Asymptotic Distribution of χ2-type Statistics 1. Introduction ... - CiteSeerX

Asymptotic Distribution of χ2 -type Statistics Friedrich G¨otze Fakult¨ at f¨ ur Mathematik, Universit¨ at Bielefeld Vladimir V. Ulyanov Faculty of Computational Mathematics and Cybernetics, Moscow State University

Abstract Let Z 1 , Z 2 , . . . be independent identically distributed lattice random vectors in Rk with mean zero and nonsingular covariance matrix V . We construct for Sn = n−1/2 (Z 1 + . . . + Z n ) an approximation of the form P(SnT V −1 Sn < x) − Gk (x) = J1 + O(n−1 ) which holds uniformly for all real x, provided E|Z 1 |4 < ∞, where Gk (x) denotes the distribution function of a chi-square random variable with k degrees of freedom. The term J1 is of order O(n−1 ) when k ≥ 5. Moreover, we prove non-uniform bounds for the difference P(SnT V −1 Sn < x) − Gk (x) which are small when x is large. Furthermore, we discuss applications of these general results to the family of goodness-of-fit statistics.

1. Introduction and Main Results. Let Z 1 , Z 2 , . . . denote i.i.d. random vectors in Rk with mean zero and nonsingular covariance matrix V . Assume that P(Z 1 ∈ U ) = 1, where U denotes a lattice given by {a + m : m ∈ Zk } for some a ∈ Rk . Define Sn = n−1/2 (Z 1 + . . . + Z n ). If E|Z 1 |s < ∞ then it is well-known that (see Bikyalis (1969) and Bhattacharya and Rao (1986), Theorem 22.1) n

k/2

P(Sn = ym,n ) = ϕ(ym,n ) +

s−2 X

n−j/2 Pj (−ϕ(ym,n )) + o(n−(s−2)/2 )

(1)

j=1

uniformly in ym,n = n−1/2 (m + na), where ϕ(x) denotes the density function of the multivariate normal distribution N (0, V ). For the definition of the 1

Edgeworth expansion terms Pj (−ϕ(x)) obtained by partial derivatives of ϕ(x) weighted with partial moments of Z 1 see e.g. in Bhattacharya and Rao (1986), ch.2 sect.7. In the following we provide expressions for Pj (−ϕ(x)) for some special cases only. Summing up both sides of (1) for all points ym,n from a Borel set A one obtains (see Bikyalis (1969), Theorem 1) ) ( s−2 Z Y k v X (−1)v √ ∂ n−v/2 Bv ( ny` − na` ) v H(y) P(Sn ∈ A) = d v! ∂y` A `=1 v=0 +o(n−(s−2)/2 )

(2)

uniformly for all Borel sets A, where B0 (x) ≡ 1, Bv (x) = −v!

∞ X

e2πimx (2πim)v m=−∞

f or v = 1, 2, . . .

and H(y) =

s−2 X

n−j/2 Pj (−Φ(y))

j=0

with Φ(y) denotes the multivariate normal distribution function of N (0, V ). In particular, if E|Z 1 |3 < ∞ we conclude for the multivariate distribution function Fn (x) of Sn 1 Fn (x) = Φ(x) + √ P1 (−Φ(x)) n k √ 1 X S1 ( nxi − nai )Di Φ(x) + o(n−1/2 ) − √ n i=1

(3)

uniformly in x ∈ Rk . Here ∂ Φ(x), ∂xi 1 S1 (u) = [u] − u + , 2 Di Φ(x) =

where [u] denotes the integer part of u ∈ R1 . However in the multidimensional cases one rarely needs the approximations given by (3) or (2) for general Borel sets A. It is desirable to determine explicit approximations for the probabilities P(Sn ∈ B) when B is a so-called 2

extended convex set. Following Yarnold (1972) we call B an extended convex set if B has the following representation for every r ∈ {1, . . . , k}  B = x = (x1 , . . . , xk )T : λr (x1 ) < xr < θr (x1 ) and x1 = (x1 , . . . , xr−1 , xr+1 , . . . , xk )T ∈ Br ,

where Br ⊂ Rk−1 and λr and θr denote continuous real functions on Rk−1 . Yarnold (1972), Theorem 2, obtained a more compact expression for the Stieltjes integral in (2) provided that the set B is an extended convex. Further simplifications are possible in case of the probabilities P(SnT V −1 Sn < c), that  is when B = B(c) = x : xT V −1 x < c is an ellipsoid. In order to formulate the result we introduce some additional notation. Denote by N (nc) the number of integer vectors m in the ellipsoid (m + na)T V −1 (m + na) < nc with center at −na and by V (nc) the volume of this ellipsoid. Note that V (nc) = (πnc)k/2 |V |1/2 /Γ(k/2 + 1). Recall that a matrix A is called rational if there exists a real number λ 6= 0 such that the matrix λA has integer entries only, otherwise it is called irrational. Using introduced notations we have Theorem 1.

