- Email: [email protected]
Complete asymptotic expansions for normal extremes
Statistics and Probability Letters 103 (2015) 127–133
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Complete asymptotic expansions for normal extremes Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, UK
article
abstract
info
Article history: Received 23 January 2015 Received in revised form 13 April 2015 Accepted 22 April 2015 Available online 29 April 2015 Keywords: Bell polynomials Expansions Normal distribution
Hall (1979) was the first to derive the rate of convergence for normal extremes. Many authors have followed up the work of Hall, but complete asymptotic expansions have not been known for normal extremes. Here, we derive such expansions for the first time. The expansions are single infinite sums of terms involving Bell polynomials and Stirling numbers. The usefulness of the expansions over the results in Hall is illustrated computationally. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Let Φ (·) denote the cumulative distribution function (cdf) of a standard normal random variable. It is well known that Φ (·) belongs to the max domain of attraction of the Gumbel extreme value distribution, i.e.,
Φ n (an x + bn ) → exp {− exp(−x)}
(1)
as n → ∞, where an = (2 log n)−1/2 ,
an 1 [log log n + log(4π )] b n = a− n − 2
(2)
for n ≥ 1. Many authors have been interested in convergence aspects of (1) because of the universality of the normal distribution. Hall (1979) was the first to investigate convergence aspects of (1). He established the convergence rate of (1). He chose an and bn slightly differently to satisfy 1 an = b − n ,
2π b2n exp b2n = n2
(3)
for n ≥ 1. Since Hall’s seminal paper, many other authors have considered convergence aspects of (1). We mention Hall (1980), Nair (1981), Cohen (1982a,b), Rootzén (1983) and Gomes (1984). To the best of our knowledge, complete asymptotic expansions for Φ n (an x + bn ) have not been known to date. By a complete asymptotic expansion, we mean the following: suppose that Fn (x) → P (x) uniformly as n → ∞. Suppose that we have an asymptotic expansion of P (x), so we can write P (x) =
∞
ci ei (x),
i=0
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2015.04.026 0167-7152/© 2015 Elsevier B.V. All rights reserved.
128
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
where ci is a constant and ei is a function. Write P n ( x) =
n
ci ei (x)
i=0
and define ∆n (x) = Fn (x) − Pn (x). The complete asymptotic expansion is an expansion for ∆n (x) of the form
∆n (x) =
∞
Ci,n πi,n (x).
(4)
i=1
This expansion can be derived for any Fn (x). Some recent examples of complete asymptotic expansions are those due to Hashorva (2009a,b, 2010), Debicki et al. (2014) and Hashorva et al. (2014). An expansion such as (4) could have both practical and theoretical appeal. In a practical sense, it could lead to better approximations, see Section 3 for an illustration. Theoretically, such an expansion can be used to derive expansions for the corresponding probability density function (pdf), moments, cumulants, quantiles, etc. The aim of this note is to derive complete asymptotic expansions for Φ n (an x + bn ). The derived expansions are single infinite sums. The terms of the infinite sums involve Bell polynomials and the Stirling number of the first kind. In-built routines for Bell polynomials and the Stirling number of the first kind are available in most computer algebra packages. For example, see BellY and StirlingS1 in Mathematica. So, the expansions given will be accessible to most practitioners. The complete asymptotic expansions for Φ n (an x + bn ) are given in Section 2. The corresponding proof is given in Section 4. Computational issues relating to the expansions in Section 2 are discussed in Section 3. Our results in Sections 2–4 can in principle be extended to any other cdf belonging to the max domain of attraction of an extreme value distribution. We have illustrated our results for the normal distribution because of its universality. For x = (x1 , x2 , . . .), we shall let Brk (x) denote the partial exponential Bell polynomial defined by
∞
k
xr t /r ! r
k! =
∞
Brk (x)t r /r !.
(5)
r =k
r =1
This polynomial is tabled on p. 307 of Comtet (1974) for r ≤ 12. We shall use the notation (a)k = a(a + 1) · · · (a + k − 1) (j) to denote the ascending factorial and the notation Si to denote the Stirling number of the first kind. 2. Main results Our main results are Theorems 1 and 2. Theorem 1 gives a complete asymptotic expansion for n (−1)k exp(−kx)
∆n (x) = Φ n (an x + bn ) −
(6)
k!
k=0
as n → ∞ when an and bn are given by (2). Theorem 2 gives a complete asymptotic expansion for Dn (x) = Φ n (an x + bn ) −
n (−1)k exp(−kx)
(7)
k!
k=0
as n → ∞ when an and bn are given by (3). The proof of Theorem 1 is given in Section 4. The proof of Theorem 2 is similar to that of Theorem 1. Theorem 1. With an and bn given by (2), we have
∆ n ( x) =
∞ (−2)r B0r (c)
r!
r =1
+
(2 log n)−r ∆n,0,r (x)
∞ n n (−1)k (−2)r Br0 (c) r =1 k=1
+ n!
r!
k
∞ n k (−1)k (−2)r Br ℓ (c) r =n k=1 ℓ=1
+ 2n!
