Comparing distributions in ℝn

Consider two independent finite samples {x1i} i=1k and {x 2i} i=1k of size k from distributions F1 and F2 in ℝn. Their sample covariances at point q ∈ M, which we refer to as an observation point, are

∑k -→ -→ ˆΣ (q ) = 1- (qxi)(qxi)′, s = 1,2. s k s s i=1
Let Σs(q) and Ω ∈ ℝn2×n2 be the mean and covariance of Z = (-→ qX)(-→ qX)′, for X ~ F s. Thus, we assume that the mean and covarince of the tensor valued trandom variable Z exist. As an application of the Central Limit Theorem we obtained the asymptotic
√-- k (ˆΣ1(q)ˆΣ -1(q) - In) ⇝ Nn ×n(0,2Ω ⊗ Σ -1(q)), 2
provided that Σ1(q) = Σ2(q) = Σ(q). Further application of delta method gives us
√-- -1 ′ T -1 ′ ξk(q;h) := k (h(ˆΣ1(q)ˆΣ 2 (q))- h (In )) ⇝ N (0,2[h(In)] (Ω ⊗Σ (q))[h (In)]),
(1)

for similarity invariant h which gradient h′(.) ∈ ℝn2 is continuous and does not vanish at In. For example, ξk(q; tr) = √ -- k(tr(ˆ Σ1(q) ˆ Σ2-1(q)) - n) and ξk(q; det) = √ -- k(det(ˆ Σ1(q) ˆ Σ2-1(q)) - 1).

We apply bootstrapping method to utilize (1). For k′ < k, let ym,m = 1,...,M be instances of statistic ξk(q; h) based on subsamples of size k′ of the initial k-samples. The observation point q is chosen to be the sample mean of the combined x1 and x2 samples. Then, according to (1), ξ = ¯y ∕s.e.(y) goes to N(0, 1) in distribution as k →∞.

Another hypothesis of interest compares the usual covariances, defined at the mean points H2 : Σ1(μ1) = Σ2(μ2). The corresponding likelihood ratio statistics against the alternative Ha : Σ1(μ1)⁄=Σ2(μ2) is

|ˆΣ (ˆμ )|k∕2|Σˆ (ˆμ )|k∕2 λ = ---1--1------2--2----. |Σˆ1 (ˆμ1) + ˆΣ2(ˆμ2)|k
(2)

The exact distribution of λ is a product of independent Beta-distributions but can be approximated by chi-squared ones.

Comparing the peformance of ξ and λ statistics

The following applet simulates distributions from different families and calculates ξ and λ statistics. The results are reported in term of p-values, Xi-pval for ξ and L-pval for λ. The user can choose the dimension n, sample size k and whether the two samples are equally distributed or not. Sub-sample size is fixed to k′ = k∕2 and M = k∕4. Possible choices for function h are the trace, determinant and h(A) = tr(log(A)). Shown are the two k-samples in red and blue and the observation point in green.

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