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1 by Iosif Pinelis We consider Hotelling's T 2 statistic for an arbitrary d-dimensional sample. If the sampling is not too deterministic or inhomogeneous, then under zero means hypothesis, T 2 tends to χ 2 d in distribution. We are showing that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ(More)
(A) Closed and bounded centrally symmetric sets S in E n can also be characterized by the property that for each n-dimensional simplex T with vertices in S there is a translate of −T also having its vertices in S. This, of course, is a consequence of Theorem 1, since the three-point sets belonging to S are subsets of the (n + 1)-point sets with vertices in(More)
Elsewhere we developed rules for the monotonicity pattern of the ratio r := f /g of two differentiable functions on an interval (a, b) based on the monotonicity pattern of the ratio ρ := f /g of the derivatives. Those rules are applicable even more broadly than l'Hospital's rules for limits, since in general we do not require that both f and g, or either of(More)
The most widely used evolutionary model for phylogenetic trees is the equal-rates Markov (ERM) model. A problem is that the ERM model predicts less imbalance than observed for trees inferred from real data; in fact, the observed imbalance tends to fall between the values predicted by the ERM model and those predicted by the(More)
Let η 1 , η 2 ,. .. be independent (but not necessarily identically distributed) zero-mean random variables (r.v.'s) such that |η i | 1 almost surely for all i, and let Z stand for a standard normal r.v. Let a 1 , a 2 ,. .. be any real numbers such that a 2 1 + a 2 2 + · · · = 1. It is shown that then for all x > 0 P(|a 1 η 1 + a 2 η 2 +. .. | x) < P(|Z| x(More)
Let BS 1 ,. .. , BSn be independent identically distributed random variables each having the standardized Bernoulli distribution with parameter p ∈ (0, 1). Let m * (p) := (1 + p + 2 p 2)/(2 p − p 2 + 4 p 2) if 0 < p 1 2 and m * (p) := 1 if 1 2 p < 1. Let m m * (p). Let f be such a function that f and f ′′ are nondecreasing and convex. Then it is proved that(More)