(n − k)!r !(k − ℓ)!
n (−1)r r =1
+
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
n r k (−1)k (−2)r Br ℓ (c) r =1 k=1 ℓ=1
+
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
r!
(n − k)!r !(k − ℓ)!
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
exp(−rx) ∆n,r ,0 (x) − 1
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m) r =1 ℓ=1 m=0
ℓ!(r − ℓ)!
ℓ
exp(−rx)nm−r ∆n,r ,0 (x)
(8)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
129
as n → ∞, where
∆n,k,r (x) =
1+
x 2 log n
−
kx2 · exp −
4 log n
log log n + log 4π
−k−2r
4 log n
+k
log log n + log 4π
−k
4 log n
(log log n + log 4π )2
16 log n
and c = {i!(1/2)i , i ≥ 1}. As we see, (10) involves six summations. The first is a single infinite sum of terms involving Bell polynomials. The second is a single infinite sum of a single finite sum of terms involving Bell polynomials. The third is a single infinite sum of a double finite sum of terms involving Bell polynomials. The fourth is a triple finite sum of terms involving Bell polynomials. The fifth is a single finite sum of elementary terms. The sixth is a triple finite sum of terms involving Stirling numbers. Overall, the expansion in Theorem 1 is a single infinite sum of a finite number of terms involving Bell polynomials and Stirling numbers. Note that ∆n,k,r (x) → 1 as n → ∞. In fact, one can write
∆n,k,r (x) =
∞ −k − 2r ∞ kr
·
∞
p
∞ (−k)q
·
q!
q =0
x2q q
4q log n
∞ (−k)s
·
s!
s=0
·
(−1)
∞ kr
r
r ! 4r log n v=0
w=0
w
q
x2
4 log n
s
16 log n p−t
r r
1
(−k)q q!
(log log n + log 4π )2
22p−t logp n u=0
2s 2s
(16 log n)s
q =0
xt
p−t
r =0
1
(−k)s s!
s=0
t
t =0
p ∞
4 log n
r ∞
4 log n
p −k − 2r p
p=0
=
log log n + log 4π
r!
r =0
=
log log n + log 4π
−
2 log n
p
p=0
x
v
p−t
u
(log log n)u (log 4π )p−t −u
(log log n)v (log 4π )v−r
(log log n)w (log 4π )2s−w
p p−t ∞ ∞ ∞ r ∞ 2s −k − 2r p p − t r 2s p
p=0 t =0 u=0 q=0 r =0 v=0 s=0 w=0
· 2t −2p−2q−2r −4s (log 4π )p−t −u+v−r +2s−w
t kq+r +s q!r !s!
v
u
xt +2q
w
(log log n)u+v+w logp+q+r +s n
(−1)p+q+s−t
.
(9)
Note that the term of (9) corresponding to p = q = r = s = 0 is equal to one. Each of the remaining terms approaches zero as n → ∞. Theorem 2. With an and bn given by (3), we have D n ( x) =
∞ (−2)r B0r (c)
r!
r =1
+ n!
a2r n ∆n,0,r (x) +
+ 2n!
(n − k)!r !(k − ℓ)!
r =1 k=1 ℓ=1
ℓ−1 r
(n − k)!r !(k − ℓ)!
(−1)
r =1 ℓ=1 m=0
exp(−kx)n−k a2r n ∆n,k,r (x)
exp(−kx)n−k a2r n ∆n,k,r (x) +
n (−1)r r =1
(m) r )r −ℓ Sℓ
(1 − ℓ!(r − ℓ)!
r +m+ℓ
r!
exp(−kx)n−k a2r n ∆n,k,r (x)
n r k (−1)k (−2)r Br ℓ (c)
n
k
r =1 k=1
∞ n k (−1)k (−2)r Br ℓ (c) r =n k=1 ℓ=1
+
∞ n n (−1)k (−2)r Br0 (c)
exp(−rx)nm−r ∆n,r ,0 (x)
r!
exp(−rx) ∆n,r ,0 (x) − 1
(10)
as n → ∞, where
∆n,k,r (x) = (2π )
k 2
1+
−k−2r a2n x
exp −
ka2n x2
2
and c = {i!(1/2)i , i ≥ 1}. The expansion in Theorem 2 is also a single infinite sum of a finite number of terms involving Bell polynomials and Stirling numbers.
