Variance Inequality Research Articles

Invariances in variance estimates

We provide variants and improvements of the Brascamp–Lieb variance inequality which takes into account the invariance properties of the underlying measure. This is applied to spectral gap estimates for log-concave measures with many symmetries and to non-interacting conservative spin systems.

A simple variance inequality for [formula omitted]-statistics of a Markov chain with applications

We establish a simple variance inequality for U-statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large numbers for U-statistics of a Markov Chain under conditions which are close to being optimal.

Robust Means Modeling

This study proposes robust means modeling (RMM) approaches for hypothesis testing of mean differences for between-subjects designs in order to control the biasing effects of nonnormality and variance inequality. Drawing from structural equation modeling (SEM), the RMM approaches make no assumption of variance homogeneity and employ robust estimation/rescaling strategies in order to alleviate reliance on normality. A Monte Carlo simulation is conducted to compare the Type I error rate and the power of the proposed six RMM test statistics to five analysis of variance (ANOVA)-based statistics, the latter of which have also employed trimmed means and Winsorized variances to enhance robustness. Various simulation factors manipulated include variance inequality, sample-size pairings with group variances, degree of nonnormality, alpha level for hypothesis tests, and effect size. Results show that the proposed RMM methods are indeed superior to the traditional ANOVA-based methods.

Barycenters in Alexandrov spaces of curvature bounded below

We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metric spaces of curvature bounded above. We prove the reverse variance inequality, and show that the push forward of a measure to the tangent cone at its barycenter has the flat support.

On matrix variance inequalities

Olkin and Shepp [2005, A matrix variance inequality. J. Statist. Plann. Inference 130, 351–358] presented a matrix form of Chernoff's inequality for Normal and Gamma (univariate) distributions. We extend and generalize this result, proving Poincaré-type and Bessel-type inequalities, for matrices of arbitrary order and for a large class of distributions.

An asymptotic variance inequality for instrumental variable estimators signaling proportional bias increases

An asymptotic variance inequality for instrumental variable (IV) estimators is proposed, which suggests a critical variance that signals proportional increases in the bias of IV estimators through the augmentation of a set of instruments.

Revealing Hidden Einstein-Podolsky-Rosen Nonlocality

Steering is a form of quantum nonlocality that is intimately related to the famous Einstein-Podolsky-Rosen (EPR) paradox that ignited the ongoing discussion of quantum correlations. Within the hierarchy of nonlocal correlations appearing in nature, EPR steering occupies an intermediate position between Bell nonlocality and entanglement. In continuous variable systems, EPR steering correlations have been observed by violation of Reid's EPR inequality, which is based on inferred variances of complementary observables. Here we propose and experimentally test a new criterion based on entropy functions, and show that it is more powerful than the variance inequality for identifying EPR steering. Using the entropic criterion our experimental results show EPR steering, while the variance criterion does not. Our results open up the possibility of observing this type of nonlocality in a wider variety of quantum states.

Properties of a randomization test for multifactor comparisons of groups

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.

Poisson process Fock space representation, chaos expansion and covariance inequalities

We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris–FKG-inequalities for monotone functions of η.

Variance inequalities using first derivatives

We develop variance inequalities on functions of random variables using mild information such as the first derivatives. Specifically, when f and g are absolutely continuous functions, we show that Var [ f ( X ) ] ≤ Var [ g ( X ) ] for any random variable X if and only if f is a function of g and | f ′ | ≤ | g ′ | almost everywhere. Our results provide a form of master inequality from which other variance inequalities can be obtained.

Concentration inequalities for Markov processes via coupling

We obtain moment and Gaussian bounds for general coordinate-wise Lipschitz functions evaluated along the sample path of a Markov chain. We treat Markov chains on general (possibly unbounded) state spaces via a coupling method. If the first moment of the coupling time exists, then we obtain a variance inequality. If a moment of order $1+a$ $(a > 0)$ of the coupling time exists, then depending on the behavior of the stationary distribution, we obtain higher moment bounds. This immediately implies polynomial concentration inequalities. In the case that a moment of order $1+ a$ is finite, uniformly in the starting point of the coupling, we obtain a Gaussian bound. We illustrate the general results with house of cards processes, in which both uniform and non-uniform behavior of moments of the coupling time can occur.

