1,812 research outputs found
Local linear spatial quantile regression
Copyright @ 2009 International Statistical Institute / Bernoulli Society for Mathematical Statistics and Probability.Let {(Yi,Xi), i ∈ ZN} be a stationary real-valued (d + 1)-dimensional spatial processes. Denote by x →
qp(x), p ∈ (0, 1), x ∈ Rd , the spatial quantile regression function of order p, characterized by P{Yi ≤
qp(x)|Xi = x} = p. Assume that the process has been observed over an N-dimensional rectangular domain
of the form In := {i = (i1, . . . , iN) ∈ ZN|1 ≤ ik
≤ nk, k = 1, . . . , N}, with n = (n1, . . . , nN) ∈ ZN. We
propose a local linear estimator of qp. That estimator extends to random fields with unspecified and possibly
highly complex spatial dependence structure, the quantile regression methods considered in the context of
independent samples or time series. Under mild regularity assumptions, we obtain a Bahadur representation
for the estimators of qp and its first-order derivatives, from which we establish consistency and asymptotic
normality. The spatial process is assumed to satisfy general mixing conditions, generalizing classical time
series mixing concepts. The size of the rectangular domain In is allowed to tend to infinity at different
rates depending on the direction in ZN (non-isotropic asymptotics). The method provides muchAustralian Research Counci
Semiparametrically Efficient Inference Based on Signs and Ranks for Median Restricted Models
Since the pioneering work of Koenker and Bassett (1978), econometric models involving median and quantile rather than the classical mean or conditional mean concepts have attracted much interest.Contrary to the traditional models where the noise is assumed to have mean zero, median-restricted models enjoy a rich group-invariance structure.In this paper, we exploit this invariance structure in order to obtain semiparametrically efficient inference procedures for these models.These procedures are based on residual signs and ranks, and therefore insensitive to possible misspecification of the underlying innovation density, yet semiparametrically efficient at correctly specified densities.This latter combination is a definite advantage of these procedures over classical quasi-likelihood methods.The techniques we propose can be applied, without additional technical difficulties, to both cross-sectional and time-series models.They do not require any explicit tangent space calculation nor any projections on these.models;regression analysis;econometrics
A Class of Simple Distribution-free Rank-based Unit Root Tests (Revision of DP 2010-72)
We propose a class of distribution-free rank-based tests for the null hypothesis of a unit root. This class is indexed by the choice of a reference density g, which needs not coincide with the unknown actual innovation density f. The validity of these tests, in terms of exact finite sample size, is guaranteed, irrespective of the actual underlying density, by distribution-freeness. Those tests are locally and asymptotically optimal under a particular asymptotic scheme, for which we provide a complete analysis of asymptotic relative efficiencies. Rather than asymptotic optimality, however, we emphasize finitesample performances. Finite-sample performances of unit root tests, however, depend quite heavily on initial values. We therefore investigate those performances as a function of initial values. It appears that our rank-based tests significantly outperform the traditional Dickey-Fuller tests, as well as the more recent procedures proposed by Elliot, Rothenberg, and Stock (1996), Ng and Perron (2001), and Elliott and M¨uller (2006), for a broad range of initial values and for heavy-tailed innovation densities. As such, they provide a useful complement to existing techniques.Unit root;Dickey-Fuller test;Local Asymptotic Normality;Rank test
Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields
We prove the asymptotic normality of the kernel density estimator (introduced
by Rosenblatt (1956) and Parzen (1962)) in the context of stationary strongly
mixing random fields. Our approach is based on the Lindeberg's method rather
than on Bernstein's small-block-large-block technique and coupling arguments
widely used in previous works on nonparametric estimation for spatial
processes. Our method allows us to consider only minimal conditions on the
bandwidth parameter and provides a simple criterion on the (non-uniform) strong
mixing coefficients which do not depend on the bandwith.Comment: 16 page
Signal detection in high dimension: The multispiked case
This paper deals with the local asymptotic structure, in the sense of Le
Cam's asymptotic theory of statistical experiments, of the signal detection
problem in high dimension. More precisely, we consider the problem of testing
the null hypothesis of sphericity of a high-dimensional covariance matrix
against an alternative of (unspecified) multiple symmetry-breaking directions
(\textit{multispiked} alternatives). Simple analytical expressions for the
asymptotic power envelope and the asymptotic powers of previously proposed
tests are derived. These asymptotic powers are shown to lie very substantially
below the envelope, at least for relatively small values of the number of
symmetry-breaking directions under the alternative. In contrast, the asymptotic
power of the likelihood ratio test based on the eigenvalues of the sample
covariance matrix is shown to be close to that envelope. These results extend
to the case of multispiked alternatives the findings of an earlier study
(Onatski, Moreira and Hallin, 2011) of the single-spiked case. The methods we
are using here, however, are entirely new, as the Laplace approximations
considered in the single-spiked context do not extend to the multispiked case
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