Asymptotic distribution of the OLS estimator for a mixed spatial model

We find the asymptotic distribution of the OLS estimator of the parameters β and ρ in the mixed spatial model with exogenous regressors Y_n=X_nβ+ρW_nY_n+V_n. The exogenous regressors may be bounded or growing, like polynomial trends. The assumption about the spatial matrix W_n is appropriate for the situation when each economic agent is influenced by many others. The error term is a short-memory linear process. The key finding is that in general the asymptotic distribution contains both linear and quadratic forms in standard normal variables and is not normal.     

The restricted EM algorithm under inequality restrictions on the parameters

One of the most powerful algorithms for maximum likelihood estimation for many incomplete-data problems is the EM algorithm. The restricted EM algorithm for maximum likelihood estimation under linear restrictions on the parameters has been handled by Kim and Taylor (J. Amer. Statist. Assoc. 430 (1995) 708-716). This paper proposes an EM algorithm for maximum likelihood estimation under inequality restrictions A₀β?0, where β is the parameter vector in a linear model W=Xβ+ε and ε is an error variable distributed normally with mean zero and a known or unknown variance matrix Σ>0. Some convergence properties of the EM sequence are discussed. Furthermore, we consider the consistency of the restricted EM estimator and a related testing problem.     

Adaptive minimax testing in the discrete regression scheme

We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0,1]. We wish to test the null hypothesis that the regression function belongs to a given functional class (parametric or even nonparametric) against a composite nonparametric alternative. The functions under the alternative are separated in the L₂-norm from any function in the null hypothesis. We assume that the regression function belongs to a wide range of Hölder classes but as the smoothness parameter of the regression function is unknown, an adaptive approach is considered. It leads to an optimal and unavoidable loss of order Open image in new window in the minimax rate of testing compared with the non-adaptive setting. We propose a smoothness-free test that achieves the optimal rate, and finally we prove the lower bound showing that no test can be consistent if in the distance between the functions under the null hypothesis and those in the alternative, the loss is of order smaller than the optimal loss.     

Robustness of A-optimal designs

Suppose that Y=(Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j),X=(X_ij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Y_i are dependent, i.e., for i≠i′, the covariance of Y_i,Y_i^′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(X_ij), where X_ij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight b_j being weighed on the left and right of a chemical balance respectively.     

Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W_p(m_i,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Q_k(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images ρ_k(Σ⁺) of the Moore-Penrose inverse Σ⁺ of Σ are mutually orthogonal: ρ_k(Σ⁺)ρ_?(Σ⁺)=0 for k≠?.     

On the distribution of the ratio of the largest eigenvalue to the trace of a Wishart matrix

The ratio of the largest eigenvalue divided by the trace of a p×p random Wishart matrix with n degrees of freedom and an identity covariance matrix plays an important role in various hypothesis testing problems, both in statistics and in signal processing. In this paper we derive an approximate explicit expression for the distribution of this ratio, by considering the joint limit as both p,n→∞ with p/n→c. Our analysis reveals that even though asymptotically in this limit the ratio follows a Tracy-Widom (TW) distribution, one of the leading error terms depends on the second derivative of the TW distribution, and is non-negligible for practical values of p, in particular for determining tail probabilities. We thus propose to explicitly include this term in the approximate distribution for the ratio. We illustrate empirically using simulations that adding this term to the TW distribution yields a quite accurate expression to the empirical distribution of the ratio, even for small values of p,n.     

Kernel estimation of density level sets

Let f be a multivariate density and f_n be a kernel estimate of f drawn from the n-sample X₁,…,X_n of i.i.d. random variables with density f. We compute the asymptotic rate of convergence towards 0 of the volume of the symmetric difference between the t-level set {f?t} and its plug-in estimator {f_n?t}. As a corollary, we obtain the exact rate of convergence of a plug-in-type estimate of the density level set corresponding to a fixed probability for the law induced by f.     

On the false discovery proportion convergence under Gaussian equi-correlation

We study the convergence of the false discovery proportion (FDP) of the Benjamini-Hochberg procedure in the Gaussian equi-correlated model, when the correlation ρ_m converges to zero as the hypothesis number m grows to infinity. In this model, the FDP converges to the false discovery rate (FDR) at rate {min(m,1/ρ_m)}^1/2, which is different from the standard convergence rate m^1/2 holding under independence.     

The estimation of a multivariate linear relation

A multivariate linear relation η_n = β₀ξ_n is considered, in which ξ_n and η_n are observed subject to white noise errors, with covariance matrices σ₀, ω₀ respectively. If their elements lie in the null space of a suitable vector function, β₀, σ₀, ω₀ may be uniquely defined by second-order functions of the data. The asymptotic properties of estimates of β₀, σ₀, ω₀ are established under relatively mild conditions. We explore the possibility that explicit formulas for consistent estimates of β₀, σ₀, ω₀ may be available.     

