such that ϕ ′ (τa ) = a. Hence, ϕ ′′ (τa ) = 2a2 > 0 and σ2 Assumption (A.2) is verified. As for Assumption (A.3), we have for all |t | ≥ δ ,
− 21 log(1 − 2σ 2 t ) and H (t ) =
2
1 2
2
n 2 φn (τa ) 4σ 4 δ 2 n− n−1 1 ≥ n log log 1 + 2 . n 2 φn (τa + it ) 4n 1 − 2σ τa n− 1 n 2 4σ 4 δ 2 ( n− ) n −1 1 Then, (4) holds with Ln (δ) = 4n log 1 + (1−2σ 2 τ n )2 . Therefore, we have nLn (n−λ ) = O(n1−2λ ), with 1 − 2λ > 0, and −1
a n−1
(A.3) is satisfied. Finally, there exists l = 4 such that
n 2 − 4(n4n−1) ∞ φn (τa + it ) l/n 4σ 4 t 2 n− 1 1+ dt , 2 φ (τ ) dt = n n a −∞ −∞ 1 − 2σ 2 τa n− 1 which implies (A.4). Observing that H (τa ) = ϕ ∗ (a) = 21 σa2 − log σa2 − 1 , the asymptotic approximation (11) therefore
∞
follows from the application of Theorem 1.
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C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
Now we present a small numerical example to illustrate the quality of the approximation. Here, we take n = 100, σ 2 = 2 and we plot the exact density of Zn (‘‘solid line’’) and the large deviation approximation (11) (‘‘dashed line’’), for values of a ranging from 1 to 3.
3.2. The Wilcoxon–Mann–Whitney test Let X1 , . . . , Xm and Y1 , . . . , Yn be two independent random samples from continuous distribution functions F and G. The Mann–Whitney statistic can be defined as follows: Mm,n =
m n
I{Yj
i =1 j =1
This statistic is used to test the null hypothesis: F = G. Let N = m + n and assume that there exists p0 ∈ (0, 1) such that m/N = p0 and n/N = 1 − p0 . Let us define Zm,n = Mm,n /N 2 . The statistic NZm,n is lattice-valued with displacement 0 and span sN = 1/N. Jin and Robinson (1999, 2003) established saddle point expansions for the probability mass function of Mm,n . Under the null hypothesis, by applying Theorem 2 with bn = N, we have the following asymptotic approximation for the probability mass function of Zm,n . Corollary 2. Let Zm,n be defined as above. Then for a real aN ∈]b0 , b′0 [, where p0 (1 − p0 )/2 < b0 < b′0 < p0 (1 − p0 ), such that NaN belongs to the range of Mm,n /N, and for N large enough,
P(Zm,n = aN ) =
1/2
1 2π
where τaN > 0, is such that
N3
1 0
ϕMW (τaN ) ′′
hτaN (x)dx − p20
e−N ϕMW (aN )+HMW (τaN ) 1 + O(N −1 ) ∗
1 0
hp0 τaN (x)dx − (1 − p0 )2
1 0
(12)
h(1−p0 )τaN (x)dx = aN , with ht (x) = xetx /(etx − 1).
′′ Furthermore, HMW (t ) = 2−1 [log{(et − 1)/(etp0 − 1)} − log{(et (1−p0 ) − 1)/(p0 (1 − p0 )t )}], ϕMW (t ) =
1 l (x)dx − 0 t ∗ 3 1 p0 0 lp0 t (x)dx − (1 − p0 ) 0 l(1−p0 )t (x)dx and ϕMW (aN ) = τaN aN − ϕMW (τaN ) where ϕMW (t ) = p0 log(p0 ) + (1 − p0 ) log(1 − 1 1 1 p0 ) + 0 gt (x)dx − p0 0 gp0 t (x)dx − (1 − p0 ) 0 g(1−p0 )t (x)dx, with gt (x) = log{(etx − 1)/x} and lt (x) = −x2 etx /(etx − 1)2 . 3 1
Remark 2. The above result assumes that aN > b0 > p0 (1 − p0 )/2 but the same kind of result also holds if aN < b0 < p0 (1 − p0 )/2 (in this case τaN is negative). In the case aN = p0 (1 − p0 )/2 (assuming that NaN belongs to the range of Mm,n /N), we have τaN = 0 and one can use the ′′ ∗ approximation (12) with ϕMW (0) = p0 (1 − p0 )/12 and ϕMW (2−1 p0 (1 − p0 )) = HMW (0) = 0. Proof. First, note that for a real aN > b0 > ϕ ′ (0) = p0 (1 − p0 )/2, there exists τaN > κ0 , with κ0 > 0, such that ϕ ′ (τaN ) = aN . Then, by the proof of Joutard (2013c, Corollary 3.1), Assumption (A.1) is satisfied and (1) holds with bn = N, H (t ) = p0 )
1 0
1 2
log
et −1 e −1 tp0
− log
et (1−p0 ) −1 p0 (1−p0 )t
, ϕ(t ) = p0 log(p0 ) + (1 − p0 ) log(1 − p0 ) +
g(1−p0 )t (x)dx. One can easily verify that (2)–(3) also hold with
ϕ ′ (t ) =
1
ht (x)dx − p20 0
0
1
hp0 t (x)dx − (1 − p0 )2
1
h(1−p0 )t (x)dx 0
1 0
gt (x)dx − p0
1 0
gp0 t (x)dx − (1 −
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
105
and
ϕ ′′ (t ) =
1
lt (x)dx − p30
1
lp0 t (x)dx − (1 − p0 )3
l(1−p0 )t (x)dx 0
0
0
1
where ht (x) = xetx /(etx − 1) and lt (x) = −x2 etx /(etx − 1)2 . Besides, ϕ ′′ (τaN ) ≥ β with β > 0 and (A′ .2) is therefore satisfied. Now regarding Assumption (A′ .3), we will apply Lemma 2.7 of Chaganty and Sethuraman (1985) which implies Condition C ′ of Chaganty and Sethuraman (1985). This condition plays the same role as Assumption (A′ .3) and is mainly used in the proof of Theorem 2 to show that
