A weak Grothendieck compactness principle

The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In this article, an analogue of the Grothendieck compactness principle is considered when the norm topology of a Banach space is replaced by its weak topology. It is shown that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property.     

Cesaro mean convergence of martingale differences in rearrangement invariant spaces

We study the class of r.i. spaces in which Cesaro means of any weakly null martingale difference sequence is strongly null. This property is related to the Banach-Saks property. We show that in classical (separable) r.i. spaces (such as Orlicz, Lorentz and Marcinkiewicz spaces) these properties coincide but this is no longer true for general r.i. spaces. We locate also a class of r.i. spaces having this property where an analogue of the classical Dunford-Pettis characterization of relatively weakly compact subsets in L ₁ holds. Research was partially supported by the ARC and NSF grant DMS-0244515.     

Pre-compact families of finite sets of integers and weakly null sequences in Banach spaces

In this paper we use the Nash-Williams theory of fronts and barriers to study weakly null sequences in Banach spaces. Specifically, we show how barriers relate to the classical fact that C(K) with K a countable compactum is c₀-saturated. Another result relates the notion of a barrier to the Maurey-Rosenthal example of a weakly null sequence with no unconditional subsequences. In particular, we construct examples of weakly-null sequences which are α-unconditional but not β-unconditional.     

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This paper is focused on testing the parameters of the quantile regression models. For complete observation, it is shown in literature that the test statistics, based on empirical likelihood (EL) method and smoothed empirical likelihood (SEL) method, both converge weakly to the standard Chi-square distribution $\chi_M^2$ under the null hypothesis. For right censored data, the statistics in literature, by the EL method, have a weighted Chi-square limiting distribution, but the weights are unknown. In this paper, we show that the statistics based on the EL method and the SEL method also converge weakly to $\chi_M^2$ under the null hypothesis, so there is no need to estimate any weights. As its estimating function is smoothed, the SEL method can be Bartlett corrected. Numerical results show that the SEL method, via Bartlett correction, outperforms some recent methods.     

Unbounded continuous operators and unbounded Banach–Saks property in Banach lattices

Positivity - Motivated by the equivalent definition of a continuous operator between Banach spaces in terms of weakly null nets, we introduce unbounded continuous operators by replacing weak...     

Elton near unconditionality of arrays

There are many notions of partial unconditionality defined for a weakly null basic sequence in a Banach space. In 2008 the idea of Schreier unconditionality was extended to a structure in Banach spaces called arrays. Here we extend the idea of Elton near unconditionality to arrays.     

Onc 0 sequences in Banach spaces

A Banach space has property (S) if every normalized weakly null sequence contains a subsequences equivalent to the unit vector basis ofc ₀. We show that the equivalence constant can be chosen “uniformly”, i.e., independent of the choice of the normalized weakly null sequence. Furthermore we show that a Banach space with property (S) has property (u). This solves in the negative the conjecture that a separable Banach space with property (u) not containingl ₁ has a separable dual. This is part of this author's Ph.D. dissertation prepared at The University of Texas at Austin under the supervision of H. P. Rosenthal.     

Weighted nucleoli

Cooperative games in characteristic function form (TU games) are considered. We allow for variable populations or carriers. Weighted nucleoli are defined via weighted excesses for coalitions. A solution satisfies the Null Player Out (NPO) property, if elimination of a null player does not affect the payoffs of the other players. For any single-valued and efficient solution, the NPO property implies the null player property. We show that a weighted nucleolus has the null player property if and only if the weights of multi-player coalitions are weakly decreasing with respect to coalition inclusion. Weighted nucleoli possessing the NPO-property can be characterized by means of a multiplicative formula for the weights of the multi-player coalitions and a restrictive condition on the weights of one-player coalitions. Received: March 1997/Final version: November 1998     

Correcting for biases in affected sib-pair linkage analysis caused by uncertainty in sibling relationship

When there is uncertainty in sibling relationship,the classical affected sib-pair(ASP) linkage tests may be severely biased.This can happen,for example,if some of the half sib-pairs are mixed with full sib-pairs.The genomic control method has been used in association analysis to adjust for population structures.We show that the same idea can be applied to ASP linkage analysis with uncertainty in sibling relationship.Assuming that,in addition to the candidate marker,null markers that are unlinked to the disease locus are also genotyped,we may use the information on these loci to estimate the proportion of half sib-pairs and to correct for the bias and variance distortion caused by the heterogeneity of sibling relationship.Unlike in association studies,the null loci are not required to be matched with the candidate marker in allele frequency for ASP linkage analysis.This makes our approach flexible in selecting null markers.In our simulations,using a number of 30 or more null loci can effectively remove the bias and variance distortion.It is also shown that,even the null loci are weakly linked to the disease locus,the proposed method can also provide satisfactory correction.     