Quantitative relation between noise sensitivity and influences

A Boolean function f: {0,1} ⁿ → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ _p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand’s lemma, which easily generalizes in various directions, including non-monotone functions.     

On the unlinking conjecture of independent polynomial functions

The celebrated U-conjecture states that under the N_n(0,I_n) distribution of the random vector X=(X₁,…,X_n) in Rⁿ, two polynomials P(X) and Q(X) are unlinkable if they are independent [see Kagan et al., Characterization Problems in Mathematical Statistics, Wiley, New York, 1973]. Some results have been established in this direction, although the original conjecture is yet to be proved in generality. Here, we demonstrate that the conjecture is true in an important special case of the above, where P and Q are convex nonnegative polynomials with P(0)=0.     

Error bounds for asymptotic expansions of Wilks’ lambda distribution

Let Λ=|S_e|/|S_e+S_h|, where S_h and S_e are independently distributed as Wishart distributions W_p(q,Σ) and W_p(n,Σ), respectively. Then Λ has Wilks’ lambda distribution Λ_p,q,n which appears as the distributions of various multivariate likelihood ratio tests. This paper is concerned with theoretical accuracy for asymptotic expansions of the distribution of T=-nlogΛ. We derive error bounds for the approximations. It is necessary to underline that our error bounds are given in explicit and computable forms.     

Decay property of solutions for damped wave equations with space-time dependent damping term

We consider the Cauchy problem for the damped wave equation with space-time dependent potential b(t,x) and absorbing semilinear term |u|^ρ−1u. Here, with b₀>0, α,β?0 and α+β∈[0,1). Using the weighted energy method, we can obtain the L² decay rate of the solution, which is almost optimal in the case ρ>ρc(N,α,β):=1+2/(N−α). Combining this decay rate with the result that we got in the paper [J. Lin, K. Nishihara, J. Zhai, L²-estimates of solutions for damped wave equations with space-time dependent damping term, J. Differential Equations 248 (2010) 403-422], we believe that ρc(N,α,β) is a critical exponent. Note that when α=β=0, ρc(N,α,β) coincides to the Fujita exponent ρF(N):=1+2/N. The new points include the estimate in the supercritical exponent and for not necessarily compactly supported data.     

Effective PCA for high-dimension, low-sample-size data with noise reduction via geometric representations

In this article, we propose a new estimation methodology to deal with PCA for high-dimension, low-sample-size (HDLSS) data. We first show that HDLSS datasets have different geometric representations depending on whether a ρ-mixing-type dependency appears in variables or not. When the ρ-mixing-type dependency appears in variables, the HDLSS data converge to an n-dimensional surface of unit sphere with increasing dimension. We pay special attention to this phenomenon. We propose a method called the noise-reduction methodology to estimate eigenvalues of a HDLSS dataset. We show that the eigenvalue estimator holds consistency properties along with its limiting distribution in HDLSS context. We consider consistency properties of PC directions. We apply the noise-reduction methodology to estimating PC scores. We also give an application in the discriminant analysis for HDLSS datasets by using the inverse covariance matrix estimator induced by the noise-reduction methodology.     

Empirical likelihood confidence region for parameter in the errors-in-variables models

This paper proposes a constrained empirical likelihood confidence region for a parameter β₀ in the linear errors-in-variables model: Y_i=x_i^τβ₀+ε_i,X_i=x_i+u_i,(1?i?n), which is constructed by combining the score function corresponding to the squared orthogonal distance with a constrained region of β₀. It is shown that the coverage error of the confidence region is of order n⁻¹, and Bartlett corrections can reduce the coverage errors to n⁻². An empirical Bartlett correction is given for practical implementation. Simulations show that the proposed confidence region has satisfactory coverage not only for large samples, but also for small to medium samples.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

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.     

No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix

We consider a class of matrices of the form , where X_n is an n×N matrix consisting of i.i.d. standardized complex entries, is a nonnegative definite square root of the nonnegative definite Hermitian matrix A_n, and B_n is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of A_n and B_n converge to proper probability distributions as , the empirical spectral distribution of C_n converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of A_n and B_n, with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n. The problem is motivated by applications in spatio-temporal statistics and wireless communications.     

The Optimal Discretization of Stochastic Differential Equations

We study pathwise approximation of scalar stochastic differential equations. The mean squared L₂-error and the expected number n of evaluations of the driving Brownian motion are used for the comparison of arbitrary methods. We introduce an adaptive discretization that reflects the local properties of every single trajectory. The corresponding error tends to zero like c·n^−1/2, where c is the average of the diffusion coefficient in space and time. Our method is justified by the matching lower bound for arbitrary methods that are based on n evaluations on the average. Hence the adaptive discretization is asymptotically optimal. The new method is very easy to implement, and about 7 additional arithmetical operations are needed per evaluation of the Brownian motion. Hereby we can determine the complexity of pathwise approximation of stochastic differential equations. We illustrate the power of our method already for moderate accuracies by means of a simulation experiment.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.