Comparing Sensory Experience in Bitter Taste Perception of Phenylthiocarbamide within and between Human Twins and Singletons: Intrapair Differences in Thresholds and Genetic Variance Estimates

Quantitative genetic studies revealed that not all of the phenotypic variance in PTC taste perception is heritable. To study quantitative variations in PTC tasting ability in twins and to estimate heritability of PTC taste perception on the taste of twin data on males and females sexes separately. The data for PTC taste sensitivity following the classic method of Harris & Kalmus (1949) were collected on a sample of 141 twin pairs (66 MZ and 75 DZ) and 275 singletons (128 males and 147 females) from Chandigarh, India. Genetic analyses were performed following Christian (1979), Donner (1986) and Sham (1998). Frequency of non-tasters was similar in twins (33 %) and singletons (32 %), but significant sex differences were observed. No differences were found between zygosities for mean thresholds. Similarly, no evidence of variance heterogeneity and environmental covariance was seen between zygosities. Since no basic assumption of the twin method was found violated, within-pair estimates of genetic variance would be unbiased. These estimates were highly significant in both males and females. However, dominance and additive components of genetic variance were found to differ between sexes. PTC thresholds do not seem to be significantly affected by environmental factors as no variance inequality was observed between twin zygosities. Intensity of bitterness (scalar dimensions) of PTC is a separate trait having no commonality with the genetic basis of recognition threshold for PTC tasting ability. The receptors recognizing bitter taste are different from the receptors determining intensity of taste. The absolute difference between co-twins in PTC thresholds can be used as a simple tool in the twin zygosity diagnosis. The results show that none of the MZ co-twins had manifested difference of more than 3 in their PTC threshold.

A matrix version of Chernoff inequality

An interesting result from the point of view of upper variance bounds is the inequality of Chernoff [Chernoff, H., 1981. A note on an inequality involving the normal distribution. Annals of Probability 9, 533–535]. Namely, that for every absolutely continuous function g with derivative g ′ such that Var { | g ( ξ ) | } < ∞ , and for standard normal r.v. ξ , Var ( | g ( ξ ) | ) ≤ E { ( g ′ ( ξ ) ) 2 } . Both the usefulness and simplicity of this inequality have generated a great deal of extensions, as well as alternative proofs. Particularly, Olkin and Shepp [Olkin, I., Shepp, L., 2005. A matrix variance inequality. Journal of Statistical Planning and Inference 130, 351–358] obtained an inequality for the covariance matrix of k functions. However, all the previous papers have focused on univariate function and univariate random variable. We provide here a covariance matrix inequality for multivariate function of multivariate normal distribution.

Local tail bounds for functions of independent random variables

It is shown that functions defined on (0, 1, . . r - 1} n satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger local sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand's [Ann. Probab. 22 (1994) 1576-1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on (0, 1, ..., r - 1) n for r ≥ 2.

Bell Inequalities for Continuous-Variable Correlations

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at least second moments. The derivation stems from a simple variance inequality by setting local commutators to zero. We show that above a constant detector efficiency threshold, the continuous-variable Bell violation can survive even in the macroscopic limit of large n. This method can be used to derive other well-known Bell inequalities, shedding new light on the importance of non-commutativity for violations of local realism.

A Matrix Variance Inequality for k-Functions

In this paper a course of solving variational problem is considered. [2] obtained what appears to be specialized inequality for a variance, namely, that for a standard normal variable X , Var[g(x)] \ge E[g'(x)]2. However both of the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. [5] had focused on a result of two random variables for the normal and gamma distribution. They obtained the result of normal distribution with k functions, without proving and the proof is presented here. This paper also extend the result obtained by [5] to the k functions for the gamma distribution. Keywords: Normal Distribution; Gamma Distribution; Laguerre Family; Hermite Polynomials.

90.88 Poisson’s other distribution and a variance inequality

90.88 Poisson’s other distribution and a variance inequality

A variance inequality ensuring that a pre-distance matrix is Euclidean

We derive a variance inequality on the squares of the pre-distances between n objects ensuring that the corresponding pre-distance matrix is Euclidean. The result is discussed in relation with some transformations making a pre-distance matrix Euclidean.

Matrix variance inequalities for multivariate distributions

We derive generalizations of the Hoeffding identity for multivariate random vectors and study some measures of dependence in the multidimensional case. In addition, we derive bounds on the covariance matrix for some multivariate distributions in the sense of Loewner ordering.

A matrix variance inequality

In the course of solving a variational problem Chernoff (Ann. Probab. 9 (1981) 533) obtained what appears to be a specialized inequality for a variance, namely, that for a standard normal variable X , Var [ g ( X ) ] ⩾ E [ g ′ ( X ) ] 2 . However, both the simplicity and usefulness of the inequality has generated a plethora of extensions, as well as alternative proofs. All previous papers have focused on a single function. We provide here an inequality for the covariance matrix of k functions, which leads to a matrix inequality in the sense of Loewner.