On the Szekely-Mori Asymmetry Criterion Statistics for Binary Vectors with Independent Components

For random binary vectors the first two moments and limit distributions of statistics in a recently proposed by Székely and Móri criterion of asymmetry of a distribution are investigated.


Introduction
Let X, X 1 , X 2 , . . .be i.i.d.R d -valued random vectors and X denotes a Euclidian norm of vector X.It was shown by Szekely and Mori (2001) that E( X 1 +X 2 − X 1 −X 2 ) ≥ 0 and that E( X 1 + X 2 − X 1 − X 2 ) = 0 if and only if X is symmetrically distributed (i.e., if the distributions of X and −X coincide).
A sequence of statistics was proposed by Szekely and Mori (2001) as a base of a consistent test for symmetry against general alternatives.According to Szekely and Mori (2001) if E( X ) < ∞ and where H 0 is the set of all symmetrical distributions in R d .
Here the equality holds for two-point symmetric distributions where Pr{X 1 = a} = Pr{X 1 = −a} = 1/2 for some a ∈ R d \{0}.Hence, and According to the deMoivre-Laplace theorem which corresponds to (1).

Main Results
We will consider the case when the distribution of the vector X is concentrated on a vertex set (2) PROOF.In view of ( 2) In our case The set A m consists of C m d 2 d elements, the set of all possible pairs (B, C) consists of 2 2d elements.Consequently, where ξ is a random variable with the binomial distribution Bin(d, 1/2). and (The set of possible values of θ d was widened to be valid for all where ξ d has a binomial distribution Bin(d, 1/2).

Notice that the function u
It is easy to check that k(t) for each t ∈ (−∞, ∞) due to the deMoivre-Laplace theorem.By means of Theorem 1 we may find two first moments of the U -statistics T n for uniform distribution on V d .We have Austrian Journal of Statistics, Vol. (2008), No. 1, 137-144 Since for independent vectors X 1 , X 2 , . . .with symmetrical distribution on V d for any a ∈ V d we have Due to (3) U -statistics T n are degenerate ones.Applying the results of Gregory (1977) (see also Korol'uk and Borovskih (1989)) to our case we obtain that if d = const and n → ∞ then distributions of U -statistics T n converge to the distribution of 2 d k=1 c k ν 2 k −1, where ν 1 , ν 2 , . . .are independent random variables with standard Gaussian distribution, c k ≥ 0, c k = 1 and the coefficients c k are the eigenvalues of operator S : Szekely and Mori, 2001).The exact formulas for these coefficients in the case of general d are under investigation.
This results may be used to construct a goodness-of-fit test for generators of random or pseudorandom bits.Now we consider a class of nonuniform distributions on V d corresponding to random vectors with independent components.Theorem 3. If random vectors X = (x 1 , . . ., x d ), Y = (y 1 , . . ., y d ) with values in V d are independent identically distributed with independent components, O. Menshenin and A. M. Zubkov 141 So, It follows from Lyapunov's theorem and conditions of Theorem 3 that 1 d ξ d is asymptotically normal with parameters (5) Consequently, the random variable 2( and Theorem 3 is proved.Theorem 2 is a particular case of Theorem 3, but its statement is simpler.Theorem 4. If the conditions of Theorem 3 are satisfied then there exists a constant PROOF.We will use notations introduced in the proof of Theorem 3.According to (4) The function s(x), x ∈ [0, 1], has quadratic lower and upper bounds: where .
By means of these estimates we obtain Inequality ( 6) is a consequence of ( 8), ( 10) and E( 1 It follows from Theorem 3 that there exists a sequence {α d } such that α d → 0 as To obtain upper bounds we use (9) as follows: where C * = max{C 1 , C 2 } and θ is a random variable, Pr{|θ| ≤ 1} = 1.Therefore, Further, according to the Lyapunov inequality and condition b and equality (7) and Theorem 4 are proven.If X = (x 1 , . . ., x d ), X 1 , X 2 , . . .are independent identically distributed random vectors with values in V d with independent components, then their distribution is asymmetric, U -statistics are nondegenerate and according to Hoeffding (1948) distributions of T n as n → ∞ are asymptotically normal with parameters For finite d and fixed ε 1 , . . ., ε d the parameters of asymptotic normality take concrete values.
Let the conditions of Theorem 3 be now fulfilled.Then we may use the results of Mihailov (1975) (in this paper the central limit theorem for U -statistics was proven by the method of moments under the assumption that the distributions of X i and the form of the kernels may depend on n).In this case T n are asymptotically normal with parameters We omit the proofs of these formulas.