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.     

Likelihood ratio test for a piecewise continuous Weibull model with an unknown change point

In this paper, we consider the likelihood ratio test for the scale and shape parameters in a piecewise continuous Weibull model with an unknown change point. Under the null hypothesis of no change in scale and shape parameters, we derive that the likelihood ratio process converges weakly to the squared Euclidian norm of a weighted mean zero Gaussian vector process.     

Martin’s conjecture and strong ergodicity

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.     

Testing linearity against threshold effects: uniform inference in quantile regression

This paper develops a uniform test of linearity against threshold effects in the quantile regression framework. The test is based on the supremum of the Wald process over the space of quantile and threshold parameters. We establish the limiting null distribution of the test statistic for stationary weakly dependent processes, and propose a simulation method to approximate the critical values. The proposed simulation method makes the test easy to implement. Monte Carlo experiments show that the proposed test has good size and reasonable power against non-linear threshold models.     

Some test statistics based on the martingale term of the empirical distribution function

Summary It is proved that the martingale term of the empirical distribution function converges weakly to a Gaussian process inD[0, 1]. Some statistics for goodness-of-fit tests based on the martingale term of the empirical distribution function are proposed. Asymptotic distributions of these statistics under the null hypothesis are given. The approximate Bahadur efficiencies of the statistics to the Kolmogorov-Smirnov statistic and to the Cramér-von Mises statistic are also calculated. The Institute of Statistical Mathematics     

Unconditional structures of weakly null sequences

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of , or there exists a subsequence which is boundedly convexly complete.

     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

Restricted Likelihood Ratio Testing for Zero Variance Components in Linear Mixed Models

The goal of our article is to provide a transparent, robust, and computationally feasible statistical platform for restricted likelihood ratio testing (RLRT) for zero variance components in linear mixed models. This problem is nonstandard because under the null hypothesis the parameter is on the boundary of the parameter space. Our proposed approach is different from the asymptotic results of Stram and Lee who assumed that the outcome vector can be partitioned into many independent subvectors. Thus, our methodology applies to a wider class of mixed models, which includes models with a moderate number of clusters or nonparametric smoothing components. We propose two approximations to the finite sample null distribution of the RLRT statistic. Both approximations converge weakly to the asymptotic distribution obtained by Stram and Lee when their assumptions hold. When their assumptions do not hold, we show in extensive simulation studies that both approximations outperform the Stram and Lee approximation and the parametric bootstrap. We also identify and address numerical problems associated with standard mixed model software. Our methods are motivated by and applied to a large longitudinal study on air pollution health effects in a highly susceptible cohort. Relevant software is posted as an online supplement.     

Norm optimization problem for linear operators in classical Banach spaces

The main result of the paper shows that, for 1 < p < ∞ and 1 ≤ q < ∞, a linear operator T: ℓ _p → ℓ _q attains its norm if, and only if, there exists a not weakly null maximizing sequence for T (counterexamples can be easily constructed when p = 1). For 1 < p ≠ q < ∞, as a consequence of the previous result we show that any not weakly null maximizing sequence for a norm attaining operator T: ℓ _p → ℓ _q has a norm-convergent subsequence (and this result is sharp in the sense that it is not valid if p = q). We also investigate lineability of the sets of norm-attaining and non-norm attaining operators.     

The Banach-Saks p-property

We obtain general theorems which enable the calculation of the Banach-Saks index set for rearrangement invariant function spaces in terms of their Boyd indices. For many important classes of rearrangement invariant spaces, the index set is non-trivial if and only if the Boyd indices of the underlying space are non-trivial, although the latter equivalence does not hold in full generality. We also study Banach-Saks index sets defined for weakly null sequences of (identically distributed) independent random variables and their connections with the classical Banach-Saks index set.Research supported by the RFFI and “Universities of Russia” grant.Research supported by the Australian Research Council.     

Banach Spaces Admitting a Separating Polynomial and L P Spaces

 We prove that in a Banach space admitting a separating polynomial, each weakly null normalized sequence has a subsequence which is equivalent to the usual basis of some , p an even integer. We show that for each even integer p, the Schatten class admits a separating polynomial. For a space with a basis admitting a 4-homogeneous separating polynomial, we relate the unconditionality of the basis with the existence of certain type of polynomials defined in terms of infinite symmetric matrices. We find relations between the properties of the entries of these matrices and the geometrical structure of the space. Finally we study the existence of convex 4-homogeneous separating polynomials.