If E|Z 1 |4 < ∞ then P(SnT V −1 Sn < c) − Gk (c) = J1 + O(n−1 )

(4)

uniformly in c, where J1 = (N (nc) − V (nc))

exp(−c/2) . (2πn)k/2 |V |1/2

Moreover, if the dimension of the space k : 1 ≤ k ≤ 4, then J1 = O(n−k/(k+1) ).

(5)

If k ≥ 5 then there exists a positive constant c1 (k) depending on k only such that  k+1   σ1 c1 (k) σ1 1 + log , (6) |J1 | ≤ n σk σk where σ1 ≥ σ2 ≥ . . . ≥ σk denote the eigenvalues of V −1 . 3

If k ≥ 5 and the matrix V −1 is irrational then J1 = o(n−1 ).

(7)

Proof. The proof of the approximation (4) follows from Theorems 2 and 5 in Yarnold (1972) and Theorem 1 in Bikyalis (1969). Note that (4) is obtained under the weakest moment conditions possible necessary for a remainder term of order O(n−1 ). The bound for J1 is connected with the difficult problem to determine optimal bounds in lattice point problems for ellipsoids in number theory. The bound (5) follows from a seminal result by Esseen (1945) which implies that J1 = O(n−k/(k+1) ) for all k ≥ 1. In the bound (6) we use the improvement of Esseen’s result which is follows from the bounds on lattice point rests for ellipsoids in G¨otze (2000), Theorem 1.5. Moreover, (6) gives the optimal bound for J1 with respect to the dependence on n provided we do not require any additional properties on V . That means that there are examples of ellipsoids where n|J1 | tends to +∞ as n → +∞ when k : 1 ≤ k ≤ 4. hey follow from known lower bounds in the lattice point problem in number theory by Hardy (1916) for k = 2, by Szeg¨o (1926) for k = 3 and by Walfisz (1927) for k = 4. Note that these lower bounds do not coincide with the upper bounds of order O(n−k/(k+1) ) given in Esseen (1945). For example, in diemnsion k = 2 the lower bound is of the order n−3/4 log log n. At the same time assuming additionally that the matrix V −1 is irrational, (7) shows that J1 becomes negligible compared with the remainder term in (4). For further details see G¨otze (2000). Remark 1. It follows from Theorem 1 that J1 is not always the leading term in the asymptotic expansion (4) refining the limiting distribution Gk (c) as Yarnold and many authors following him erroneously conjectured. Remark 2. Since P(SnT V −1 Sn < c) and Gk (c) both tend to 1 as c → +∞ the difference P(SnT V −1 Sn < c)−Gk (c) is small for large values of c. However Theorem 1 does not reflect this behavior. In order to overcome this gap and to refine Theorem 1 we prove the following theorem involving non-uniform error bounds. Theorem 2. If k > 12 then there exists a positive constant c2 (k) depending on k only such that |P(SnT V −1 Sn < c) − Gk (c)| ≤ 4

c2 (k)E|Z1T V −1/2 |4 . (k 2 + c2 )

(8)

