We investigate the number of samples required for testing the monotonicity of a distribution with respect to an arbitrary underlying partially ordered set. Our first result is a nearly linear lower bound for the sample complexity of testing monotonicity with respect to the poset consisting of a directed perfect matching. This is the first nearly linear lower bound known for a natural non-symmet...

A general overview of hypothesis testing is given. The Bayesian and distribution-free framework to multiple hypothesis testing and to null hypothesis testing are discussed. Some practical algorithms are introduced, together with associated performance bounds.

are known as Lieb-Thirring bounds and hold true with finite constants Lγ,d if and only if γ ≥ 1/2 for d = 1, γ > 0 for d = 2 and γ ≥ 0 for d ≥ 3. Here and in the following, A± = (|A| ± A)/2 denote the positive and negative parts of a self-adjoint operator A. The case γ > (1 − d/2)+ was shown by Lieb and Thirring in [21]. The critical case γ = 0, d ≥ 3 is known as the Cwikel-Lieb-Rozenblum inequ...

Consider the Schrödinger operators H± = −d/dx ± V (x). We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of H±.

In this article, we derive a new covariance estimate. The estimate has a similar structure as the Brascamp-Lieb inequality and is optimal for ferromagnetic Gaussian measures. It can be naturally applied to deduce decay of correlations of lattice systems of continuous spins. We also discuss the relation of the new estimate with known estimates like a weighted estimate due to Helffer & Ledoux. Th...

Objective To find lower and upper bounds of HIV prevalence in Malawi under mild and intuitive assumptions to assess the importance of the refusal issue in the estimation of HIV prevalence. Methods We derive bounds based on the following two key assumptions: (i) Among those who have never taken an HIV test before, those who refuse to take an HIV test (hereafter “refusers”) have at least as much ...

So far we have talked about the expectation of random variables, and moved on to the variance or the second moment. However, neither of these methods is strong enough to produce the tight bounds needed for some applications. We have already stated that if all you are given is the variance of a r.v., then Chebyshev’s Inequality is essentially the best you can do. Thus, more information is needed...

The bounds testing approach of Pesaran et al. (2001) allows examination of levels relationships between series when their orders of integration are either unknown or uncertain. In this paper it is shown that when the order of integration of a series is uncertain due to the presence of structural change, the bounds testing procedure is subject to severe size distortion and consequently detects s...

We prove tight bounds of Θ(k logk) queries for non-adaptively testing whether a function f : {0,1}→{0,1} is a k-parity or far from any k-parity. The lower bound combines a recent method of Blais, Brody and Matulef [4] to get lower bounds for testing from communication complexity with an Ω(k logk) lower bound for the one-way communication complexity of k-disjointness.

2. Query Uncertainty 4 2.1. Derivation of Special Case: Query Dependent on a Subset . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. Derivation of Special Case: Independent Representation Elements . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3. Linear Query with Independent Elements Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4. Empirical Ent...