In1 =
bn ϕn′′ (τaN )
1/2
2π
−λ bn ≤|t |≤π/sn
exp(−bn Gn (t ))dt ,
where the expression of Gn is given in (6), goes exponentially fast to zero. Here, we consider only the particular cases m = n and τaN = 0. If m ̸= n and τaN > 0, the proof is similar but is long and tedious so we shall not consider it here. Let fm,n (t ) = Real(Gm,n (t )). By (6) with τaN = 0, fm,n (t ) = Real(Gm,n (t )) = −N −1 log(|φm,n (it )|) where φm,n is the m.g.f. of Mm,n /N. Assuming that m = n, we have N = 2n and
2n eit (k/2n) − 1 2n −1 k=n+1 fn,n (t ) = −(2n)−1 log n it (k/2n) n e − 1 k=1
+n ) n k sin t (k4n . = −(2n)−1 log sin tk k + n k=1
4n
sin(k tk/4n) Note that the function t → sin(0tk/4n) , where k0 = (k + n)/k > 1, is decreasing on (0, η1 ), η1 = 2π , and increasing on (−η1 , 0). Consequently, fn,n is increasing on (0, η1 ) and decreasing on (−η1 , 0). This verifies the condition (i) of Chaganty and Sethuraman (1985, Lemma 2.7). Now, for the condition (ii), we have
t (k+n) n sin 4 + (2n)−1 log 2n . inf fn,n (t ) = −(2n)−1 sup log sin tk 2π≤|t |≤2nπ n 2π n−1 ≤|t |≤2π k=1 4
(13)
Note that for all 2π n−1 ≤ t ≤ 2π ,
t (k+n) n sin 4 n sin tk ≤ M0 k=1 4
(14)
where M0 < 4. Moreover, we have n
−1
log
2n
n
√ ≈ log(4) − n−1 log( π n)
(15)
as n → ∞. Therefore, combining (13)–(15), for n large enough there exists ϵ > 0 such that inf2π≤|t |≤2nπ fn,n (t ) > ϵ. This implies condition (ii) of the lemma. Finally, it remains to verify condition (iii). For η > 0, we have: sup fn,n (t ) =
|t |≤η
sup −(4n)−1 |t |≤ηn−1
n
log
k=1
sin2
sin2
t (k+n) 4
tk 4
k2
(k + n)2
.
Using Taylor expansions, for η small enough and n large enough, we have for |t | ≤ ηn−1 ,
−n − 1
n k=1
log
sin2
t (k+n) 4
2 tk
sin
4
k2
(k + n)2
2 2 ≈ n t .
24
106
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107 Table 1 Approximations of P(Mm,n = l) for the Mann–Whitney for m = n = 5. l
15
16
17
18
19
Exact
0.07143
0.06349
0.05556
0.04365
0.03571
ALD ASP l
0.07472 0.07095 20
0.06603 0.06353 21
0.05620 0.05472 22
0.04598 0.04523 23
0.03608 0.03577 24
Exact ALD ASP
0.02778 0.02709 0.02696
0.01984 0.01942 0.01925
0.01190 0.01327 0.01293
0.00794 0.00883 0.00807
0.00397 0.00687 0.00463
Table 2 Approximations of P(Mm,n = l) for the Mann–Whitney for m = 10, n = 6. l
50
51
52
53
54
Exact ALD ASP l
0.00437 0.00424 0.00429 55
0.00325 0.00325 0.00330 56
0.00250 0.00245 0.00248 57
0.00175 0.00180 0.00183 58
0.00137 0.00130 0.00131 59
Exact ALD ASP
0.000874 0.000910 0.000911
0.000624 0.000622 0.000614
0.000375 0.000414 0.000398
0.000250 0.000275 0.000246
0.000125 0.000217 0.000145
Hence, there exists η < 2π such that, for n large enough, sup|t |≤η fn,n (t ) < ϵ. The condition (iii) is satisfied and we can apply Lemma 2.7 of Chaganty and Sethuraman (1985) to verify Assumption (A′ .3). Finally, according to Chaganty and Sethuraman (1985, Remark 2.16), Assumption (A′ .4) follows from the fact that 1/|sN | = N and Theorem 2 can then be applied to obtain (12). Remark 3. Note that it should also be possible to apply Theorem 2 to the Jonckheere–Terpstra statistic (see Jonckheere, 1954; Terpstra, 1952). In fact, the Jonckheere–Terpstra test can be seen as an extension of the Wilcoxon–Mann–Whitney test to more than two samples and so the proof is similar to that of Corollary 2 regarding Assumptions (A.1)–(A.2). The main difficulty lies in the verification of Assumption (A′ .3). 4. Numerical results In this section we compare the (exact) probabilities P(Mm,n = l) with the large deviation approximation (12) and a saddle point approximation derived by Jin and Robinson (2003) (these approximations are denoted, respectively, by ALD and ASP in Tables 1 and 2). The comparisons are carried out on two examples that were also studied in Jin and Robinson (2003): m = n = 5 and m = 10, n = 6. The saddle point approximation of P(Mm,n = l) can be expressed as
P(Mm,n = l) =
exp(NK (u) − lu) 1 + N −1 P2 (u) + O(N −3/2 ) , √ ′′ 2π NK (u)
(16)
where K (u) = N −1 log E{exp(uMm,n )}, u is such that K ′ (u) = l/N and P2 (u) =
3K (4) (u) 24(K ′′ (u))2
−
15(K (3) (u))2 72(K ′′ (u))3
.
