Weighted-mean regions of a probability distribution

In this paper we investigate a new class of central regions for probability distributions on R^d, called weighted-mean regions. Their restrictions to an empirical distribution are the weighted-mean trimmed regions investigated by Dyckerhoff and Mosler (2011) for d-variate data. Furthermore a new class of stochastic orderings of variability, the weighted-mean orderings, is introduced.     

Integral trimmed regions

We define a new family of central regions with respect to a probability measure. They are induced by a set or a family of sets of functions and we name them integral trimmed regions. The halfspace trimming and the zonoid trimming are particular cases of integral trimmed regions. We focus our work on the derivation of properties of such integral trimmed regions from conditions satisfied by the generating classes of functions. Further we show that, under mild conditions, the population integral trimmed region of a given depth can be characterized in terms of certain regions based on empirical distributions.     

One-dimensional p-p plots and precedence tests for point processes on ℝ d

Given two univariate distributions F and G, the hypothesis that F = G can be tested against the alternative that F < G using a precedence test statistic, or more generally the procentile-procentile plot of G against F. In this paper we propose a d-dimensional generalization of the precedence test and the corresponding p-p plot that is appropriate for a test of F = G against the alternative that F ≺ G for various stochastic orders on ℝ ^d . The test statistic is consistent and the p-p plot process is shown to converge to a Gaussian limit. Under the null hypothesis, the p-p plot can be regarded as a two-sample version of Kendall’s process, and in one case the resulting test statistic has the novelty of being distribution free.      

Multivariate Gini Indices

Two extensions of the univariate Gini index are considered:R_D, based on expected distance between two independent vectors from the same distribution with finite meanμ ^d; andR_V, related to the expected volume of the simplex formed fromd+1 independent such vectors. A new characterization ofR_Das proportional to a univariate Gini index for a particular linear combination of attributes relates it to the Lorenz zonoid. TheLorenz zonoidwas suggested as a multivariate generalization of the Lorenz curve.R_Vis, up to scaling, the volume of the Lorenz zonoid plus a unit cube of full dimension. Whend=1, bothR_DandR_Vequal twice the area between the usual Lorenz curve and the line of zero disparity. Whend>1, they are different, but inherit properties of the univariate Gini index and are related via the Lorenz zonoid:R_Dis proportional to the average of the areas of some two-dimensioned projections of the lift zonoid, whileR_Vis the average of the volumes of projections of the Lorenz zonoid over all coordinate subspaces.     

Stochastic Comparisons and Dependence among Concomitants of Order Statistics

Let (X_i, Y_i) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X_(r) denotes the rth ordered X-variate then the Y-variate, Y_[r], paired with X_(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and dependence among concomitants of order statistics under different types of dependence between the parent random variables X and Y. The results obtained apply to any distribution with monotone dependence between X and Y. In particular, when X and Y are likelihood ratio dependent, it is shown that the successive concomitants of order statistics are increasing according to likelihood ratio ordering and they are TP₂ dependent in pairs. If we assume that the conditional hazard rate of Y given X=x is decreasing in x, then the concomitants are increasing according to hazard rate ordering and are dependent according to the right corner set increasing property. Finally, it is proved that if Y is stochastically increasing in X, then the concomitants of order statistics are stochastically increasing and are associated. Analogous results are obtained when the variables X and Y are negatively dependent. We also prove that if the hazard rate of the conditional distribution of Y given X=x is decreasing in x and y, then the concomitants have DFR (decreasing failure rate) distributions and are ordered according to dispersive ordering.     

Weighted-mean trimming of multivariate data

A general notion of trimmed regions for empirical distributions in d-space is introduced. The regions are called weighted-mean trimmed regions. They are continuous in the data as well as in the trimming parameter. Further, these trimmed regions have many other attractive properties. In particular they are subadditive and monotone which makes it possible to construct multivariate measures of risk based on these regions. Special cases include the zonoid trimming and the ECH (expected convex hull) trimming. These regions can be exactly calculated for any dimension. Finally, the notion of weighted-mean trimmed regions extends to probability distributions in d-space, and a law of large numbers applies.     

Best-possible bounds on sets of bivariate distribution functions

The fundamental best-possible bounds inequality for bivariate distribution functions with given margins is the Fréchet–Hoeffding inequality: If H denotes the joint distribution function of random variables X and Y whose margins are F and G, respectively, then max(0,F(x)+G(y)−1)H(x,y)min(F(x),G(y)) for all x,y in [−∞,∞]. In this paper we employ copulas and quasi-copulas to find similar best-possible bounds on arbitrary sets of bivariate distribution functions with given margins. As an application, we discuss bounds for a bivariate distribution function H with given margins F and G when the values of H are known at quartiles of X and Y.     

On the asymptotic maximal density of a set avoiding solutions to linear equations modulo a prime

Given a finite family F\mathcal{F} of linear forms with integer coefficients, and a compact abelian group G, an F\mathcal{F}-free set in G is a measurable set which does not contain solutions to any equation L(x)=0 for L in F\mathcal{F}. We denote by d_F(G)d_{\mathcal{F}}(G) the supremum of μ(A) over F\mathcal{F}-free sets A⊂G, where μ is the normalized Haar measure on G. Our main result is that, for any such collection F\mathcal{F} of forms in at least three variables, the sequence d_F(\mathbb Z_p)d_{\mathcal{F}}({\mathbb {Z}}_{p}) converges to d_F(\mathbb R/\mathbb Z)d_{\mathcal{F}}({\mathbb {R}}/{\mathbb {Z}}) as p→∞ over primes. This answers an analogue for ℤ _p of a question that Ruzsa raised about sets of integers.     

