Learn More
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)
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)
The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account positive-part third moments, and at that significantly improved by using, instead of the class of all increasing exponential functions, a much larger class of generalized moment functions. The resulting bounds have certain optimality(More)