High-dimensional asymptotic expansions for the distributions of canonical correlations

This paper examines asymptotic distributions of the canonical correlations between and with q≤p, based on a sample of size of N=n+1. The asymptotic distributions of the canonical correlations have been studied extensively when the dimensions q and p are fixed and the sample size N tends toward infinity. However, these approximations worsen when q or p is large in comparison to N. To overcome this weakness, this paper first derives asymptotic distributions of the canonical correlations under a high-dimensional framework such that q is fixed, m=n−p→∞ and c=p/n→c₀∈[0,1), assuming that and have a joint (q+p)-variate normal distribution. An extended Fisher’s z-transformation is proposed. Then, the asymptotic distributions are improved further by deriving their asymptotic expansions. Numerical simulations revealed that our approximations are more accurate than the classical approximations for a large range of p,q, and n and the population canonical correlations.     

Partial autocorrelation functions of the fractional ARIMA processes with negative degree of differencing

Let be a fractional ARIMA(p,d,q) process with partial autocorrelation function α(·). In this paper, we prove that if d∈(−1/2,0) then |α(n)|∼|d|/n as n→∞. This extends the previous result for the case 0<d<1/2.     

Wavelet bases on a manifold

Using the decomposition method, we present in this paper constructions of multiresolution analyses on a compact Riemannian manifold M of dimension n(n∈N). These analyses are generated by a finite number of basic functions and are adapted to the study of the Sobolev spaces H¹(M) and .     

Geometry and marginals

It is known that observations of a real-valued random variable defined over a smooth manifold M can be used to make inferences about M, at least when M is a curve or surface. We refine and extend the underlying asymptotic results and remove the condition . New examples of nonsmoothness in marginals are described in detail for , and methods are given for calculations in general.     

Multivariate stress-strength reliability model and its evaluation for coherent structures

Consider a system which has n independent components (or subsystems) each consisting of m dependent elements. Let , i=1,2,…,n denote the random strength vector of the ith component, where denotes the random strength of the jth element of the ith component. The elements of the components are subjected to a common random stress over time. In this paper, we setup a multivariate stress-strength model based on the conditional ordering between s and and evaluate the reliability of coherent structures in this setup.     

Some families of strongly clean rings

A ring R with identity is called strongly clean if every element of R is the sum of an idempotent and a unit that commute with each other. For a commutative local ring R and for an arbitrary integer n?2, the paper deals with the question whether the strongly clean property of M_n(R[[x]]), , and M_n(RC₂) follows from the strongly clean property of M_n(R). This is ‘Yes’ if n=2 by a known result.     

Stabilizers and orbits of smooth functions

Let be a smooth function such that f(0)=0. We give a condition J(id) on f when for arbitrary preserving orientation diffeomorphism such that ?(0)=0 the function ?○f is right equivalent to f, i.e. there exists a diffeomorphism such that ?○f=f○h at 0∈Rm. The requirement is that f belongs to its Jacobi ideal. This property is rather general: it is invariant with respect to the stable equivalence of singularities, and holds for non-degenerated, simple, and many other singularities.We also globalize this result as follows. Let M be a smooth compact manifold, a surjective smooth function, DM the group of diffeomorphisms of M, and the group of diffeomorphisms of R that have compact support and leave [0,1] invariant. There are two natural right and left-right actions of DM and on C^∞(M,R). Let SM(f), SMR(f), OM(f), and OMR(f) be the corresponding stabilizers and orbits of f with respect to these actions. We prove that if f satisfies J(id) at each critical point and has additional mild properties, then the following homotopy equivalences hold: SM(f)≈SMR(f) and OM(f)≈OMR(f). Similar results are obtained for smooth mappings M→S¹.     

Embeddings of homology equivalent manifolds with boundary

We prove a theorem on equivariant maps implying the following two corollaries:(1) Let N and M be compact orientable n-manifolds with boundaries such that M⊂N, the inclusion M→N induces an isomorphism in integral cohomology, both M and N have (n−d−1)-dimensional spines and . Then the restriction-induced map Embm(N)→Embm(M) is bijective. Here Embm(X) is the set of embeddings X→Rm up to isotopy (in the PL or smooth category).(2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N?D³ (or for its special 2-spine N) there exists an equivariant map , although N does not embed into R³.The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map implies embeddability of N into Rm? An answer was known for each pair (m,n) except (3,3) and (3,2).     