Now regarding the approximation (12), we have P(Zm,n = aN ) = P(Mm,n = l), where l = aN ∗ N 2 . For efficiency reasons, we use aN = (l + 1/2)/N 2 instead of aN = l/N 2 on the right-hand side of (12) since it turns out to give better results as far as these numerical comparisons are concerned. We see from Tables 1 and 2 that the approximation (12) is good at most values and sometimes it can even be better than the approximation (16) (e.g. l = 20 or 21 in Table 1 and l = 51, 53 or 56 in Table 2) despite its larger relative error of order N −1 . Note, however, that this is possible because of the small values of m and n. For large values of m and n, the approximation (16) will usually give better results than (12) since its relative error of order N −3/2 is smaller. We can also observe that the two approximations (in particular (12)) are not accurate for values near the endpoints of the support (e.g. l = 24 in Table 1 and l = 59 in Table 2). Note that Jin and Robinson (1999) obtained saddle point approximations that can be accurate near the endpoints of the support. In conclusion, these two examples show that the large deviation approximation can give good results and sometimes even perform (slightly) better than the saddle point approximation. But as mentioned above, the latter approximation is
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more accurate for larger values of m and n. However, Jin and Robinson (1999, 2003) mainly deal with Mann–Whitney, while Theorems 1 and 2 concern more general statistics. Furthermore, according to Remark 3, Corollary 2 may also be seen as a first step toward an approximation for the probability mass function of the Jonckheere–Terpstra statistic. Acknowledgment The author would like to thank the anonymous referee for helpful comments which have led to an improved version of this paper. References Bean, R., Froda, S., van Eeden, C., 2004. The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney nulldistribution: a numerical comparison. J. Nonparametr. Stat. 16, 279–288. Bercu, B., Gamboa, F., Lavielle, M., 2000. Sharp large deviations for Gaussian quadratic forms with applications. ESAIM Probab. Stat. 3, 1–29. Blackwell, D., Hodges, J.L., 1959. The probability in the extreme tail of a convolution. Ann. Math. Stat. 30, 1113–1120. Buckle, N., Kraft, C., van Eeden, C., 1969. An approximation to the Wilcoxon–Mann–Whitney distribution. J. Amer. Statist. Assoc. 64, 591–599. Chaganty, N.R., Sethuraman, J., 1985. Large deviation local limit theorems for arbitrary sequences of random variables. Ann. Probab. 13, 97–114. Jin, R., Robinson, J., 2003. Saddlepoint approximations of the two-sample Wilcoxon statistic. Lecture Notes-Monograph Series, pp. 149–158. Jin, R., Robinson, J., 1999. Saddlepoint approximation near the endpoints of the support. Statist. Probab. Lett. 295–303. Jonckheere, A.R., 1954. A distribution-free k-sample test against ordered alternatives. Biometrika 41, 133–145. Joutard, C., 2013c. Large deviation approximations for the Mann–Whitney statistic and the Jonckheere–Terpstra statistic. J. Nonparametr. Stat. 25, 873–888. Joutard, C., 2013a. Strong large deviations for arbitrary sequences of random variables. Ann. Inst. Statist. Math. 1, 49–67. Joutard, C., 2013b. A strong large deviation theorem. Math. Methods Statist. 22, 155–164. Mann, H.B., Whitney, D.R., 1947. On a test of whether one or two random variables is stochastically larger than the other. Ann. Math. Statist. 18, 50–60. Richter, W., 1957. Local limit theorems for large deviations. Theory Probab. Appl. 2, 206–219. Terpstra, T.J., 1952. The asymptotic normality and consistency of Kendall’s test against trend, when ties are present in one ranking. Indag. Math. 14, 327–333. Wilcoxon, F., 1945. Individual comparison by ranking methods. Biometrics 1, 80–83.
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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Asymptotic approximation for the probability density function of an arbitrary sequence of random variables Cyrille Joutard ∗ Université Montpellier 3 and Institut de Mathématiques et de Modélisation de Montpellier (I3M), Route de Mende, 34199 Montpellier, France
article
info
Article history: Received 30 September 2013 Received in revised form 20 March 2014 Accepted 20 March 2014 Available online 26 March 2014
abstract We establish a large deviation approximation for the density function of an arbitrary sequence of random variables. The results are analogous to those obtained by Chaganty and Sethuraman (1985). We apply our theorems to the sample variance and the Mann–Whitney two-sample statistic. © 2014 Elsevier B.V. All rights reserved.
