Adaptive-to-Model Hybrid of Tests for Regressions

Li, Lingzhu; Zhu, Xuehu; Zhu, Lixing

doi:10.1080/01621459.2021.1941052

Cited by 3 publications

(6 citation statements)

References 26 publications

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“…This property implies that it can detect local alternatives with a deviation term slower than n −k/(2k+1) . Unlike the n −1/2 threshold shown in [22], there is a shrinkage of critical order of δ n for functional data, which is the price we have to pay to use reproducing kernel based estimator. However, since k = m + s + 1, we can set a larger m to get the critical order closer to n −1/2 .…”

Section: Indicative Dimensionmentioning

confidence: 98%

“…A2 regularity condition guarantees that • K is well defined. It also implies that the dimension of a subspace in H will be preserved after being applied by C. A3 and A4 are usually known as the linearity condition and constant variance condition under the SDR framework, see [12,13,29] for more information and see [22] for finite-dimensional case. With these assumptions, We imitate the convex combined matrix proposed in [22] and develop the indicative operator on H as…”

Section: Indicative Dimensionmentioning

confidence: 99%

“…We propose an adaptive model checking test for FLM and illuminate its asymptotic properties in different underlying models to address the challenges. Our test is motivated by the adaptive-to-model hybrid test proposed in [22]. We are interested in the incomplete nature of collected data.…”

Section: Introductionmentioning

confidence: 99%

“…The first component simply uses a moment-based sum of weighted residuals as a test statistic. It shares asymptotic behaviors with global smoothing methods [31,8] that can achieve the fastest possible convergence rate [16,22] while having a tractable null distribution according to the inference results in [28]. The second component adjusts the typical conditional moment-based test proposed by [39] for functional data.…”

Section: Introductionmentioning

confidence: 99%

“…The indicative dimension borrows ideas from sufficient dimension reduction (SDR) theory [6,41,21]. It has been applied to building adaptive-to-model tests [16,33,22] which can alleviate the curse of dimensionality. In the past decades, many efforts have been devoted to functional SDR [12,13,18,11,23,29,30], paving the way to build the hybrid tests for FLM.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive model checking test for functional linear model

Shi¹,

Liu²,

Sun³

et al. 2022

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Numerous studies have been devoted to the estimation and inference problems for functional linear models (FLM). However, few works focus on model checking problem that ensures the reliability of results. Limited tests in this area do not have tractable null distributions or asymptotic analysis under alternatives. Also, the functional predictor is usually assumed to be fully observed, which is impractical. To address these problems, we propose an adaptive model checking test for FLM. It combines regular moment-based and conditional moment-based tests, and achieves model adaptivity via the dimension of a residual-based subspace. The advantages of our test are manifold. First, it has a tractable chi-squared null distribution and higher powers under the alternatives than its components. Second, asymptotic properties under different underlying models are developed, including the unvisited local alternatives. Third, the test statistic is constructed upon finite grid points, which incorporates the discrete nature of collected data. We develop the desirable relationship between sample size and number of grid points to maintain the asymptotic properties. Besides, we provide a data-driven approach to estimate the dimension leading to model adaptivity, which is promising in sufficient dimension reduction. We conduct comprehensive numerical experiments to demonstrate the advantages the test inherits from its two simple components.

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Section: Indicative Dimensionmentioning

confidence: 98%