Asymptotic properties of penalized spline estimators in concave extended linear models: Rates of convergence

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree, n, because of the fact that this space, M(n), can be realised as a kind of high-dimensional algebraic surface of dimension n(2s + m) where s and m are the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determining n and then the appropriate element of M(n). However, the fact that M(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record, T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

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Functional Linear Regression

10.1093/oxfordhb/9780199568444.013.2 ◽

2018 ◽

Author(s):

Hervé Cardot ◽

Pascal Sarda

Keyword(s):

Linear Regression ◽

Least Squares ◽

Linear Models ◽

Estimation Error ◽

Asymptotic Properties ◽

Principal Component Regression ◽

Principal Component ◽

Penalized Least Squares ◽

Open Problems ◽

Functional Linear Regression

This article presents a selected bibliography on functional linear regression (FLR) and highlights the key contributions from both applied and theoretical points of view. It first defines FLR in the case of a scalar response and shows how its modelization can also be extended to the case of a functional response. It then considers two kinds of estimation procedures for this slope parameter: projection-based estimators in which regularization is performed through dimension reduction, such as functional principal component regression, and penalized least squares estimators that take into account a penalized least squares minimization problem. The article proceeds by discussing the main asymptotic properties separating results on mean square prediction error and results on L2 estimation error. It also describes some related models, including generalized functional linear models and FLR on quantiles, and concludes with a complementary bibliography and some open problems.

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Robust Statistical Inference in Generalized Linear Models Based on Minimum Renyi’s Pseudodistance Estimators

Entropy ◽

10.3390/e24010123 ◽

2022 ◽

Vol 24 (1) ◽

pp. 123

Author(s):

María Jaenada ◽

Leandro Pardo

Keyword(s):

Generalized Linear Models ◽

Influence Function ◽

Regression Models ◽

Linear Models ◽

Asymptotic Properties ◽

Significant Loss ◽

Superior Performance ◽

Test Statistics ◽

Linear Regression Models ◽

Type Test

Minimum Renyi’s pseudodistance estimators (MRPEs) enjoy good robustness properties without a significant loss of efficiency in general statistical models, and, in particular, for linear regression models (LRMs). In this line, Castilla et al. considered robust Wald-type test statistics in LRMs based on these MRPEs. In this paper, we extend the theory of MRPEs to Generalized Linear Models (GLMs) using independent and nonidentically distributed observations (INIDO). We derive asymptotic properties of the proposed estimators and analyze their influence function to asses their robustness properties. Additionally, we define robust Wald-type test statistics for testing linear hypothesis and theoretically study their asymptotic distribution, as well as their influence function. The performance of the proposed MRPEs and Wald-type test statistics are empirically examined for the Poisson Regression models through a simulation study, focusing on their robustness properties. We finally test the proposed methods in a real dataset related to the treatment of epilepsy, illustrating the superior performance of the robust MRPEs as well as Wald-type tests.

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‘Asymptotic properties of penalized spline estimators’

Biometrika ◽

10.1093/biomet/asy024 ◽

2018 ◽

Vol 105 (2) ◽

pp. 503-503 ◽

Cited By ~ 1

Author(s):

G Claeskens ◽

T Krivobokova ◽

J D Opsomer

Keyword(s):

Asymptotic Properties ◽

Penalized Spline

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Multivariate linear time series models

Advances in Applied Probability ◽

10.1017/s0001867800022710 ◽

1984 ◽

Vol 16 (03) ◽

pp. 492-561 ◽

Cited By ~ 11

Author(s):

E. J. Hannan ◽

L. Kavalieris

Keyword(s):

Limit Theorems ◽

Analysis Of Algorithms ◽

Linear Time ◽

Asymptotic Properties ◽

Real Data ◽

Rates Of Convergence ◽

Structure Theory ◽

Time Series Models ◽

Transfer Model ◽

Rational Transfer

This paper is in three parts. The first deals with the algebraic and topological structure of spaces of rational transfer function linear systems—ARMAX systems, as they have been called. This structure theory is dominated by the concept of a space of systems of order, or McMillan degree,n,because of the fact that this space,M(n), can be realised as a kind of high-dimensional algebraic surface of dimensionn(2s+m) wheresandmare the numbers of outputs and inputs. In principle, therefore, the fitting of a rational transfer model to data can be considered as the problem of determiningnand then the appropriate element ofM(n). However, the fact thatM(n) appears to need a large number of coordinate neighbourhoods to cover it complicates the task. The problems associated with this program, as well as theory necessary for the analysis of algorithms to carry out aspects of the program, are also discussed in this first part of the paper, Sections 1 and 2.The second part, Sections 3 and 4, deals with algorithms to carry out the fitting of a model and exhibits these algorithms through simulations and the analysis of real data.The third part of the paper discusses the asymptotic properties of the algorithm. These properties depend on uniform rates of convergence being established for covariances up to some lag increasing indefinitely with the length of record,T. The necessary limit theorems and the analysis of the algorithms are given in Section 5. Many of these results are of interest independent of the algorithms being studied.

Download Full-text