Keywords: Density function Large deviations Sample variance Mann–Whitney statistic
1. Introduction In this paper, we obtain large deviation results analogous to Chaganty and Sethuraman (1985) who derived large deviation local limit theorems for the probability density function of an arbitrary sequence of random variables n−1 Tn . More precisely, we generalize their results by establishing asymptotic approximations of large deviation type for the probability 1 density function (or the probability mass function) of an arbitrary sequence of random variables Zn = b− n Tn , where the rate bn is a sequence of real positive numbers going to infinity as n → ∞. These approximations are obtained, in particular, by assuming an asymptotic expansion for the normalized cumulant generating function (c.g.f.) of bn Zn . As in Chaganty and Sethuraman (1985), we distinguish the case where Zn is lattice-valued from the case where Zn is absolutely continuous. Unlike the results of Chaganty and Sethuraman (1985), the asymptotic expansion of the normalized c.g.f. allows us to obtain, in the absolutely continuous case, an explicit asymptotic expression for the density function of Zn that depends on n only through bn . In particular, the real τa , used to make an exponential change of measure (see Section 2), does not depend on n. Our results can also be useful when the (normalized) c.g.f. of bn Zn is intractable and we have to use its asymptotic function instead to make the exponential change of measure (see, e.g., Bercu et al., 2000). The proofs of the results follow the same lines as in Chaganty and Sethuraman (1985). The main difference lies in the use of an approximation for the c.g.f. 1 of bn Zn , so that basically ϕn is replaced with its approximation ϕ + b− n H (see Section 2). Once again, this allows us to deal with the case of intractable c.g.f. We illustrate our theorems with two statistical applications, namely the sample variance for the absolutely continuous case and the Mann–Whitney two-sample statistic for the lattice case. Note that the techniques used in the proofs of the corollaries of Section 3 are different from the ones of Chaganty and Sethuraman (1985), in particular when we want to show, in the Mann–Whitney case, that the conditions of Chaganty and Sethuraman (1985, Lemma 2.7) are satisfied. We eventually
∗
Tel.: +33 672356596. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.spl.2014.03.018 0167-7152/© 2014 Elsevier B.V. All rights reserved.
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
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carry out some numerical comparisons for the Mann–Whitney statistic Mm,n (see Mann and Whitney, 1947 and Wilcoxon, 1945). More precisely, we compare the (exact) probabilities P(Mm,n = l) with the large deviation approximation and a saddle point approximation derived by Jin and Robinson (2003). The results of Chaganty and Sethuraman (1985) can be seen as an extension of Richter (1957) who obtained an asymptotic approximation for the probability density function of the sample mean. Around the same time, Blackwell and Hodges (1959) established large deviation approximations for the probability mass function of the sample mean, for lattice-valued random variables. In the particular case of the Mann–Whitney statistic, Jin and Robinson (1999, 2003) obtained saddle point approximations for the probability mass function of this statistic. Regarding the approximations for the tail probabilities of the Mann–Whitney statistic, one can refer to the works of Bean et al. (2004), Buckle et al. (1969), Jin and Robinson (2003) and Joutard (2013c). The paper is organized as follows. In Section 2, we introduce the assumptions before presenting the main results. Section 3 deals with the two statistical applications and Section 4 is devoted to the numerical comparisons. 2. Theorems In this section, we give asymptotic approximations of large deviation type for the density function (or the mass function) of an arbitrary sequence of random variables. The context is the following. Let (Zn ) be a sequence of real random variables and let (bn ) be a sequence of real positive numbers such that limn→∞ bn = ∞. Let φn be the moment generating function (m.g.f.) 1 of bn Zn , that is, φn (t ) = E{exp(tbn Zn )}, and let ϕn be the normalized c.g.f. of bn Zn , namely ϕn (t ) = b− n log E{exp(tbn Zn )}. Assume that there exists a differentiable function ϕ defined on an interval (−α, α), α > 0, such that limn→∞ ϕn (t ) = ϕ(t ) for all t ∈ (−α, α). Let a be real such that there exists τa ∈ (−α, α) satisfying ϕ ′ (τa ) = a. Furthermore, we assume the following assumptions: (A.1) The c.g.f. ϕn is analytic in DC := {z ∈ C : |z | < α}, and there exists M > 0 such that |ϕn (z )| < M for all z ∈ DC and n large enough. (A.2) There exists H, a continuous function at τa , such that for n large enough,
−2 1 ϕn (τa ) = ϕ(τa ) + b− , n H (τa ) + O bn
(1)
1 ϕn′ (τa ) = ϕ ′ (τa ) + O(b− n ),
(2)
ϕn (τa ) = ϕ (τa ) + O(bn ), ′′
′′
−1
(3)
where the function ϕ is twice differentiable at τa , and ϕ (τa ) > 0. (A.3) There exist γ , δ1 > 0 such that for 0 < δ < δ1 , ′′
1 φn (τa ) inf b− n log |t |≥δ φn (τa + it )
≥ Ln (δ),
(4) γ
where Ln is a function satisfying bn Ln (b−λ n ) = O(bn ) for some 1/3 < λ < 1/2. (A.4) There exist p, l > 0 such that
φn (τa + it ) l/bn p φ (τ ) dt = O(bn ). n a −∞ ∞
Below is the theorem dealing with absolutely continuous random variables. Theorem 1. Let assumptions (A.1)–(A.4) hold. Then for n large enough, we have the following asymptotic approximation for the probability density function kn of Zn , kn (a) =
bn
1/2
2π ϕ ′′ (τa )
1 exp(−bn ϕ ∗ (a) + H (τa )) 1 + O(b− n )
(5)
where τa is such that ϕ ′ (τa ) = a, and ϕ ∗ (a) = τa a − ϕ(τa ). Proof. The proof follows the same lines as in Chaganty and Sethuraman (1985). First, set Gn (t ) = −ϕn (τa + it ) + ϕn (τa ) + ita.
(6)
Then, by (2) and noting that ϕ (τa ) = a, one can find η0 > 0 small enough such that for |t | < η0 , ′
Gn (t ) =
t 2 ′′ it 3 (3) 1 ϕn (τa ) + ϕn (τa ) − Rn (τa + it ) − itO(b− n ), 2 3!
(7)
and by Assumption (A.1), there exists a constant M0 > 0 such that the remainder term Rn satisfies
|Rn (τa + it )| ≤ M0 t 4 .