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S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
3. Computational issues Here, we assess the behaviors of ∆n (x), Dn (x) in (6), (7) versus n and x. We also compare ∆n (x) versus approximations to Φ n (x) suggested by Theorem 3 in Hall (1980). Fig. 1 shows how ∆n (x), Dn (x) vary with respect to n and x. For small values of x (x = 0.1, 0.2, 0.5), ∆n (x) initially increases from being negative to a positive value. Thereafter ∆n (x) decreases to zero as n increases. For not small values of x (x = 2), ∆n (x) decreases from a positive value to zero as n increases. For small values of x (x = 0.1, 0.2), Dn (x) initially increases from being positive to another positive value. Thereafter Dn (x) decreases to zero as n increases. For other values of x (x = 0.5, 2), Dn (x) decreases from a positive value to zero as n increases. In general the values of ∆n (x) are smaller than those of Dn (x). So, it is better to choose an and bn as those given by (2). Hall (1980) suggested various approximations to Φ n (x). The two approximations suggested by his Theorem 3 are
Qn,1 (x) = exp −zn (x) 1 − x
−2
+ 3x
−4
+
zn (x)(n − 1)
2
and Qn,2 (x) = exp −zn (x) 1 − x−2
,
where zn (x) = φ(x)n/x and φ(·) denotes the standard normal pdf. Hall (1980) claimed that these approximations are ‘‘very good’’ for n ≥ 10. Let
∆n,2 (x) = Φ n (x) − Qn,1 (x) and
∆n,3 (x) = Φ n (x) − Qn,2 (x) denote the corresponding errors. Fig. 2 plots ∆n (x), ∆n,2 (x) and ∆n,3 (x) versus x = 0.01, 0.02, . . . , 2 for n = 5, 10, 15, 20, 25, 30. We see that ∆n (x) gives the best performance for
• • • • • •
n n n n n n
= 5 when x is approximately between 0.5 and 2; = 10 when x is approximately between 0.7 and 2; = 15 when x is approximately between 0.8 and 2; = 20 when x is approximately between 0.9 and 2; = 25 when x is approximately between 0.95 and 2; = 30 when x is approximately between 1 and 2.
Hence, the approximation for Φ n (x) suggested by Theorem 1 can be of better use than others. 4. Proof of Theorem 1 Using the facts
Φ (x) =
1
1 + erf
2
x
√
,
2 where erf(·) denotes the error function defined by 2 erf(x) = √
x
π
exp −t 2 dt ,
0
and erf(x) =
2 1 1 exp −x 2 F0 1, ; ; − 2
1
1
−√ 2 x πx as x → ∞, where 2 F0 (a, b; ; x) denotes a hypergeometric function defined by ∞ (a)k (b)k xk , 2 F0 (a, b; ; x) = k! k=0 x
we can write
Φ (x) = n
1
1 + erf
x
n
√ 2 n φ(x) 1 2 = 1− ;;− 2 2 F0 1, 2n
x
=
n n k=0
k
2
(−1)k
φ k (x) xk
x
2 F0
1,
1 2
;;−
2 x2
k (11)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
Fig. 1. ∆n (x), Dn (x) in (6), (7) versus n = 2, 3, . . . , 1000 for x = 0.1, 0.2, 0.5, 2. The x axis is plotted in log scale.
Fig. 2. ∆n (x), ∆n,2 (x) and ∆n,3 (x) versus x = 0.01, 0.02, . . . , 2 for n = 5, 10, 15, 20, 25, 30.
131
132
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
as x → ∞. Using the definition of Bell polynomials, we can write
2 F0
1,
1
;;−
2
2
i k ∞ i!(1/2)i 2 − 2 i! x i=0 i k ∞ i!(1/2)i 2 1+ − 2 i! x i=1 i ℓ ∞ k k i!(1/2)i 2 − 2 ℓ i! x i=1 ℓ=0 r ∞ k Br ℓ (c) 2 k! − 2 . (k − ℓ)! r =ℓ r ! x ℓ=0
k =
x2
=
=
=
(12)
Substituting (12) into (11), we obtain
Φ n (x) = n!
n k ∞ (−1)k+r 2r Br ℓ (c) φ k (x) (n − k)!r !(k − ℓ)! xk+2r k=0 ℓ=0 r =ℓ
as x → ∞. Let un = an x + bn . It is not difficult to see that
2 2 1 a x + 2an bn x + b2n φ (un ) = √ exp − n 2 2π √ 2 log n x2 log log n + log 4π (log log n + log 4π )2 = exp −x − + − n
4 log n
4 log n
16 log n
and i u− n =
√
x 2 log n
= (2 log n)
+
−i/2
2 log n −
1+
x 2 log n
log log n + log 4π
−i
√
2 2 log n
−
log log n + log 4π 4 log n
−i
.
