Problems arising from jackknifing the estimate of a Kaplan-Meier integral

Stute and Wang (1994) considered the problem of estimating the integral S^θ = ∫ θ dF, based on a possibly censored sample from a distribution F, where θ is an F-integrable function. They proposed a Kaplan-Meier integral to approximate S^θ and derived an explicit formula for the delete-1 jackknife estimate . differs from only when the largest observation, X_(n), is not censored (δ_(n) = 1 and next-to-the-largest observation, X_(n-1), is censored (δ_(n-1) = 0). In this note, it will pointed out that when X_(n) is censored is based on a defective distribution, and therefore can badly underestimate . We derive an explicit formula for the delete-2 jackknife estimate . However, on comparing the expressions of and , their difference is negligible. To improve the performance of and , we propose a modified estimator according to Efron (1980). Simulation results demonstrate that is much less biased than and and .     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

Sequences that omit a box (modulo 1)

Let be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj?σ>0. For an open box I in [0,1d⁾, we write It is shown that the Hausdorff dimension of is d−1 whenever The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S₁,…,Sd be strictly increasing in N. Define to be the set of x in [0, 1) for which A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vr−ur→∞ and1+vr−ur?C#(Br). Kaufman has shown that is countable whenever S₁,…,Sd fulfill condition D(C). Here it is shown that is finite under this hypothesis. An upper bound for is provided.     

Backward Shift Invariant Operator Ranges

Results on first order Ext groups for Hilbert modules over the disk algebra are used to study certain backward shift invariant operator ranges, namely de Branges–Rovnyak spaces and a more general class called (W; B) spaces. Necessary and sufficient conditions are given for the groups Ext¹_A()(, (W; B)) to vanish whereis thedualof the vector-valued Hardy module, H². One condition involves an extension problem for the Hankel operator with symbolB,Γ_B, but viewed as a module map from H²into (W; B). The group Ext¹_A()(, (W; B))=(0) precisely whenΓ_Bextends to a module map from L²into (W; B) and this in turn is equivalent to the injectivity of (W; B) in the category of contractive HilbertA()-modules. This result applied to the de Branges–Rovnyak spaces yields a connection between the extension problem for the HankelΓ_B and the operator corona problem.     

An Erdős–Ko–Rado Theorem for Direct Products

Letn_i, k_ibe positive integers,i=1, ..., d,satisfyingn_i≥2k_i.LetX₁, ..., X_dbe pairwise disjoint sets with |X_i| =n_i.Letbe the family of those (k₁+···+k_d)-element sets which have exactlyk_ielements inX_i, i=1,..., d.It is shown that if⊂is an intersecting family then ||/||≤max_ik_i/n_i,and this is best possible. The proof is algebraic, although in thed=2 case a combinatorial argument is presented as well.     

Edge choosability and total choosability of planar graphs with no 3-cycles adjacent 4-cycles

Two cycles are said to be adjacent if they share a common edge. Let G be a planar graph without triangles adjacent 4-cycles. We prove that if Δ(G)≥6, and and if Δ(G)≥8, where and denote the list edge chromatic number and list total chromatic number of G, respectively.     

More Efficient Algorithm for Ordered Tree Inclusion

Given two ordered trees and , the tree inclusion problem is to determine whether it is possible to obtain from by deleting nodes. Recently, this problem has been recognized as an important primitive in query processing for structured text databases. In this paper we present anO(|leaves()| ||) time andO(|leaves()|min(depth(), |leaves()|)) space algorithm for ordered tree inclusion, by means of a sophisticated bottom-up-matching strategy. Our algorithm improves the previous best one (Kilpeläinen, 1992, Ph.D. thesis, Dept. Computer Science, Univ. Helsinki) that requiresO(|| ||) time andO(||min(depth(), |leaves()|)) space.     

A further refinement of a three critical points theorem

In this paper, we complete the refinement process, made by Ricceri (2009) [4], of a result established by Ricceri (2000) [1], which is one of the most applied abstract multiplicity theorems in the past decade. A sample of application of our new result is as follows.Let (n≥3) be a bounded domain with smooth boundary and let .Then, for each ?>0 small enough, there exists λ_?>0 such that, for every compact interval , there exists ρ>0 with the following property: for every λ∈[a,b] and every continuous function satisfying for some , there exists δ>0 such that, for each ν∈[0,δ], the problem has at least three weak solutions whose norms in are less than ρ.     