(8)
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C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
Using an exponential change of measure, let us define dHn (y) = exp(τa y − bn ϕn (τa ))dKn (y), where Kn is the distribution function of bn Zn . By the inversion formula, the density function kn of Zn is then given by kn (y) =
∞
bn
φn (τa + it )e−bn (τa +it )y dt .
2π
−∞
bn
Next, by (3), kn (a) =
1/2
2π ϕ ′′ (τa )
1 exp(−bn ϕ ∗ (a) + H (τa ))In [1 + O(b− n )]
where ϕ ∗ (a) = sups∈(−α,α) {sa − ϕ(s)} = τa a − ϕ(τa ) and
In =
bn ϕn′′ (τa )
1/2
2π
=
e−H (τa )
∞
exp(bn [ϕ ∗ (a) − (τa + it )a])φn (τa + it )dt −∞
bn ϕn′′ (τa )
1/2
−H (τa )
∞
exp(bn [ϕn (τa + it ) − ϕn (τa ) − ita
e
2π
−∞
1 −2 −1 + b− n H (τa ) + O(bn )])dt = (In1 + In2 )[1 + O(bn )] ′′ 1/2 ′′ 1/2 b ϕ (τ ) b ϕ (τ ) e−bn Gn (t ) dt , In2 = n 2nπ a e−bn Gn (t ) dt . Note that we have used (1) and λ is chosen with In1 = n 2nπ a |t |≥b−λ |t |
goes exponentially fast to zero. Now, as for the second term In2 , by (7) we have
In2 =
bn ϕn′′ (τa )
1/2
2π
1/2−λ
|s|
is
bn
s
+ √ O(1) + bn Rn τa + i √
s2 ′′ ϕn (τa ) 2
is3 1 − √ ϕn(3) (τa ) 6 bn
exp −
bn
+ Ln (s) ds
(9)
is where Ln (s) = ezn − 1 − zn , with zn = − 6√ ϕn(3) (τa ) + √isb O(1) + bn Rn (τa + i √sb ). On the right-hand side of (9), the first b 3
n
1/2−λ
n
√
n
2 integral is 1 + O(b− ϕ ′′ (τa )) (Φ is the standard normal distribution) goes exponentially fast to n ) since the term Φ (−bn zero. The second and third integrals are zero since the integrands are odd functions. As in Chaganty and Sethuraman (1985), 1 −λ √is using (8), one can easily see that the fourth integral is O(b− n ). Finally, noting that | b O(1)| ≤ O(bn ) → 0, as n → ∞, for n
1/2−λ
| s| ≤ bn , we can show that |zn | < 1/2 and prove as in Chaganty and Sethuraman (1985) that the fifth integral is also 1 −1 O(b− n ). Therefore, In2 = 1 + O(bn ). This completes the proof of Theorem 1. Now we assume that Zn is lattice-valued. We recall that a random variable is said to be lattice-valued if its range lies in a subset of the lattice set {d0 + ks, k ∈ Z}, where the real d0 is called the displacement and the positive real s is the span. Denote by dn and sn the displacement and the span of bn Zn . We also assume that the real an is such that bn an belongs to the range of bn Zn , and there exists τan ∈ (−α1 , α1 ), with 0 < α1 < α , such that ϕ ′ (τan ) = an . The versions of assumptions (A.2)–(A.4) for lattice-valued random variables are given below. (A′ .2) There exists H, a continuous function in (−α1 , α1 ), such that for n large enough, (1)–(3) hold (with a replaced by an ) wherein the function ϕ is twice differentiable in (−α1 , α1 ) and there exists β > 0 such that ϕ ′′ (τan ) ≥ β . (A′ .3) There exist γ , δ1 > 0 such that for 0 < δ < δ1 , inf
−1
δ≤|t |≤π/sn
bn
φn (τan ) ≥ Ln (δ), log φn (τan + it ) γ
where Ln is a function satisfying bn Ln (b−λ n ) = O(bn ) for some 1/3 < λ < 1/2.
(A .4) There exist p, l > 0 such that ′
sup
τ ∈(−α,α)
φn (τ + it ) l/bn p φ (τ ) dt = O(bn ). n −π/sn π/sn
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The theorem dealing with lattice-valued random variables is as follows. Theorem 2. Let assumptions (A.1) and (A′ .2)–(A′ .4) hold. Then for a real an such that bn an belongs to the range of the latticevalued random variable bn Zn and for n large enough,
P(Zn = an ) = |sn |
1/2
1 2π bn ϕ ′′ (τan )
1 exp(−bn ϕ ∗ (an ) + H (τan )) 1 + O(b− n )
(10)
where τan is such that ϕ ′ (τan ) = an , and ϕ ∗ (an ) = τan an − ϕ(τan ). Proof. The proof is similar to that of Theorem 1 and we shall not consider it here.
3. Statistical applications In this section, we give an application of Theorem 1 to illustrate the case of absolutely continuous random variables and an application of Theorem 2 to illustrate the case of lattice-valued random variables. 3.1. Example 1. The sample variance We consider the sample variance Zn =
n (Xi − X )2 ,
1
n − 1 i=1 where the random variables X1 , X2 , . . . are independent and identically distributed (i.i.d.). Assuming that the Xi ’s have a n normal distribution N (µ; σ 2 ), σ 2 > 0, we know that σ −2 i=1 (Xi − X )2 follows a chi-square distribution with n − 1 degrees of freedom. Large deviation approximations were obtained in Joutard (2013a,b) for the tail probabilities of Zn . Here, by applying Theorem 1 to Zn with bn = n, we obtain a large deviation approximation for the density function of Zn . Corollary 1. Let Zn be defined as above. Then for a real a > 0 and n large enough, the following asymptotic approximation holds for the probability density function kn of Zn , kn (a) =
where ISV (a) =
1 2
n 1
π 2a a
σ2
exp(−(n − 1)ISV (a)) 1 + O(n−1 ) ,
− log
a
σ2
(11)
−1 .
