Let f and g be differentiable functions on an interval (a, b), and let the derivative g be positive on (a, b). The main result of the paper implies that, if f (a+) = g(a+) = 0 and f g is increasing on (a, b) , then f g is increasing on (a, b). Let −∞ ≤ a < b ≤ ∞. Let f and g be differentiable functions on the interval (a, b). Assume also that the derivative… (More)
An algorithmic description of the dependence of the oscillation pattern of the ratio f g of two functions f and g on the oscillation pattern of the ratio f g of their derivatives is given. This tool is then used in order to refine and extend the Yao-Iyer inequality, arising in bioequivalence studies. The convexity conjecture by Topsøe concerning information… (More)
Let (S 0 , S 1 ,. . .) be a supermartingale relative to a nondecreasing sequence of σ-algebras
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)
This paper continues a series of results begun by a l'Hospital type rule for mono-tonicity, which is used here to obtain refinements of the Eaton-Pinelis inequalities for sums of bounded independent random variables.
(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)
Based on a " monotonicity " analogue of the l'Hospital Rule, monotonicity properties of the relative error are established for a Padé approximation of Mills' ratio.
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)