¯1, . . . , X ¯ n the Proof. We need some additional notation. Denote by X ¯ truncations of the random elements Z1 , . . . , Zn respectively, i.e. X1 = Z1 I(|Z1 | < √ n), where I(A) is the indicator function of the event A. We introduce independent identically distributed random elements Z¯1 , . . . , Z¯n such that a ¯ 1 − EX ¯ 1 ) + Y1 . Here distribution of Z¯1 coincides with the distribution of ζ1 (X Y1 has the Gaussian distribution N(0, V /2), ζ1 is a real random variable such that Eζ1 = 0, Eζ12 = 1/2, Eζ13 = 1. For example, we can take ( √ √ − 3−√6 6 with probability 1/2 + 6/6, √ √ ζ1 = 3+ √ 6 6/6. with probability 1/2 − 6 Let ξ be a random variable with density function p(x) = (c8 /8) · [sin(x/8)/(x/8)]8 R with c8 = ( (sin x/x)8 dx)−1 . Then E|ξ|5 < ∞ and characteristic function of ξ equals zero outside of [−1, 1]. Moreover there exists a number α > 1 such that E|ξ|ν ≤ α for ν = 1, . . . , 4. ¯ 1 +. . .+X ¯ n ), H ¯ n = n−1/2 (Z¯1 +. . .+Z¯n ), Ci = {r : r > i}, Put S¯n = n−1/2 (X ¯ n |2 < r 2 ) , δ(i) = sup r 4 P(|S¯n |2 < r 2 ) − P(|H (9) Ci

¯ n |2 + ξT −1 < r 2 ) , δT (r) = P(|S¯n |2 + ξT −1 < r 2 ) − P(|H δT = sup r 4 · δT (r).

(10)

C1

Here |S¯n | denotes the usual norm of S¯n in Rk and T ≥ 1. The following inequality (cf. Lemma 4 in Rotar’ (1970) and Lemma 12 in Sazonov et al. (1988)) plays a crucial role in the proof of Theorem 2: if δ(0) = δ(2) = δ then there exist a positive constant c such that for any T ≥ 1 we have ¯ 3 (V )), δ ≤ c(δT + T −1 · Λ

(11)

where for any integer i > 0 we put ¯ i (V ) = Λ

i Y j=1

5

λ−1 j

!(i−1)/i

,

and λ21 ≥ · · · ≥ λ2i · · · denote the eigenvalues of V . We prove now this inequality. It follows from the definition of δ that there exists r0 ∈ C2 such that ¯ n |2 < r 2 ) ≥ δ/2. r04 P(|S¯n |2 < r02 ) − P(|H 0

There are two possible cases. Case A:   ¯ n |2 < r 2 ) ≥ δ/2 r04 P(|S¯n |2 < r02 ) − P(|H 0 or

Case B:

  ¯ n |2 < r 2 ) − P(|S¯n |2 < r 2 ) ≥ δ/2. r04 P(|H 0 0

Before we consider these two cases separately we construct upper bound for

¯ n − a|2 < r 2 + h2 ), P(r 2 − h1 < |H

where 0 < h1 , h2 < 1/2, r ∈ C2 , h1 < r 2 . ¯ n |2 . Obviously we can write H ¯ n in the form Let p(x) be the density of |H ¯ n = Y + S˜n , where Y ∈ N(0, V /2) and S˜n is a normed sum of i.i.d. random H elements. Then for any x > 0 we have Z ∞ 1 p(x) = e−itx E exp{it|Y + S˜n |2 }dt 2π −∞   √ √ 2 = ES˜n I(|S˜n | < x/2) + I(|S˜n | ≥ x/2) π Z ∞ × e−4itx EY exp{it|2Y + 2S˜n |2 }dt. −∞

Applying now Chebyshev’s inequality and the well-known results for the densities of the norm of Gaussian elements in a Hilbert space (see, e.g. Corollary 13 in Sazonov et al. (1988)) we get for x > 0 ¯ 3 (V )x−2 . p(x) ≤ c˜1 Λ Therefore, Z

r 2 +h2 r 2 −h1

  ¯ 3 (V ) min{(r 2 − h1 )2 , (r 2 + h2 )2 } −1 . (12) p(x)dx ≤ c˜1 (h1 + h2 )Λ

¯ n |4 . Put By Chebyshev’s inequality we have δ ≤ E|S¯n |4 + E|H ¯ n |4 ). h0 = δ(λ1 λ2 λ3 )2/3 /˜ c , where c˜ is a constant such that c˜ > 4(E|S¯n |4 + E|H 6

It implies, in particular, that h0 < 1/4. Moreover, let c˜ be sufficiently large such that c˜ > 2048 c˜1 . Now we consider the case A. If r ∈ C2 , then for h ≤ min{1, r 2 } we have 2 (r ± h)1/2 ∈ C1 . Hence  Z h 2 2 P(|S¯n |2 < r02 + h0 − u) δT ≥ (r0 + h0 ) |u|