Remark 1. Note that in this particular example assuming a normality assumption, the exact distribution of the statistic is known, which allows to obtain an explicit expression for the c.g.f. of nZn . It would be more interesting to drop the normality assumption and assume, for example, a sub-Gaussian distribution for X1 . Unfortunately in this case, one cannot derive an expression for the c.g.f. ϕn that satisfies (1). This therefore remains an open problem and this example can be considered merely as an illustration of Theorem 1. Proof. By the proof of Joutard (2013a, Corollary 2.2), Assumption (A.1) is satisfied and (1) holds with bn = n, ϕ(t ) =
log(1 − 2σ 2 t ) + 1−σ2σt 2 t . Besides, one can easily verify that (2)–(3) hold with ϕ ′ (t ) = 1−σ2σ 2 t 4 −σ 2 and ϕ ′′ (t ) = (1−22σσ 2 t )2 . From Joutard (2013a), we have τa = a2a such that ϕ ′ (τa ) = a. Hence, ϕ ′′ (τa ) = 2a2 > 0 and σ2 Assumption (A.2) is verified. As for Assumption (A.3), we have for all |t | ≥ δ ,
− 21 log(1 − 2σ 2 t ) and H (t ) =
2
1 2
2
n 2 φn (τa ) 4σ 4 δ 2 n− n−1 1 ≥ n log log 1 + 2 . n 2 φn (τa + it ) 4n 1 − 2σ τa n− 1 n 2 4σ 4 δ 2 ( n− ) n −1 1 Then, (4) holds with Ln (δ) = 4n log 1 + (1−2σ 2 τ n )2 . Therefore, we have nLn (n−λ ) = O(n1−2λ ), with 1 − 2λ > 0, and −1
a n−1
(A.3) is satisfied. Finally, there exists l = 4 such that
n 2 − 4(n4n−1) ∞ φn (τa + it ) l/n 4σ 4 t 2 n− 1 1+ dt , 2 φ (τ ) dt = n n a −∞ −∞ 1 − 2σ 2 τa n− 1 which implies (A.4). Observing that H (τa ) = ϕ ∗ (a) = 21 σa2 − log σa2 − 1 , the asymptotic approximation (11) therefore
∞
follows from the application of Theorem 1.
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C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
Now we present a small numerical example to illustrate the quality of the approximation. Here, we take n = 100, σ 2 = 2 and we plot the exact density of Zn (‘‘solid line’’) and the large deviation approximation (11) (‘‘dashed line’’), for values of a ranging from 1 to 3.
3.2. The Wilcoxon–Mann–Whitney test Let X1 , . . . , Xm and Y1 , . . . , Yn be two independent random samples from continuous distribution functions F and G. The Mann–Whitney statistic can be defined as follows: Mm,n =
m n
I{Yj
i =1 j =1
This statistic is used to test the null hypothesis: F = G. Let N = m + n and assume that there exists p0 ∈ (0, 1) such that m/N = p0 and n/N = 1 − p0 . Let us define Zm,n = Mm,n /N 2 . The statistic NZm,n is lattice-valued with displacement 0 and span sN = 1/N. Jin and Robinson (1999, 2003) established saddle point expansions for the probability mass function of Mm,n . Under the null hypothesis, by applying Theorem 2 with bn = N, we have the following asymptotic approximation for the probability mass function of Zm,n . Corollary 2. Let Zm,n be defined as above. Then for a real aN ∈]b0 , b′0 [, where p0 (1 − p0 )/2 < b0 < b′0 < p0 (1 − p0 ), such that NaN belongs to the range of Mm,n /N, and for N large enough,
P(Zm,n = aN ) =
1/2
1 2π
where τaN > 0, is such that
N3
1 0
ϕMW (τaN ) ′′
hτaN (x)dx − p20
e−N ϕMW (aN )+HMW (τaN ) 1 + O(N −1 ) ∗
1 0
hp0 τaN (x)dx − (1 − p0 )2
1 0
(12)
h(1−p0 )τaN (x)dx = aN , with ht (x) = xetx /(etx − 1).
