Relations between the first four moments
Let m_j denote the jth moment of a random variable X. One of the results is the inequality m_3 \leq c m_4^{3/4} with the best possible factor c over all random variables X with a non-positive mean.
math.PR↗
arXiv subjects
Iosif Pinelis
Publications and source records attributed to Iosif Pinelis.
Optimal re-centering bounds, with applications to Rosenthal-type concentration of measure inequalities
For any nonnegative Borel-measurable function f such that f(x)=0 if and only if x=0, the best constant c_f in the inequality E f(X-E X) \leq c_f E f(X) for all random variables X with a finite mean is obtained. Properties of the constant c_f in the case when f=|.|^p are studied. Applications to concentration of measure in the form of Rosenthal-type bounds on the moments of separately Lipschitz functions on product spaces are given.
math.PR↗
Improved nonuniform Berry--Esseen-type bounds
New nonuniform Berry--Esseen-type bounds for sums of independent random variables are obtained, motivated by recent studies concerning such bounds for nonlinear statistics. The proofs are based on the Chen--Shao concentration techniques within the framework of the Stein method.
math.PR↗
An asymptotically Gaussian bound on the Rademacher tails
An explicit upper bound on the tail probabilities for the normalized Rademacher sums is given. This bound, which is best possible in a certain sense, is asymptotically equivalent to the corresponding tail probability of the standard normal distribution, thus affirming a longstanding conjecture by Efron. Applications to sums of general centered uniformly bounded independent random variables and to the Student test are presented.
math.PR↗
Tchebycheff systems and extremal problems for generalized moments: a brief survey
A brief presentation of basics of the theory of Tchebycheff and Markov systems of functions and its applications to extremal problems for integrals of such functions is given. The results, as well as all the necessary definitions, are stated in most common terms. This work is motivated by specific applications in probability and statistics. A few related questions are also briefly discussed, including the one on the existence of a Tchebycheff system on a given topological space.
math.OC↗
Geometrically convergent sequences of upper and lower bounds on the Wallis ratio and related expressions
Sequences of algebraic upper and lower bounds on the Wallis ratio are given with the relative errors that converge to 0 geometrically and uniformly on any interval of the form [x_0,\infty) for x_0>-\frac12; moreover, the relative and absolute errors converge to 0 as x\to\infty. These conclusions are based on corresponding results for the digamma function ψ:=\Ga'/\Ga. Relations with other relevant results are discussed, as well as the corresponding computational aspects. This work was motivated by studies of exact bounds involving the Student probability distribution.
math.CA↗
Between Chebyshev and Cantelli
A family of exact upper bounds interpolating between Chebyshev's and Cantelli's is presented.
math.PR↗
Schur^2-concavity properties of Gaussian measures, with applications to hypotheses testing
The main results imply that the probability P(\ZZ\in A+þ) is Schur-concave/Schur-convex in (þ_1^2,\dots,þ_k^2) provided that the indicator function of a set A in \R^k is so, respectively; here, þ=(þ_1,\dots,þ_k) in \R^k and \ZZ is a standard normal random vector in \R^k. Moreover, it is shown that the Schur-concavity/Schur-convexity is strict unless the set A is equivalent to a spherically symmetric set. Applications to testing hypotheses on multivariate means are given.
math.PR↗
Asymptotic efficiency of p-mean tests for means in high dimensions
The asymptotic efficiency, ARE_{p,2}, of the tests for multivariate means theta in \R^d based on the p-means relative to the standard 2-mean, (approximate) likelihood ratio test (LRT), is considered for large dimensions d. It turns out that these p-mean tests for p>2 may greatly outperform the LRT while never being significantly worse than the LRT. For instance, ARE_{p,2} for p=3 varies from about 0.96 to \infty, depending on the direction of the alternative mean vector theta_1, for the null hypothesis H_0: theta=\0. These results are based on a complete characterization, under certain general and natural conditions, of the varying pairs (n,theta_1) for which the values of the power of the p-mean test for theta=\0 and theta=theta_1 tend, respectively, to prescribed values alpha and beta. The proofs use such classic results as the Berry-Esseen bound in the central limit theorem and the conditions of convergence to a given infinitely divisible distribution, as well as a recent result by the author on the Schur^2-concavity properties of Gaussian measures.
math.ST↗
On the Non-degeneracy of Kendall's and Spearman's Correlation Coefficients
Hoeffding proved that Kendall's and Spearman's nonparametric measures of correlation between two continuous random variables X and Y are each asymptotically normal with an asymptotic variance of the form sigma^2/n -- provided the non-degeneracy condition sigma^2>0 holds, where sigma^2 is a certain (always nonnegative) expression which is determined by the joint distribution (say mu) of X and Y. Sufficient conditions for sigma^2>0 in terms of the support set (say S) of mu are given, the same for both correlation statistics. One of them is that there exist a rectangle with all its vertices in S, sides parallel to the X and Y axes, and an interior point also in S. Another sufficient condition is that the Lebesgue measure of S be nonzero.
