Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.     

On rate-optimal nonparametric wavelet regression with long memory moving average errors

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes $B^{s}_{p,q}$ B p , q s . Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples.     

Asymptotic results in segmented multiple regression

This paper studies the asymptotic behavior of the least squares estimators in segmented multiple regression. For a model with more than one partitioning variable, each of which has one or more change-points, we study the asymptotic properties of the estimated change-points and regression coefficients. Using techniques in empirical process theory, we prove the consistency of the least squares estimators and also establish the asymptotic normality of the estimated regression coefficients. For the estimated change-points, we obtain their consistency at the rates of or 1/n, with or without continuity constraints, respectively. The change-points estimated under the continuity constraints are also shown to asymptotically have a multivariate normal distribution. For the case where the regression mean functions are not assumed to be continuous at the change-points, the asymptotic distribution of the estimated change-points involves a step function process, whose distribution does not follow a well-known distribution.     

Signed-rank tests for location in the symmetric independent component model

The so-called independent component (IC) model states that the observed p-vector X is generated via X=ΛZ+μ, where μ is a p-vector, Λ is a full-rank matrix, and the centered random vector Z has independent marginals. We consider the problem of testing the null hypothesis H₀:μ=0 on the basis of i.i.d. observations X₁,…,X_n generated by the symmetric version of the IC model above (for which all ICs have a symmetric distribution about the origin). In the spirit of [M. Hallin, D. Paindaveine, Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks, Annals of Statistics, 30 (2002), 1103-1133], we develop nonparametric (signed-rank) tests, which are valid without any moment assumption and are, for adequately chosen scores, locally and asymptotically optimal (in the Le Cam sense) at given densities. Our tests are measurable with respect to the marginal signed ranks computed in the collection of null residuals , where is a suitable estimate of Λ. Provided that is affine-equivariant, the proposed tests, unlike the standard marginal signed-rank tests developed in [M.L. Puri, P.K. Sen, Nonparametric Methods in Multivariate Analysis, Wiley & Sons, New York, 1971] or any of their obvious generalizations, are affine-invariant. Local powers and asymptotic relative efficiencies (AREs) with respect to Hotelling’s T² test are derived. Quite remarkably, when Gaussian scores are used, these AREs are always greater than or equal to one, with equality in the multinormal model only. Finite-sample efficiencies and robustness properties are investigated through a Monte Carlo study.     

Confidence intervals for marginal parameters under fractional linear regression imputation for missing data

Item nonresponse occurs frequently in sample surveys and other applications. Imputation is commonly used to fill in the missing item values in a random sample {Y_i;i=1,…,n}. Fractional linear regression imputation, based on the model with independent zero mean errors ?_i, is used to create one or more imputed values in the data file for each missing item Y_i, where {X_i,i=1,…,n}, is observed completely. Asymptotic normality of the imputed estimators of the mean μ=E(Y), distribution function θ=F(y) for a given y, and qth quantile θ_q=F^-1(q),0<q<1 is established, assuming that Y is missing at random (MAR) given X. This result is used to obtain normal approximation (NA)-based confidence intervals on μ,θ and θ_q. In the case of θ_q, a Bahadur-type representation and Woodruff-type confidence intervals are also obtained. Empirical likelihood (EL) ratios are also obtained and shown to be asymptotically scaled variables. This result is used to obtain asymptotically correct EL-based confidence intervals on μ,θ and θ_q. Results of a simulation study on the finite sample performance of NA-based and EL-based confidence intervals are reported.     

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.     

Superefficient estimation of the marginals by exploiting knowledge on the copula

We consider the problem of estimating the marginals in the case where there is knowledge on the copula. If the copula is smooth, it is known that it is possible to improve on the empirical distribution functions: optimal estimators still have a rate of convergence n^−1/2, but a smaller asymptotic variance. In this paper we show that for non-smooth copulas it is sometimes possible to construct superefficient estimators of the marginals: we construct both a copula and, exploiting the information our copula provides, estimators of the marginals with the rate of convergence logn/n.     

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.     

Microstructure noise in the continuous case: The pre-averaging approach

This paper presents a generalized pre-averaging approach for estimating the integrated volatility, in the presence of noise. This approach also provides consistent estimators of other powers of volatility — in particular, it gives feasible ways to consistently estimate the asymptotic variance of the estimator of the integrated volatility. We show that our approach, which possesses an intuitive transparency, can generate rate optimal estimators (with convergence rate n^−1/4$n^{-1/4}$).     

Residual variance estimation using a nearest neighbor statistic

In this paper we consider the problem of estimating E[(Y−E[Y∣X])²] based on a finite sample of independent, but not necessarily identically distributed, random variables . We analyze the theoretical properties of a recently developed estimator. It is shown that the estimator has many theoretically interesting properties, while the practical implementation is simple.     

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].     

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . On the basis of data consisting of n observations on all p+q characteristics and an additional N−n observations on the last q characteristics, where all observations are mutually independent, denote by the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in . For the problem of shrinking to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.     

Nonparametric comparison of regression functions

In this work, we provide a new methodology for comparing regression functions m₁ and m₂ from two samples. Since apart from smoothness no other (parametric) assumptions are required, our approach is based on a comparison of nonparametric estimators and of m₁ and m₂, respectively. The test statistics incorporate weighted differences of and computed at selected points. Since the design variables may come from different distributions, a crucial question is where to compare the two estimators. As our main results we obtain the limit distribution of (properly standardized) under the null hypothesis H₀:m₁=m₂ and under local and global alternatives. We are also able to choose the weight function so as to maximize the power. Furthermore, the tests are asymptotically distribution free under H₀ and both shift and scale invariant. Several such ’s may then be combined to get Maximin tests when the dimension of the local alternative is finite. In a simulation study we found out that our tests achieve the nominal level and already have excellent power for small to moderate sample sizes.     

Strong convergence in nonparametric regression with truncated dependent data

In this paper we derive rates of uniform strong convergence for the kernel estimator of the regression function in a left-truncation model. It is assumed that the lifetime observations with multivariate covariates form a stationary α-mixing sequence. The estimation of the covariate’s density is considered as well. Under the assumption that the lifetime observations are bounded, we show that, by an appropriate choice of the bandwidth, both estimators of the covariate’s density and regression function attain the optimal strong convergence rate known from independent complete samples.     

Progressive Type II censored order statistics for multivariate observations

For a sequence of independent and identically distributed random vectors , i=1,2,…,n, we consider the conditional ordering of these random vectors with respect to the magnitudes of , where N is a p-variate continuous function defined on the support set of X₁ and satisfying certain regularity conditions. We also consider the Progressive Type II right censoring for multivariate observations using conditional ordering. The need for the conditional ordering of random vectors exists for example, in reliability analysis when a system has n independent components each consisting of p arbitrarily dependent and parallel connected elements. Let the vector of life lengths for the ith component of the system be , where denotes the life length of the jth element of the ith component. Then the first failure in the system occurs at time , and for this case . In this paper we introduce the conditionally ordered and Progressive Type II right-censored conditionally ordered statistics for multivariate observations and to study their distributional properties.