So, we can write un−k−2r φ k (un ) = exp(−kx)n−k (2 log n)−r ∆n,k,r (x) and
Φ n (un ) = n!
n k ∞ (−1)k+r 2r Br ℓ (c) exp(−kx)n−k (2 log n)−r ∆n,k,r (x) ( n − k )! r !( k − ℓ)! k=0 ℓ=0 r =ℓ
(13)
as x → ∞. We now consider the term of (13) corresponding to ℓ = 0 and r = 0, which is n!
n k=0
(−1)k exp(−kx)∆n,k,0 (x). k!(n − k)!nk
Using the facts
(a + b)m = m!
m (a)k (b)m−k k=0
k!(m − k)!
and
(a)m =
m (−1)k+m Sm(k) ak , k =0
(14)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
133
we can write (14) as n (−1)k (n − k + 1)k
k!nk
k=0
=
n (−1)k k=0
nk
(−1)k
exp(−kx)∆n,k,0 (x)
exp(−kx)∆n,k,0 (x)
ℓ=0
n
=
k=0
=
nk
n (−1)r r =1
=
r!
ℓ!(k − ℓ)!
ℓ (1 − k)k−ℓ k
exp(−kx)∆n,k,0 (x)
ℓ=0
exp(−rx)∆n,r ,0 (x) +
ℓ!(k − ℓ)!
k!
k=0
+
n (−1)r r =1
r!
m=0
ℓ!(r − ℓ)!
r =1 ℓ=1 m=0
ℓ
exp(−rx)nm−r ∆n,r ,0 (x).
exp(−rx) ∆n,r ,0 (x) − 1
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m)
ℓ!(r − ℓ)!
(−1)m+ℓ Sℓ(m) nm
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m) r =1 ℓ=1 m=0
n (−1)k exp(−kx)
+
k (n)ℓ (1 − k)k−ℓ
ℓ
exp(−rx)nm−r ∆n,r ,0 (x).
(15)
Note that (15) contains the last two summations of (10). The first summation of (10) corresponds to the terms of (13) with {k = 0, ℓ = 0, r ≥ 1}. The second summation of (10) corresponds to the terms of (13) with {1 ≤ k ≤ n, ℓ = 0, r ≥ 1}. The third and fourth summations of (10) follow by noting n k ∞
:=
k=1 ℓ=1 r =ℓ
∞ n k r =n k=1 ℓ=1
+2
n r k
.
r =1 k=1 ℓ=1
The proof is complete. Acknowledgments The author thanks the editor and the two referees for careful reading and comments which greatly improved the paper. References Cohen, J.P., 1982a. Convergence rates for the ultimate and pentultimate approximations in extreme value theory. Adv. Appl. Probab. 14, 833–854. Cohen, J.P., 1982b. The penultimate form of approximation to normal extremes. Adv. Appl. Probab. 14, 324–339. Comtet, L., 1974. Advanced Combinatorics. Reidel, Dordrecht. Debicki, K., Hashorva, E., Ji, L., Tabis, K., 2014. On the probability of conjunctions of stationary Gaussian processes. Statist. Probab. Lett. 88, 141–148. Gomes, M.I., 1984. Penultimate limiting forms in extreme value theory. Ann. Inst. Statist. Math. 36, 71–85. Hall, P., 1979. On the rate of convergence of normal extremes. J. Appl. Probab. 16, 433–439. Hall, P., 1980. Estimating probabilities for normal extremes. Adv. Appl. Probab. 12, 491–500. Hashorva, E., 2009a. Asymptotics for Kotz type III elliptical distributions. Statist. Probab. Lett. 79, 927–935. Hashorva, E., 2009b. Conditional limit results for type I polar distributions. Extremes 12, 239–263. Hashorva, E., 2010. Asymptotics of the norm of elliptical random vectors. J. Multivariate Anal. 101, 926–935. Hashorva, E., Ling, C.X., Peng, Z., 2014. Tail asymptotic expansions for L-statistics. Sci. China Math. 57, 1993–2012. Nair, K.A., 1981. Asymptotic distribution and moments of normal extremes. Ann. Probab. 9, 150–153. Rootzén, H., 1983. The rate of convergence of extremes of stationary normal sequences. Adv. Appl. Probab. 15, 54–80.