Centrality as a gradual notion: A new bridge between fuzzy sets and statistics

This paper aims at formalizing the intuitive idea that some points are more central in a probability distribution than others. Our proposal relies on fuzzy events to define a fuzzy set of central points for a distribution (or a family of distributions, including imprecise probability models). This framework has a natural interpretation in terms of fuzzy logic and unifies many known notions from statistics, including the mean, median and mode, interquantile intervals, the Lorenz curve, the halfspace median, the zonoid and lift zonoid, the coverage function and several expectations and medians of random sets, and the Choquet integral against an infinitely alternating or infinitely monotone capacity.     

Homogeneous Nilpotent Matrices in Two Variables

We give a general geometrical procedure to construct nilpotent morphisms Φ : F → F(d), with F a vector bundle on ?¹, obtaining an analog of the Jordan canonical form. We investigate the possible splitting types of F in dependence on the degeneration behavior of Φ. Applications to nilpotent matrices with an arbitrary number of variables are also given.     

Groups with given properties of finite subgroups

Suppose that in every finite even order subgroup F of a periodic group G, the equality [u, x]² = 1 holds for any involution u of F and for an arbitrary element x of F. Then the subgroup I generated by all involutions in G is locally finite and is a 2-group. In addition, the normal closure of every subgroup of order 2 in G is commutative.     

Multi-party computation with conversion of secret sharing

Classical results in unconditionally secure multi-party computation (MPC) protocols with a passive adversary indicate that every n-variate function can be computed by n participants, such that no set of size t < n/2 participants learns any additional information other than what they could derive from their private inputs and the output of the protocol. We study unconditionally secure MPC protocols in the presence of a passive adversary in the trusted setup (‘semi-ideal’) model, in which the participants are supplied with some auxiliary information (which is random and independent from the participant inputs) ahead of the protocol execution (such information can be purchased as a “commodity” well before a run of the protocol). We present a new MPC protocol in the trusted setup model, which allows the adversary to corrupt an arbitrary number t < n of participants. Our protocol makes use of a novel subprotocol for converting an additive secret sharing over a field to a multiplicative secret sharing, and can be used to securely evaluate any n-variate polynomial G over a field F, with inputs restricted to non-zero elements of F. The communication complexity of our protocol is O(ℓ · n ²) field elements, where ℓ is the number of non-linear monomials in G. Previous protocols in the trusted setup model require communication proportional to the number of multiplications in an arithmetic circuit for G; thus, our protocol may offer savings over previous protocols for functions with a small number of monomials but a large number of multiplications.     

Transitive permutation groups and equipotent voting rules

Let F a two-alternative voting rule and G_F the subgroup of permutations of the voters under which F is invariant. Group theoretic properties of G_F provide information about the voting rule F. In particular, sets of imprimitivity of G_F describe the ‘committee decomposition’ structure of F and permutation group transitivity of G_F (equipotency) is shown to be closely connected with equal distribution of power among the voters. If equipotency replaces anonymity in the hypotheses of May's theorem, voting rules other than simple majority are possible. By combining equipotency with two additional social choice conditions a new characterization of simple majority rule is obtained. Equipotency is proposed as an important alternative to the more restrictive anonymity as a fairness criterion in social choice.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Galois Stability, Integrality and Realization Fields for Representations of Finite Abelian Groups

For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d and finite Abelian subgroups GGL _n (E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider the fields E=F(G) that are obtained via adjoining all matrix coefficients of all matrices gG to F. It is proved that under some reasonable restrictions for n, any E can be realized as F(G), while if all coefficients of matrices in G are algebraic integers, there are only finitely many fields E=F(G) for prescribed integers n and t or prescribed n and d.     

Forest embeddings in regular graphs of large girth

This paper deals with the enumeration of distinct embeddings (both induced and partial) of arbitrary graphs in regular graphs of large girth. A simple explicit recurrence formula is presented for the number of embeddings of an arbitrary forest F in an arbitrary regular graph G of sufficiently large girth. This formula (and hence the number of embeddings) depends only on the order and degree of regularity of G, and the degree sequence and component structure (multiset of component orders) of F. A concept called c-subgraph regularity is introduced which generalizes the familiar notion of regularity in graphs. (Informally, a graph is c-subgraph regular if its vertices cannot be distinguished on the basis of embeddings of graphs of order less than or equal to c.) A central result of this paper is that if G is regular and has girth g, then G is (g ? 1)-subgraph regular.     

Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s‐Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 ‐ o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n^{1+Ω(1/log log n)}. Recall that the Ising model with inverse temperature β defined on a graph G = (V,E) is the distribution over {±}^Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < ∞ and the Ising model defined on G (n, d/n), there exists a β_d > 0, such that for all β < β_d with probability going to 1 as n →∞, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd?s‐Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub‐graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and β < β_d, where d tanh(β_d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Inequalities for the Extremal Coefficients of Multivariate Extreme Value Distributions

The extremal coefficients are the natural dependence measures for multivariate extreme value distributions. For an m-variate distribution 2^m distinct extremal coefficients of different orders exist; they are closely linked and therefore a complete set of 2^m coefficients cannot take any arbitrary values. We give a full characterization of all the sets of extremal coefficients. To this end, we introduce a simple class of extreme value distributions that allows for a 1-1 mapping to the complete sets of extremal coefficients. We construct bounds that higher order extremal coefficients need to satisfy to be consistent with lower order extremal coefficients. These bounds are useful as lower order extremal coefficients are the most easily inferred from data.