Data driven rank tests for classes of tail alternatives

Tail alternatives describe the frequent occurrence of a non-constant shift in the two-sample problem with a shift function increasing in the tail. The classes of shift functions can be built up using Legendre polynomials. It is important to rightly choose the number of polynomials involved. Here this choice is based on the data, using a modification of Schwarz's selection rule. Given the data driven choice of the model, appropriate rank tests are applied. Simulations show that the new data driven rank tests work very well. While other tests for detecting shift alternatives as Wilcoxon's test may completely break down for important classes of tail alternatives, the new tests have high and stable power. The new tests have also higher power than data driven rank tests for the unconstrained two-sample problem. Theoretical support is obtained by proving consistency of the new tests against very large classes of alternatives, including all common tail alternatives. A simple but accurate approximation of the null distribution makes application of the new tests easy

Central Limit Theorem for a Class of Relativistic Diffusions

Two similar Minkowskian diffusions have been considered, on one hand by Barbachoux, Debbasch, Malik and Rivet ([BDR1], [BDR2], [BDR3], [DMR], [DR]), and on the other hand by Dunkel and H\"anggi ([DH1], [DH2]). We address here two questions, asked in [DR] and in ([DH1], [DH2]) respectively, about the asymptotic behaviour of such diffusions. More generally, we establish a central limit theorem for a class of Minkowskian diffusions, to which the two above ones belong. As a consequence, we correct a partially wrong guess in [DH1].Comment: 20 page

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. Our approach provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results and comparisons with existing results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory, number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of "weakly dependent" random variables. The large number as well as the variety of examples hint at a universality class for second order fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new section on mod-Gaussian convergence coming from the factorization of the generating function ; the multi-dimensional results have been moved to a forthcoming paper ; and the introduction has been reworke

Long-Term Followup after Electrocautery Transurethral Resection of the Prostate for Benign Prostatic Hyperplasia

Introduction. For decades, transurethral resection of the prostate (TURP) has been the “gold standard” operation for benign prostatic hyperplasia (BPH) but is based mainly on historic data. The historic data lacks use of validated measures and current TURP differs significantly from that performed 30 years ago. Methods. Men who had undergone TURP between 2001 and 2005 were reviewed. International prostate symptom score (IPSS), quality of life (QOL) and peak urinary flow rate (Qmax⁡), and postvoid residual (PVR) were recorded. Operative details and postoperative complications were documented. Patients were then invited to attend for repeat assessment. Results. 91 patients participated. Mean follow-up time was 70 months. Mean follow-up results were IPSS—7; QoL—1.5; Qmax⁡—23 mL/s; PVR—45 mL. These were an improvement from baseline of 67%, 63%, 187%, and 80%, respectively. Early complication rates were low, with no blood transfusions, TUR syndrome, or deaths occurring. Urethral stricture rate was higher than anticipated at 14%. Conclusion. This study shows modern TURP still produces durable improvement in voiding symptoms which remains comparable with historic studies. This study, however, found a marked drop in early complications but, conversely, a higher than expected incidence of urethral strictures

Palm pairs and the general mass-transport principle

We consider a lcsc group G acting properly on a Borel space S and measurably on an underlying sigma-finite measure space. Our first main result is a transport formula connecting the Palm pairs of jointly stationary random measures on S. A key (and new) technical result is a measurable disintegration of the Haar measure on G along the orbits. The second main result is an intrinsic characterization of the Palm pairs of a G-invariant random measure. We then proceed with deriving a general version of the mass-transport principle for possibly non-transitive and non-unimodular group operations first in a deterministic and then in its full probabilistic form.Comment: 26 page

On Convergence Properties of Shannon Entropy

Convergence properties of Shannon Entropy are studied. In the differential setting, it is shown that weak convergence of probability measures, or convergence in distribution, is not enough for convergence of the associated differential entropies. A general result for the desired differential entropy convergence is provided, taking into account both compactly and uncompactly supported densities. Convergence of differential entropy is also characterized in terms of the Kullback-Liebler discriminant for densities with fairly general supports, and it is shown that convergence in variation of probability measures guarantees such convergence under an appropriate boundedness condition on the densities involved. Results for the discrete setting are also provided, allowing for infinitely supported probability measures, by taking advantage of the equivalence between weak convergence and convergence in variation in this setting.Comment: Submitted to IEEE Transactions on Information Theor

Percolation in invariant Poisson graphs with i.i.d. degrees

Let each point of a homogeneous Poisson process in R^d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme that is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components

On q-Gaussians and Exchangeability

The q-Gaussians are discussed from the point of view of variance mixtures of normals and exchangeability. For each q< 3, there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables; thereby, providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of yielding process versions which are naturally q-Brownian motions. Explicit mixing distributions for q-Gaussians should facilitate applications to areas such as option pricing. The model might provide insight into the study of superstatistics.Comment: 14 page

Identification of Regions Important for Resistance and Signalling within the Antimicrobial Peptide Transporter BceAB of <em>Bacillus subtilis</em>

In the low-G+C-content Gram-positive bacteria, resistance to antimicrobial peptides is often mediated by so-called resistance modules. These consist of a two-component system and an ATP-binding cassette transporter and are characterized by an unusual mode of signal transduction where the transporter acts as a sensor of antimicrobial peptides, because the histidine kinase alone cannot detect the substrates directly. Thus, the transporters fulfill a dual function as sensors and detoxification systems to confer resistance, but the mechanistic details of these processes are unknown. The paradigm and best-understood example for this is the BceRS-BceAB module of Bacillus subtilis, which mediates resistance to bacitracin, mersacidin, and actagardine. Using a random mutagenesis approach, we here show that mutations that affect specific functions of the transporter BceAB are primarily found in the C-terminal region of the permease, BceB, particularly in the eighth transmembrane helix. Further, we show that while signaling and resistance are functionally interconnected, several mutations could be identified that strongly affected one activity of the transporter but had only minor effects on the other. Thus, a partial genetic separation of the two properties could be achieved by single amino acid replacements, providing first insights into the signaling mechanism of these unusual modules

Maximizing the Conditional Expected Reward for Reaching the Goal

The paper addresses the problem of computing maximal conditional expected accumulated rewards until reaching a target state (briefly called maximal conditional expectations) in finite-state Markov decision processes where the condition is given as a reachability constraint. Conditional expectations of this type can, e.g., stand for the maximal expected termination time of probabilistic programs with non-determinism, under the condition that the program eventually terminates, or for the worst-case expected penalty to be paid, assuming that at least three deadlines are missed. The main results of the paper are (i) a polynomial-time algorithm to check the finiteness of maximal conditional expectations, (ii) PSPACE-completeness for the threshold problem in acyclic Markov decision processes where the task is to check whether the maximal conditional expectation exceeds a given threshold, (iii) a pseudo-polynomial-time algorithm for the threshold problem in the general (cyclic) case, and (iv) an exponential-time algorithm for computing the maximal conditional expectation and an optimal scheduler.Comment: 103 pages, extended version with appendices of a paper accepted at TACAS 201