Contents lists available at ScienceDirect
Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro
Complete asymptotic expansions for normal extremes Saralees Nadarajah School of Mathematics, University of Manchester, Manchester M13 9PL, UK
article
abstract
info
Article history: Received 23 January 2015 Received in revised form 13 April 2015 Accepted 22 April 2015 Available online 29 April 2015 Keywords: Bell polynomials Expansions Normal distribution
Hall (1979) was the first to derive the rate of convergence for normal extremes. Many authors have followed up the work of Hall, but complete asymptotic expansions have not been known for normal extremes. Here, we derive such expansions for the first time. The expansions are single infinite sums of terms involving Bell polynomials and Stirling numbers. The usefulness of the expansions over the results in Hall is illustrated computationally. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Let Φ (·) denote the cumulative distribution function (cdf) of a standard normal random variable. It is well known that Φ (·) belongs to the max domain of attraction of the Gumbel extreme value distribution, i.e.,
Φ n (an x + bn ) → exp {− exp(−x)}
(1)
as n → ∞, where an = (2 log n)−1/2 ,
an 1 [log log n + log(4π )] b n = a− n − 2
(2)
for n ≥ 1. Many authors have been interested in convergence aspects of (1) because of the universality of the normal distribution. Hall (1979) was the first to investigate convergence aspects of (1). He established the convergence rate of (1). He chose an and bn slightly differently to satisfy 1 an = b − n ,
2π b2n exp b2n = n2
(3)
for n ≥ 1. Since Hall’s seminal paper, many other authors have considered convergence aspects of (1). We mention Hall (1980), Nair (1981), Cohen (1982a,b), Rootzén (1983) and Gomes (1984). To the best of our knowledge, complete asymptotic expansions for Φ n (an x + bn ) have not been known to date. By a complete asymptotic expansion, we mean the following: suppose that Fn (x) → P (x) uniformly as n → ∞. Suppose that we have an asymptotic expansion of P (x), so we can write P (x) =
∞
ci ei (x),
i=0
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.spl.2015.04.026 0167-7152/© 2015 Elsevier B.V. All rights reserved.
128
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
where ci is a constant and ei is a function. Write P n ( x) =
n
ci ei (x)
i=0
and define ∆n (x) = Fn (x) − Pn (x). The complete asymptotic expansion is an expansion for ∆n (x) of the form
∆n (x) =
∞
Ci,n πi,n (x).
(4)
i=1
This expansion can be derived for any Fn (x). Some recent examples of complete asymptotic expansions are those due to Hashorva (2009a,b, 2010), Debicki et al. (2014) and Hashorva et al. (2014). An expansion such as (4) could have both practical and theoretical appeal. In a practical sense, it could lead to better approximations, see Section 3 for an illustration. Theoretically, such an expansion can be used to derive expansions for the corresponding probability density function (pdf), moments, cumulants, quantiles, etc. The aim of this note is to derive complete asymptotic expansions for Φ n (an x + bn ). The derived expansions are single infinite sums. The terms of the infinite sums involve Bell polynomials and the Stirling number of the first kind. In-built routines for Bell polynomials and the Stirling number of the first kind are available in most computer algebra packages. For example, see BellY and StirlingS1 in Mathematica. So, the expansions given will be accessible to most practitioners. The complete asymptotic expansions for Φ n (an x + bn ) are given in Section 2. The corresponding proof is given in Section 4. Computational issues relating to the expansions in Section 2 are discussed in Section 3. Our results in Sections 2–4 can in principle be extended to any other cdf belonging to the max domain of attraction of an extreme value distribution. We have illustrated our results for the normal distribution because of its universality. For x = (x1 , x2 , . . .), we shall let Brk (x) denote the partial exponential Bell polynomial defined by
∞
k
xr t /r ! r
k! =
∞
Brk (x)t r /r !.
(5)
r =k
r =1
This polynomial is tabled on p. 307 of Comtet (1974) for r ≤ 12. We shall use the notation (a)k = a(a + 1) · · · (a + k − 1) (j) to denote the ascending factorial and the notation Si to denote the Stirling number of the first kind. 2. Main results Our main results are Theorems 1 and 2. Theorem 1 gives a complete asymptotic expansion for n (−1)k exp(−kx)
∆n (x) = Φ n (an x + bn ) −
(6)
k!
k=0
as n → ∞ when an and bn are given by (2). Theorem 2 gives a complete asymptotic expansion for Dn (x) = Φ n (an x + bn ) −
n (−1)k exp(−kx)
(7)
k!
k=0
as n → ∞ when an and bn are given by (3). The proof of Theorem 1 is given in Section 4. The proof of Theorem 2 is similar to that of Theorem 1. Theorem 1. With an and bn given by (2), we have
∆ n ( x) =
∞ (−2)r B0r (c)
r!
r =1
+
(2 log n)−r ∆n,0,r (x)
∞ n n (−1)k (−2)r Br0 (c) r =1 k=1
+ n!
r!
k
∞ n k (−1)k (−2)r Br ℓ (c) r =n k=1 ℓ=1
+ 2n!