On the Korteweg–de Vries Equation: Convergent Birkhoff Normal Form

The Korteweg–de Vries equation (KdV)[formula]is a completely integrable Hamiltonian system of infinite dimension with phase space the Sobolev spaceH^N(S¹; ), (N?1), Hamiltonian (q):=∫_S1((∂_xq(x))²+q(x)³) dx, and Poisson structure ∂/∂x. The functionq≡0 is an elliptic fixed point. We prove that for anyN?1, the Korteweg–de Vries equation (and thus the entire KdV-hierarchy) admits globally defined real analytic action-angle variables. As a consequence it follows that in a neighborhood ofq≡0 inH¹(S¹; ), the KdV-Hamiltonian (and similarly any Hamiltonian in the KdV-hierarchy) admits a convergent Birkhoff normal form; to the best of our knowledge this is the first such example in infinite dimension. Moreover, using the constructed action-angle variables, we analyze the regularity properties of the Hamiltonian vectorfield of KdV.     

Transcendence of Binomial and Lucas' Formal Power Series

The formal power series[formula]is transcendental over (X) whentis an integer ≥ 2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn = ∑n_ipⁱis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over _p(X), the field of rational functions over the Galois field _p, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over (X).     

CERES in higher-order logic

We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an (unsatisfiable) set of sequents (the characteristic sequent set) from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.     

The Real Tree

In this note, it is shown that there exists a natural metric on the set[formula]of bounded subsets of containing their infimum which endowsTwith the structure of an -tree so that, for everyt∈T, the “number” of connected components ofT\{t} coincides with the cardinality #() of the set of subsets of . In addition, the set of ends ofTis explicitly determined, and various further features ofTare discussed, too.     

Second order, multi-point problems with variable coefficients

In this paper, we consider the eigenvalue problem consisting of the equation where and λ∈R, together with the multi-point boundary conditions where m^±?1 are integers, and, for i=1,…,m^±, , , with , . We show that if the coefficients are sufficiently small (depending on r), then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients are not sufficiently small, then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case (r≡1), but the variable coefficient case has not been considered previously (apart from the existence of ‘principal’ eigenvalues).Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for the existence of general solutions and also of nodal solutions—these results rely on the spectral properties of the linear problem.     

Logarithmic Sobolev Inequalities for Pinned Loop Groups

LetGbe a connected compact type Lie group equipped with anAd_G-invariant inner product on the Lie algebra ofG. Given this data there is a well known left invariant “H¹-Riemannian structure” on =(G)—the infinite dimensional group of continuous based loops inG. Using this Riemannian structure, we define and construct a “heat kernel”ν_T(g₀, ·) associated to the Laplace–Beltrami operator on (G). HereT>0,g₀∈(G), andν_T(g₀,·) is a certain probability measure on (G). For fixedg₀∈(G) andT>0, we use the measureν_T(g₀,·) and the Riemannian structure on (G) to construct a “classical” pre-Dirichlet form. The main theorem of this paper asserts that this pre-Dirichlet form admits a logarithmic Sobolev inequality.     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Higher order log-concavity in Euler’s difference table

For 0≤k≤n, let be the entries in Euler’s difference table and let . Dumont and Randrianarivony showed equals the number of permutations on [n] whose fixed points are contained in {1,2,…,k}. Rakotondrajao found a combinatorial interpretation of the number in terms of k-fixed-points-permutations of [n]. We show that for any n≥1, the sequence is essentially 2-log-concave and reverse ultra log-concave.     

A Representation of Projection Lattices and Their States in Euclidean Space

We propose a representationr : ∪ Ω → ^ν, where is the collection of closed subspaces of ann-dimensional real, complex, or quaternionic Hilbert space , or equivalently, the projection lattice of this Hilbert space, where Ω is the set of all states ω : → [0, 1]. The value that ω ∈ Ω takes ina ∈ is given by the scalar product of the representative points (r(a) andr(ω)). The representationr(a ∨ b) of the join of two orthogonal elementsa, b ∈ is equal tor(a) + r(b). The convex closure of the representation of Σ, the set of atoms of , is equal to the representation of Ω.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

An endpoint space–time estimate for the Schrödinger equation

We obtain endpoint estimates for the Schrödinger operator f→eitΔf in with initial data f in the homogeneous Sobolev space . The exponents and regularity index satisfy and . For n=2 we prove the estimates in the range q>16/5, and for n?3 in the range q>2+4/(n+1).