′′ Furthermore, HMW (t ) = 2−1 [log{(et − 1)/(etp0 − 1)} − log{(et (1−p0 ) − 1)/(p0 (1 − p0 )t )}], ϕMW (t ) =
1 l (x)dx − 0 t ∗ 3 1 p0 0 lp0 t (x)dx − (1 − p0 ) 0 l(1−p0 )t (x)dx and ϕMW (aN ) = τaN aN − ϕMW (τaN ) where ϕMW (t ) = p0 log(p0 ) + (1 − p0 ) log(1 − 1 1 1 p0 ) + 0 gt (x)dx − p0 0 gp0 t (x)dx − (1 − p0 ) 0 g(1−p0 )t (x)dx, with gt (x) = log{(etx − 1)/x} and lt (x) = −x2 etx /(etx − 1)2 . 3 1
Remark 2. The above result assumes that aN > b0 > p0 (1 − p0 )/2 but the same kind of result also holds if aN < b0 < p0 (1 − p0 )/2 (in this case τaN is negative). In the case aN = p0 (1 − p0 )/2 (assuming that NaN belongs to the range of Mm,n /N), we have τaN = 0 and one can use the ′′ ∗ approximation (12) with ϕMW (0) = p0 (1 − p0 )/12 and ϕMW (2−1 p0 (1 − p0 )) = HMW (0) = 0. Proof. First, note that for a real aN > b0 > ϕ ′ (0) = p0 (1 − p0 )/2, there exists τaN > κ0 , with κ0 > 0, such that ϕ ′ (τaN ) = aN . Then, by the proof of Joutard (2013c, Corollary 3.1), Assumption (A.1) is satisfied and (1) holds with bn = N, H (t ) = p0 )
1 0
1 2
log
et −1 e −1 tp0
− log
et (1−p0 ) −1 p0 (1−p0 )t
, ϕ(t ) = p0 log(p0 ) + (1 − p0 ) log(1 − p0 ) +
g(1−p0 )t (x)dx. One can easily verify that (2)–(3) also hold with
ϕ ′ (t ) =
1
ht (x)dx − p20 0
0
1
hp0 t (x)dx − (1 − p0 )2
1
h(1−p0 )t (x)dx 0
1 0
gt (x)dx − p0
1 0
gp0 t (x)dx − (1 −
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
105
and
ϕ ′′ (t ) =
1
lt (x)dx − p30
1
lp0 t (x)dx − (1 − p0 )3
l(1−p0 )t (x)dx 0
0
0
1
where ht (x) = xetx /(etx − 1) and lt (x) = −x2 etx /(etx − 1)2 . Besides, ϕ ′′ (τaN ) ≥ β with β > 0 and (A′ .2) is therefore satisfied. Now regarding Assumption (A′ .3), we will apply Lemma 2.7 of Chaganty and Sethuraman (1985) which implies Condition C ′ of Chaganty and Sethuraman (1985). This condition plays the same role as Assumption (A′ .3) and is mainly used in the proof of Theorem 2 to show that
In1 =
bn ϕn′′ (τaN )
1/2
2π
−λ bn ≤|t |≤π/sn
exp(−bn Gn (t ))dt ,
where the expression of Gn is given in (6), goes exponentially fast to zero. Here, we consider only the particular cases m = n and τaN = 0. If m ̸= n and τaN > 0, the proof is similar but is long and tedious so we shall not consider it here. Let fm,n (t ) = Real(Gm,n (t )). By (6) with τaN = 0, fm,n (t ) = Real(Gm,n (t )) = −N −1 log(|φm,n (it )|) where φm,n is the m.g.f. of Mm,n /N. Assuming that m = n, we have N = 2n and
2n eit (k/2n) − 1 2n −1 k=n+1 fn,n (t ) = −(2n)−1 log n it (k/2n) n e − 1 k=1
+n ) n k sin t (k4n . = −(2n)−1 log sin tk k + n k=1
4n
sin(k tk/4n) Note that the function t → sin(0tk/4n) , where k0 = (k + n)/k > 1, is decreasing on (0, η1 ), η1 = 2π , and increasing on (−η1 , 0). Consequently, fn,n is increasing on (0, η1 ) and decreasing on (−η1 , 0). This verifies the condition (i) of Chaganty and Sethuraman (1985, Lemma 2.7). Now, for the condition (ii), we have
t (k+n) n sin 4 + (2n)−1 log 2n . inf fn,n (t ) = −(2n)−1 sup log sin tk 2π≤|t |≤2nπ n 2π n−1 ≤|t |≤2π k=1 4
(13)
Note that for all 2π n−1 ≤ t ≤ 2π ,
t (k+n) n sin 4 n sin tk ≤ M0 k=1 4
(14)
where M0 < 4. Moreover, we have n
−1
log
2n
n
√ ≈ log(4) − n−1 log( π n)
(15)
as n → ∞. Therefore, combining (13)–(15), for n large enough there exists ϵ > 0 such that inf2π≤|t |≤2nπ fn,n (t ) > ϵ. This implies condition (ii) of the lemma. Finally, it remains to verify condition (iii). For η > 0, we have: sup fn,n (t ) =
|t |≤η
sup −(4n)−1 |t |≤ηn−1
n
log
k=1
sin2
sin2
t (k+n) 4
tk 4
k2
(k + n)2
.
Using Taylor expansions, for η small enough and n large enough, we have for |t | ≤ ηn−1 ,
−n − 1
n k=1
log
sin2
t (k+n) 4
2 tk
sin
4
k2
(k + n)2
2 2 ≈ n t .