The increment ratio statistic

We introduce a new statistic written as a sum of certain ratios of second-order increments of partial sums process of observations, which we call the increment ratio (IR) statistic. The IR statistic can be used for testing nonparametric hypotheses for d-integrated () behavior of time series X_t, including short memory (d=0), (stationary) long-memory and unit roots (d=1). If S_n behaves asymptotically as an (integrated) fractional Brownian motion with parameter , the IR statistic converges to a monotone function Λ(d) of as both the sample size N and the window parameter m increase so that N/m→∞. For Gaussian observations X_t, we obtain a rate of decay of the bias EIR-Λ(d) and a central limit theorem (N/m)^1/2(IR-EIR)→N(0,σ²(d)), in the region . Graphs of the functions Λ(d) and σ(d) are included. A simulation study shows that the IR test for short memory (d=0) against stationary long-memory alternatives has good size and power properties and is robust against changes in mean, slowly varying trends and nonstationarities. We apply this statistic to sequences of squares of returns on financial assets and obtain a nuanced picture of the presence of long-memory in asset price volatility.     

The Dirichlet Markov Ensemble

We equip the polytope of n×n Markov matrices with the normalized trace of the Lebesgue measure of Rⁿ². This probability space provides random Markov matrices, with i.i.d. rows following the Dirichlet distribution of mean (1/n,…,1/n). We show that if is such a random matrix, then the empirical distribution built from the singular values of tends as n→∞ to a Wigner quarter-circle distribution. Some computer simulations reveal striking asymptotic spectral properties of such random matrices, still waiting for a rigorous mathematical analysis. In particular, we believe that with probability one, the empirical distribution of the complex spectrum of tends as n→∞ to the uniform distribution on the unit disc of the complex plane, and that moreover, the spectral gap of is of order when n is large.     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X∼f(∥x-θ∥²) and let δ_π(X) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π(∥θ∥²), for loss ∥δ-θ∥². We show that if π(t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ₀(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e^-αtβ and e^-αt+βφ(t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .     

Density estimation by the penalized combinatorial method

Let f be an unknown multivariate density belonging to a prespecified parametric class of densities, , where k is unknown, but for all k and each has finite Vapnik-Chervonenkis dimension. Given an i.i.d. sample of size n drawn from f, we show that it is possible to select automatically, and without extra restrictions on f, an estimate with the property that . Our method is inspired by the combinatorial tools developed in Devroye and Lugosi (Combinatorial Methods in Density Estimation, Springer, New York, 2001) and it includes a wide range of density models, such as mixture models or exponential families.     

Efficient algorithms for robust generalized cross-validation spline smoothing

Generalized cross-validation (GCV) is a widely used parameter selection criterion for spline smoothing, but it can give poor results if the sample size n is not sufficiently large. An effective way to overcome this is to use the more stable criterion called robust GCV (RGCV). The main computational effort for the evaluation of the GCV score is the trace of the smoothing matrix, , while the RGCV score requires both and . Since 1985, there has been an efficient O(n) algorithm to compute . This paper develops two pairs of new O(n) algorithms to compute and , which allow the RGCV score to be calculated efficiently. The algorithms involve the differentiation of certain matrix functionals using banded Cholesky decomposition.     

On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: X_i∼N(θ_i,σ²),i=1,…,p, we consider estimating the vector θ=(θ₁,…,θ_p)^′ with loss ‖d−θ‖² under the constraint , with known τ₁,…,τ_p,σ²,m. In comparing the risk performance of Bayesian estimators δ_α associated with uniform priors on spheres of radius α centered at (τ₁,…,τ_p) with that of the maximum likelihood estimator , we make use of Stein’s unbiased estimate of risk technique, Karlin’s sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on α for δ_α to dominate . Large sample determinations of these conditions are provided. Both cases where all such δ_α’s and cases where no such δ_α’s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ_m dominates if and only if m≤k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators δ_π with π spherically symmetric and supported on the parameter space dominate whenever m≤c₁(p) with .