(n − k)!r !(k − ℓ)!
n (−1)r r =1
+
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
n r k (−1)k (−2)r Br ℓ (c) r =1 k=1 ℓ=1
+
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
r!
(n − k)!r !(k − ℓ)!
exp(−kx)n−k (2 log n)−r ∆n,k,r (x)
exp(−rx) ∆n,r ,0 (x) − 1
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m) r =1 ℓ=1 m=0
ℓ!(r − ℓ)!
ℓ
exp(−rx)nm−r ∆n,r ,0 (x)
(8)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
129
as n → ∞, where
∆n,k,r (x) =
1+
x 2 log n
−
kx2 · exp −
4 log n
log log n + log 4π
−k−2r
4 log n
+k
log log n + log 4π
−k
4 log n
(log log n + log 4π )2
16 log n
and c = {i!(1/2)i , i ≥ 1}. As we see, (10) involves six summations. The first is a single infinite sum of terms involving Bell polynomials. The second is a single infinite sum of a single finite sum of terms involving Bell polynomials. The third is a single infinite sum of a double finite sum of terms involving Bell polynomials. The fourth is a triple finite sum of terms involving Bell polynomials. The fifth is a single finite sum of elementary terms. The sixth is a triple finite sum of terms involving Stirling numbers. Overall, the expansion in Theorem 1 is a single infinite sum of a finite number of terms involving Bell polynomials and Stirling numbers. Note that ∆n,k,r (x) → 1 as n → ∞. In fact, one can write
∆n,k,r (x) =
∞ −k − 2r ∞ kr
·
∞
p
∞ (−k)q
·
q!
q =0
x2q q
4q log n
∞ (−k)s
·
s!
s=0
·
(−1)
∞ kr
r
r ! 4r log n v=0
w=0
w
q
x2
4 log n
s
16 log n p−t
r r
1
(−k)q q!
(log log n + log 4π )2
22p−t logp n u=0
2s 2s
(16 log n)s
q =0
xt
p−t
r =0
1
(−k)s s!
s=0
t
t =0
p ∞
4 log n
r ∞
4 log n
p −k − 2r p
p=0
=
log log n + log 4π
r!
r =0
=
log log n + log 4π
−
2 log n
p
p=0
x
v
p−t
u
(log log n)u (log 4π )p−t −u
(log log n)v (log 4π )v−r
(log log n)w (log 4π )2s−w
p p−t ∞ ∞ ∞ r ∞ 2s −k − 2r p p − t r 2s p
p=0 t =0 u=0 q=0 r =0 v=0 s=0 w=0
· 2t −2p−2q−2r −4s (log 4π )p−t −u+v−r +2s−w
t kq+r +s q!r !s!
v
u
xt +2q
w
(log log n)u+v+w logp+q+r +s n
(−1)p+q+s−t
.
(9)
Note that the term of (9) corresponding to p = q = r = s = 0 is equal to one. Each of the remaining terms approaches zero as n → ∞. Theorem 2. With an and bn given by (3), we have D n ( x) =
∞ (−2)r B0r (c)
r!
r =1
+ n!
a2r n ∆n,0,r (x) +
+ 2n!
(n − k)!r !(k − ℓ)!
r =1 k=1 ℓ=1
ℓ−1 r
(n − k)!r !(k − ℓ)!
(−1)
r =1 ℓ=1 m=0
exp(−kx)n−k a2r n ∆n,k,r (x)
exp(−kx)n−k a2r n ∆n,k,r (x) +
n (−1)r r =1
(m) r )r −ℓ Sℓ
(1 − ℓ!(r − ℓ)!
r +m+ℓ
r!
exp(−kx)n−k a2r n ∆n,k,r (x)
n r k (−1)k (−2)r Br ℓ (c)
n
k
r =1 k=1
∞ n k (−1)k (−2)r Br ℓ (c) r =n k=1 ℓ=1
+
∞ n n (−1)k (−2)r Br0 (c)
exp(−rx)nm−r ∆n,r ,0 (x)
r!
exp(−rx) ∆n,r ,0 (x) − 1
(10)
as n → ∞, where
∆n,k,r (x) = (2π )
k 2
1+
−k−2r a2n x
exp −
ka2n x2
2
and c = {i!(1/2)i , i ≥ 1}. The expansion in Theorem 2 is also a single infinite sum of a finite number of terms involving Bell polynomials and Stirling numbers.