24
106
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107 Table 1 Approximations of P(Mm,n = l) for the Mann–Whitney for m = n = 5. l
15
16
17
18
19
Exact
0.07143
0.06349
0.05556
0.04365
0.03571
ALD ASP l
0.07472 0.07095 20
0.06603 0.06353 21
0.05620 0.05472 22
0.04598 0.04523 23
0.03608 0.03577 24
Exact ALD ASP
0.02778 0.02709 0.02696
0.01984 0.01942 0.01925
0.01190 0.01327 0.01293
0.00794 0.00883 0.00807
0.00397 0.00687 0.00463
Table 2 Approximations of P(Mm,n = l) for the Mann–Whitney for m = 10, n = 6. l
50
51
52
53
54
Exact ALD ASP l
0.00437 0.00424 0.00429 55
0.00325 0.00325 0.00330 56
0.00250 0.00245 0.00248 57
0.00175 0.00180 0.00183 58
0.00137 0.00130 0.00131 59
Exact ALD ASP
0.000874 0.000910 0.000911
0.000624 0.000622 0.000614
0.000375 0.000414 0.000398
0.000250 0.000275 0.000246
0.000125 0.000217 0.000145
Hence, there exists η < 2π such that, for n large enough, sup|t |≤η fn,n (t ) < ϵ. The condition (iii) is satisfied and we can apply Lemma 2.7 of Chaganty and Sethuraman (1985) to verify Assumption (A′ .3). Finally, according to Chaganty and Sethuraman (1985, Remark 2.16), Assumption (A′ .4) follows from the fact that 1/|sN | = N and Theorem 2 can then be applied to obtain (12). Remark 3. Note that it should also be possible to apply Theorem 2 to the Jonckheere–Terpstra statistic (see Jonckheere, 1954; Terpstra, 1952). In fact, the Jonckheere–Terpstra test can be seen as an extension of the Wilcoxon–Mann–Whitney test to more than two samples and so the proof is similar to that of Corollary 2 regarding Assumptions (A.1)–(A.2). The main difficulty lies in the verification of Assumption (A′ .3). 4. Numerical results In this section we compare the (exact) probabilities P(Mm,n = l) with the large deviation approximation (12) and a saddle point approximation derived by Jin and Robinson (2003) (these approximations are denoted, respectively, by ALD and ASP in Tables 1 and 2). The comparisons are carried out on two examples that were also studied in Jin and Robinson (2003): m = n = 5 and m = 10, n = 6. The saddle point approximation of P(Mm,n = l) can be expressed as
P(Mm,n = l) =
exp(NK (u) − lu) 1 + N −1 P2 (u) + O(N −3/2 ) , √ ′′ 2π NK (u)
(16)
where K (u) = N −1 log E{exp(uMm,n )}, u is such that K ′ (u) = l/N and P2 (u) =
3K (4) (u) 24(K ′′ (u))2
−
15(K (3) (u))2 72(K ′′ (u))3
.
Now regarding the approximation (12), we have P(Zm,n = aN ) = P(Mm,n = l), where l = aN ∗ N 2 . For efficiency reasons, we use aN = (l + 1/2)/N 2 instead of aN = l/N 2 on the right-hand side of (12) since it turns out to give better results as far as these numerical comparisons are concerned. We see from Tables 1 and 2 that the approximation (12) is good at most values and sometimes it can even be better than the approximation (16) (e.g. l = 20 or 21 in Table 1 and l = 51, 53 or 56 in Table 2) despite its larger relative error of order N −1 . Note, however, that this is possible because of the small values of m and n. For large values of m and n, the approximation (16) will usually give better results than (12) since its relative error of order N −3/2 is smaller. We can also observe that the two approximations (in particular (12)) are not accurate for values near the endpoints of the support (e.g. l = 24 in Table 1 and l = 59 in Table 2). Note that Jin and Robinson (1999) obtained saddle point approximations that can be accurate near the endpoints of the support. In conclusion, these two examples show that the large deviation approximation can give good results and sometimes even perform (slightly) better than the saddle point approximation. But as mentioned above, the latter approximation is
C. Joutard / Statistics and Probability Letters 90 (2014) 100–107
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more accurate for larger values of m and n. However, Jin and Robinson (1999, 2003) mainly deal with Mann–Whitney, while Theorems 1 and 2 concern more general statistics. Furthermore, according to Remark 3, Corollary 2 may also be seen as a first step toward an approximation for the probability mass function of the Jonckheere–Terpstra statistic. Acknowledgment The author would like to thank the anonymous referee for helpful comments which have led to an improved version of this paper. References Bean, R., Froda, S., van Eeden, C., 2004. The normal, Edgeworth, saddlepoint and uniform approximations to the Wilcoxon–Mann–Whitney nulldistribution: a numerical comparison. J. Nonparametr. Stat. 16, 279–288. Bercu, B., Gamboa, F., Lavielle, M., 2000. Sharp large deviations for Gaussian quadratic forms with applications. ESAIM Probab. Stat. 3, 1–29. Blackwell, D., Hodges, J.L., 1959. The probability in the extreme tail of a convolution. Ann. Math. Stat. 30, 1113–1120. Buckle, N., Kraft, C., van Eeden, C., 1969. An approximation to the Wilcoxon–Mann–Whitney distribution. J. Amer. Statist. Assoc. 64, 591–599. Chaganty, N.R., Sethuraman, J., 1985. Large deviation local limit theorems for arbitrary sequences of random variables. Ann. Probab. 13, 97–114. Jin, R., Robinson, J., 2003. Saddlepoint approximations of the two-sample Wilcoxon statistic. Lecture Notes-Monograph Series, pp. 149–158. Jin, R., Robinson, J., 1999. Saddlepoint approximation near the endpoints of the support. Statist. Probab. Lett. 295–303. Jonckheere, A.R., 1954. A distribution-free k-sample test against ordered alternatives. Biometrika 41, 133–145. Joutard, C., 2013c. Large deviation approximations for the Mann–Whitney statistic and the Jonckheere–Terpstra statistic. J. Nonparametr. Stat. 25, 873–888. Joutard, C., 2013a. Strong large deviations for arbitrary sequences of random variables. Ann. Inst. Statist. Math. 1, 49–67. Joutard, C., 2013b. A strong large deviation theorem. Math. Methods Statist. 22, 155–164. Mann, H.B., Whitney, D.R., 1947. On a test of whether one or two random variables is stochastically larger than the other. Ann. Math. Statist. 18, 50–60. Richter, W., 1957. Local limit theorems for large deviations. Theory Probab. Appl. 2, 206–219. Terpstra, T.J., 1952. The asymptotic normality and consistency of Kendall’s test against trend, when ties are present in one ranking. Indag. Math. 14, 327–333. Wilcoxon, F., 1945. Individual comparison by ranking methods. Biometrics 1, 80–83.