math.ST↗
Exact lower bounds on the exponential moments of Winsorized and truncated random variables
Exact lower bounds on the exponential moments of min(y,X) and XI{X<y} are provided given the first two moments of a random variable X. These bounds are useful in work on large deviations probabilities and nonuniform Berry-Esseen bounds, when the Cramér tilt transform may be employed. Asymptotic properties of these lower bounds are presented. Comparative advantages of the Winsorization min(y,X) over the truncation XI{X<y} are demonstrated.
math.PR↗
Positive-part moments via the Fourier-Laplace transform
Integral expressions for positive-part moments E X_+^p (p>0) of random variables X are presented, in terms of the Fourier-Laplace or Fourier transforms of the distribution of X. A necessary and sufficient condition for the validity of such an expression is given. This study was motivated by extremal problems in probability and statistics, where one needs to evaluate such positive-part moments.
math.PR↗
Monotonicity properties of the asymptotic relative efficiency between common correlation statistics in the bivariate normal model
Pearson's is the most common correlation statistic, used mainly in parametric settings. Most common among nonparametric correlation statistics are Spearman's and Kendall's. We show that for bivariate normal i.i.d. samples the pairwise asymptotic relative efficiency between these three statistics depends monotonically on the population correlation coefficient. This monotonicity is a corollary to a stronger result. The proofs rely on the use of l'Hospital-type rules for monotonicity patterns.
math.ST↗
Exponential deficiency of convolutions of densities
If a probability density p(\x) (\x\in\R^k) is bounded and R(t) := \int \exp(t\ell(\x)) \d\x < \infty for some linear functional \ell and all t\in(0,1), then, for each t\in(0,1) and all large enough n, the n-fold convolution of the t-tilted density p_t(\x) := \exp(t\ell(\x)) p(\x)/R(t) is bounded. This is a corollary of a general, "non-i.i.d." result, which is also shown to enjoy a certain optimality property. Such results are useful for saddle-point approximations.
math.PR↗
On the Bennett-Hoeffding inequality
The well-known Bennett-Hoeffding bound for sums of independent random variables is refined, by taking into account truncated third moments, and at that also improved by using, instead of the class of all increasing exponential functions, the much larger class of all generalized moment functions f such that f and f" are increasing and convex. It is shown that the resulting bounds have certain optimality properties. Comparisons with related known bounds are given. The results can be extended in a standard manner to (the maximal functions of) (super)martingales.
math.PR↗
Optimal two-value zero-mean disintegration of zero-mean random variables
For any continuous zero-mean random variable (r.v.) X, a reciprocating function r is constructed, based only on the distribution of X, such that the conditional distribution of X given the (at-most-)two-point set {X,r(X)} is the zero-mean distribution on this set; in fact, a more general construction without the continuity assumption is given in this paper, as well as a large variety of other related results, including characterizations of the reciprocating function and modeling distribution asymmetry patterns. The mentioned disintegration of zero-mean r.v.'s implies, in particular, that an arbitrary zero-mean distribution is represented as the mixture of two-point zero-mean distributions; moreover, this mixture representation is most symmetric in a variety of senses. Somewhat similar representations -- of any probability distribution as the mixture of two-point distributions with the same skewness coefficient (but possibly with different means) -- go back to Kolmogorov; very recently, Aizenman et al. further developed such representations and applied them to (anti-)concentration inequalities for functions of independent random variables and to spectral localization for random Schroedinger operators. One kind of application given in the present paper is to construct certain statistical tests for asymmetry patterns and for location without symmetry conditions. Exact inequalities implying conservative properties of such tests are presented. These developments extend results established earlier by Efron, Eaton, and Pinelis under a symmetry condition.
math.PR↗
Extremal Probabilistic Problems and Hotelling's T^2 Test Under Symmetry Condition
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 show that a test for the orthant symmetry condition introduced by Efron can be constructed which does not essentially differ from the one based on χ^2_d and at the same time is applicable not only for large random homogeneous samples but for all multidimensional samples without exceptions. The main assertions have the form of inequalities, not that of limit theorems; these inequalities are exact representing the solutions to certain extremal problems. Let us also mention an auxiliary result which itself may be of interest: χ_d-(d-1)^{1/2} decreases in distribution in d to its limit N(0,1/2).
math.ST↗
Polygon Convexity: Another O(n) Test
An n-gon is defined as a sequence ¶=(V_0,...,V_{n-1}) of n points on the plane. An n-gon ¶is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of ¶coincides with the union of the edges [V_0,V_1],...,[V_{n-1},V_0]; if at that no three vertices of ¶are collinear then ¶is called strictly convex. We prove that an n-gon ¶with n\ge3 is strictly convex if and only if a cyclic shift of the sequence (\al_0,...,\al_{n-1})\in[0,2π)^n of the angles between the x-axis and the vectors V_1-V_0,...,V_0-V_{n-1} is strictly monotone. A ``non-strict'' version of this result is also proved.
cs.CG↗