Asymptotics of multivariate randomness statistics (Q1907828)

Assume for each \(i = 1, \dots, q\) a small sample of \(n(i)\) observations is collected. Let \(N = \sum^q_{i = 1} n(i)\), \(\overline F^i\) be the empirical distribution function of the \(i\)-th sample, \(\overline F^*\) be the empirical distribution function of all the samples taken together. The problem is to test if these samples are homogeneous. \textit{E. L. Lehmann} [Ann. Math. Stat. 22, 165-179 (1951; Zbl 0045.40903)] considered the problem of testing the equality of the distributions of \(q\) univariate samples. He proposed the statistic \[ \sum^q_{i = 1} \int \bigl( F^i (s) - \overline F^* (s) \bigr)^2 d \overline F^* (s) \] which is an element of the Crámer-Von Mises family. \textit{J. Kiefer} [ibid. 30, 420-447 (1959; Zbl 0134.36707)] considered the asymptotics of such statistics when \(n(i) \to \infty\) while \(q\) stays fixed. For \(q \to \infty\) while the \(n(i)\)'s stay small and for univariate observations \textit{D. McDonald} [Can. J. Stat. 19, No. 2, 209-217 (1991; Zbl 0729.60019)] introduced the following class of randomness statistics \[ S_q = \sum^q_{i = 1} \int k_q \bigl( N,i, F^i(s), \overline F^* (s) \bigr) n(i) dF^i (s). \] He showed that the statistic \((S_q - ES_q)/ \sqrt {\text{Var} (S_q)}\) converges in distribution to a standard normal when \(q\) goes to infinity while the \(n(i)\)'s remain small. This paper generalizes the class of randomness statistics to include multivariate observations. It also studies the asymptotic behavior of such statistics when \(q\) tends to infinity and the \(n(i)\)'s remain fixed.