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S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
3. Computational issues Here, we assess the behaviors of ∆n (x), Dn (x) in (6), (7) versus n and x. We also compare ∆n (x) versus approximations to Φ n (x) suggested by Theorem 3 in Hall (1980). Fig. 1 shows how ∆n (x), Dn (x) vary with respect to n and x. For small values of x (x = 0.1, 0.2, 0.5), ∆n (x) initially increases from being negative to a positive value. Thereafter ∆n (x) decreases to zero as n increases. For not small values of x (x = 2), ∆n (x) decreases from a positive value to zero as n increases. For small values of x (x = 0.1, 0.2), Dn (x) initially increases from being positive to another positive value. Thereafter Dn (x) decreases to zero as n increases. For other values of x (x = 0.5, 2), Dn (x) decreases from a positive value to zero as n increases. In general the values of ∆n (x) are smaller than those of Dn (x). So, it is better to choose an and bn as those given by (2). Hall (1980) suggested various approximations to Φ n (x). The two approximations suggested by his Theorem 3 are
Qn,1 (x) = exp −zn (x) 1 − x
−2
+ 3x
−4
+
zn (x)(n − 1)
2
and Qn,2 (x) = exp −zn (x) 1 − x−2
,
where zn (x) = φ(x)n/x and φ(·) denotes the standard normal pdf. Hall (1980) claimed that these approximations are ‘‘very good’’ for n ≥ 10. Let
∆n,2 (x) = Φ n (x) − Qn,1 (x) and
∆n,3 (x) = Φ n (x) − Qn,2 (x) denote the corresponding errors. Fig. 2 plots ∆n (x), ∆n,2 (x) and ∆n,3 (x) versus x = 0.01, 0.02, . . . , 2 for n = 5, 10, 15, 20, 25, 30. We see that ∆n (x) gives the best performance for
• • • • • •
n n n n n n
= 5 when x is approximately between 0.5 and 2; = 10 when x is approximately between 0.7 and 2; = 15 when x is approximately between 0.8 and 2; = 20 when x is approximately between 0.9 and 2; = 25 when x is approximately between 0.95 and 2; = 30 when x is approximately between 1 and 2.
Hence, the approximation for Φ n (x) suggested by Theorem 1 can be of better use than others. 4. Proof of Theorem 1 Using the facts
Φ (x) =
1
1 + erf
2
x
√
,
2 where erf(·) denotes the error function defined by 2 erf(x) = √
x
π
exp −t 2 dt ,
0
and erf(x) =
2 1 1 exp −x 2 F0 1, ; ; − 2
1
1
−√ 2 x πx as x → ∞, where 2 F0 (a, b; ; x) denotes a hypergeometric function defined by ∞ (a)k (b)k xk , 2 F0 (a, b; ; x) = k! k=0 x
we can write
Φ (x) = n
1
1 + erf
x
n
√ 2 n φ(x) 1 2 = 1− ;;− 2 2 F0 1, 2n
x
=
n n k=0
k
2
(−1)k
φ k (x) xk
x
2 F0
1,
1 2
;;−
2 x2
k (11)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
Fig. 1. ∆n (x), Dn (x) in (6), (7) versus n = 2, 3, . . . , 1000 for x = 0.1, 0.2, 0.5, 2. The x axis is plotted in log scale.
Fig. 2. ∆n (x), ∆n,2 (x) and ∆n,3 (x) versus x = 0.01, 0.02, . . . , 2 for n = 5, 10, 15, 20, 25, 30.
131
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S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
as x → ∞. Using the definition of Bell polynomials, we can write
2 F0
1,
1
;;−
2
2
i k ∞ i!(1/2)i 2 − 2 i! x i=0 i k ∞ i!(1/2)i 2 1+ − 2 i! x i=1 i ℓ ∞ k k i!(1/2)i 2 − 2 ℓ i! x i=1 ℓ=0 r ∞ k Br ℓ (c) 2 k! − 2 . (k − ℓ)! r =ℓ r ! x ℓ=0
k =
x2
=
=
=
(12)
Substituting (12) into (11), we obtain
Φ n (x) = n!
n k ∞ (−1)k+r 2r Br ℓ (c) φ k (x) (n − k)!r !(k − ℓ)! xk+2r k=0 ℓ=0 r =ℓ
as x → ∞. Let un = an x + bn . It is not difficult to see that
2 2 1 a x + 2an bn x + b2n φ (un ) = √ exp − n 2 2π √ 2 log n x2 log log n + log 4π (log log n + log 4π )2 = exp −x − + − n
4 log n
4 log n
16 log n
and i u− n =
√
x 2 log n
= (2 log n)
+
−i/2
2 log n −
1+
x 2 log n
log log n + log 4π
−i
√
2 2 log n
−
log log n + log 4π 4 log n
−i
.
