Second order optimality for estimators in time series regression models

We consider the second order asymptotic properties of an efficient frequency domain regression coefficient estimator proposed by Hannan [Regression for time series, Proc. Sympos. Time Series Analysis (Brown Univ., 1962), Wiley, New York, 1963, pp. 17-37]. This estimator is a semiparametric estimator based on nonparametric spectral estimators. We derive the second order Edgeworth expansion of the distribution of . Then it is shown that the second order asymptotic properties are independent of the bandwidth choice for residual spectral estimator, which implies that has the same rate of convergence as in regular parametric estimation. This is a sharp contrast with the general semiparametric estimation theory. We also examine the second order Gaussian efficiency of . Numerical studies are given to confirm the theoretical results.     

Quadratic prediction problems in multivariate linear models

Linear and quadratic prediction problems in finite populations have become of great interest to many authors recently. In the present paper, we mainly aim to extend the problem of quadratic prediction from a general linear model, of form , to a multivariate linear model, denoted by with . Firstly, the optimal invariant quadratic unbiased (OIQU) predictor and the optimal invariant quadratic (potentially) biased (OIQB) predictor of for any particular symmetric nonnegative definite matrix satisfying are derived. Secondly, we consider predicting and . The corresponding restricted OIQU predictor and restricted OIQB predictor for them are given. In addition, we also offer four concluding remarks. One concerns the generalization of predicting and , and the others are concerned with three possible extensions from multivariate linear models to growth curve models, to restricted multivariate linear models, and to matrix elliptical linear models.     

Single-index quantile regression

Nonparametric quantile regression with multivariate covariates is a difficult estimation problem due to the “curse of dimensionality”. To reduce the dimensionality while still retaining the flexibility of a nonparametric model, we propose modeling the conditional quantile by a single-index function , where a univariate link function g₀(⋅) is applied to a linear combination of covariates , often called the single-index. We introduce a practical algorithm where the unknown link function g₀(⋅) is estimated by local linear quantile regression and the parametric index is estimated through linear quantile regression. Large sample properties of estimators are studied, which facilitate further inference. Both the modeling and estimation approaches are demonstrated by simulation studies and real data applications.     

Testing the stability of the functional autoregressive process

The functional autoregressive process has become a useful tool in the analysis of functional time series data. It is defined by the equation , in which the observations X_n and errors ε_n are curves, and is an operator. To ensure meaningful inference and prediction based on this model, it is important to verify that the operator does not change with time. We propose a method for testing the constancy of against a change-point alternative which uses the functional principal component analysis. The test statistic is constructed to have a well-known asymptotic distribution, but the asymptotic justification of the procedure is very delicate. We develop a new truncation approach which together with Mensov’s inequality can be used in other problems of functional time series analysis. The estimation of the principal components introduces asymptotically non-negligible terms, which however cancel because of the special form of our test statistic (CUSUM type). The test is implemented using the R package fda, and its finite sample performance is examined by application to credit card transaction data.     

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.     

A majorization algorithm for constrained correlation matrix approximation

We desire to find a correlation matrix of a given rank that is as close as possible to an input matrix R, subject to the constraint that specified elements in must be zero. Our optimality criterion is the weighted Frobenius norm of the approximation error, and we use a constrained majorization algorithm to solve the problem. Although many correlation matrix approximation approaches have been proposed, this specific problem, with the rank specification and the constraints, has not been studied until now. We discuss solution feasibility, convergence, and computational effort. We also present several examples.     

Robust semiparametric M-estimation and the weighted bootstrap

M-estimation is a widely used technique for statistical inference. In this paper, we study properties of ordinary and weighted M-estimators for semiparametric models, especially when there exist parameters that cannot be estimated at the convergence rate. Results on consistency, rates of convergence for all parameters, and consistency and asymptotic normality for the Euclidean parameters are provided. These results, together with a generic paradigm for studying semiparametric M-estimators, provide a valuable extension to previous related research on semiparametric maximum-likelihood estimators (MLEs). Although penalized M-estimation does not in general fit in the framework we discuss here, it is shown for a great variety of models that many of the forgoing results still hold, including the consistency and asymptotic normality of the Euclidean parameters. For semiparametric M-estimators that are not likelihood based, general inference procedures for the Euclidean parameters have not previously been developed. We demonstrate that our paradigm leads naturally to verification of the validity of the weighted bootstrap in this setting. For illustration, several examples are investigated in detail. The new M-estimation framework and accompanying weighted bootstrap technique shed light on a universal way of investigating semiparametric models.     

Statistical inference for the index parameter in single-index models

In this paper, we are concerned with statistical inference for the index parameter in the single-index model . Based on the estimates obtained by the local linear method, we extend the generalized likelihood ratio test to the single-index model. We investigate the asymptotic behaviour of the proposed test and demonstrate that its limiting null distribution follows a χ²-distribution, with the scale constant and the number of degrees of freedom being independent of nuisance parameters or functions, which is called the Wilks phenomenon. A simulated example is used to illustrate the performance of the testing approach.     

Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation , where R is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a given matrix. Denote by ‖⋅‖ the Euclidean norm in R^d, and let F be the distribution function of R. The main result of this paper is an asymptotic expansion of the probability for F in the Gumbel or the Weibull max-domain of attraction. In the special case that is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) [1].     

Asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis when the dimension is large

This paper examines asymptotic expansions of test statistics for dimensionality and additional information in canonical correlation analysis based on a sample of size N=n+1 on two sets of variables, i.e.,  and . These problems are related to dimension reduction. The asymptotic approximations of the statistics have been studied extensively when dimensions p₁ and p₂ are fixed and the sample size N tends to infinity. However, the approximations worsen as p₁ and p₂ increase. This paper derives asymptotic expansions of the test statistics when both the sample size and dimension are large, assuming that and have a joint (p₁+p₂)-variate normal distribution. Numerical simulations revealed that this approximation is more accurate than the classical approximation as the dimension increases.     

Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

For all p>2,k>p, a size-and-reflection-shape space of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold , a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p>2. The Veronese embedding of the planar Kendall shape manifold is extended to an equivariant embedding of the size-and-shape manifold , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.     

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 .