So, we can write un−k−2r φ k (un ) = exp(−kx)n−k (2 log n)−r ∆n,k,r (x) and
Φ n (un ) = n!
n k ∞ (−1)k+r 2r Br ℓ (c) exp(−kx)n−k (2 log n)−r ∆n,k,r (x) ( n − k )! r !( k − ℓ)! k=0 ℓ=0 r =ℓ
(13)
as x → ∞. We now consider the term of (13) corresponding to ℓ = 0 and r = 0, which is n!
n k=0
(−1)k exp(−kx)∆n,k,0 (x). k!(n − k)!nk
Using the facts
(a + b)m = m!
m (a)k (b)m−k k=0
k!(m − k)!
and
(a)m =
m (−1)k+m Sm(k) ak , k =0
(14)
S. Nadarajah / Statistics and Probability Letters 103 (2015) 127–133
133
we can write (14) as n (−1)k (n − k + 1)k
k!nk
k=0
=
n (−1)k k=0
nk
(−1)k
exp(−kx)∆n,k,0 (x)
exp(−kx)∆n,k,0 (x)
ℓ=0
n
=
k=0
=
nk
n (−1)r r =1
=
r!
ℓ!(k − ℓ)!
ℓ (1 − k)k−ℓ k
exp(−kx)∆n,k,0 (x)
ℓ=0
exp(−rx)∆n,r ,0 (x) +
ℓ!(k − ℓ)!
k!
k=0
+
n (−1)r r =1
r!
m=0
ℓ!(r − ℓ)!
r =1 ℓ=1 m=0
ℓ
exp(−rx)nm−r ∆n,r ,0 (x).
exp(−rx) ∆n,r ,0 (x) − 1
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m)
ℓ!(r − ℓ)!
(−1)m+ℓ Sℓ(m) nm
n r ℓ−1 (−1)r +m+ℓ (1 − r )r −ℓ S (m) r =1 ℓ=1 m=0
n (−1)k exp(−kx)
+
k (n)ℓ (1 − k)k−ℓ
ℓ
exp(−rx)nm−r ∆n,r ,0 (x).
(15)
Note that (15) contains the last two summations of (10). The first summation of (10) corresponds to the terms of (13) with {k = 0, ℓ = 0, r ≥ 1}. The second summation of (10) corresponds to the terms of (13) with {1 ≤ k ≤ n, ℓ = 0, r ≥ 1}. The third and fourth summations of (10) follow by noting n k ∞
:=
k=1 ℓ=1 r =ℓ
∞ n k r =n k=1 ℓ=1
+2
n r k
.
r =1 k=1 ℓ=1
The proof is complete. Acknowledgments The author thanks the editor and the two referees for careful reading and comments which greatly improved the paper. References Cohen, J.P., 1982a. Convergence rates for the ultimate and pentultimate approximations in extreme value theory. Adv. Appl. Probab. 14, 833–854. Cohen, J.P., 1982b. The penultimate form of approximation to normal extremes. Adv. Appl. Probab. 14, 324–339. Comtet, L., 1974. Advanced Combinatorics. Reidel, Dordrecht. Debicki, K., Hashorva, E., Ji, L., Tabis, K., 2014. On the probability of conjunctions of stationary Gaussian processes. Statist. Probab. Lett. 88, 141–148. Gomes, M.I., 1984. Penultimate limiting forms in extreme value theory. Ann. Inst. Statist. Math. 36, 71–85. Hall, P., 1979. On the rate of convergence of normal extremes. J. Appl. Probab. 16, 433–439. Hall, P., 1980. Estimating probabilities for normal extremes. Adv. Appl. Probab. 12, 491–500. Hashorva, E., 2009a. Asymptotics for Kotz type III elliptical distributions. Statist. Probab. Lett. 79, 927–935. Hashorva, E., 2009b. Conditional limit results for type I polar distributions. Extremes 12, 239–263. Hashorva, E., 2010. Asymptotics of the norm of elliptical random vectors. J. Multivariate Anal. 101, 926–935. Hashorva, E., Ling, C.X., Peng, Z., 2014. Tail asymptotic expansions for L-statistics. Sci. China Math. 57, 1993–2012. Nair, K.A., 1981. Asymptotic distribution and moments of normal extremes. Ann. Probab. 9, 150–153. Rootzén, H., 1983. The rate of convergence of extremes of stationary normal sequences. Adv. Appl. Probab. 15, 54–80.