Second order asymptotics in nonlinear regression
type: Item
instance of: scholarly article
title: Second order asymptotics in nonlinear regression
zbMATH Open document ID: 0618.62070
DOI: 10.1016/0047-259X(86)90069-2
author: Wolfgang H. Schmidt
author: Silvelyn Zwanzig
published in: Journal of Multivariate Analysis
publication date: 1986-01-01T00:00:00Z
review text: Let \(y_ t=g(x_ t,\beta)+\sigma u_ t\), \(t=1,...,n\), be a nonlinear regression model, where g(\(\cdot,\beta): {\mathcal X}\to R^ 1\) is a known function, \(\beta \in \Xi \subset R^ p\) is an unknown parameter, \(\{u_ t\}_{t=1,2,...}\) are i.i.d. r.v. with \(Eu_ 1=0\), \(Eu^ 2_ 1=1\), \(x_ t\in R^ q\) are nonstochastic known regressors, and \(\sigma\) is an unknown parameter. Under some regularity conditions there exists a positive constant \(c_{{\mathcal K}}>0\) with \[ \sup_{\theta \in {\mathcal K}}P_{\theta}\{\| n^{1/2}({\hat \beta}-\beta)-\sum^{m- 2}_{i=0}n^{-i/2}h_ i\| >c_{{\mathcal K}}n^{-(m-1)/2}(\log n)^{m/2}\}=o(n^{-1/2}), \] where \({\hat \beta}\) is such that \[ Q({\hat \beta})=\min_{\beta \in \Xi^ c}Q(\beta),\quad Q(\beta)=(n^{-1})\sum^{n}_{i=1}(y_ t-g(x_ t,\beta))^ 2, \] \(h_ i(\theta)\) are polynomials of (\(\sigma\) /\(\sqrt{n})g^{i_ 1,...,i_{i+1}}(x_ t,\beta)\) with bounded coefficients for \(\theta\in {\mathcal K}\) and \(n>n_{{\mathcal K}}\) \((f_{\beta}^{i_ 1,...,i_{\ell}}\) denotes the partial derivative \((\partial /\partial \beta_{i_ 1},...,\partial /\partial \beta_{i_{\ell}})f(\beta)\) with \(i_ j\in \{1,...,p\}\), \(j=1,...,l)\). Similar results hold for the least-squares estimator \({\hat \sigma}^ 2\) of a parameter \(\sigma^ 2.\) Second order asymptotically efficient tests for the hypothesis \(H: \gamma\leq \gamma_ 0\) against \(K: \gamma\) \(>\gamma_ 0\) (\(\gamma\) is the first component of \(\beta\) and \(\gamma_ 0\in R^ 1)\) and for the hypothesis \(H: \sigma\) \({}^ 2\leq \sigma^ 2_ 0\) against \(K: \sigma\) \({}^ 2>\sigma^ 2_ 0\), as well as confidence regions for \(\gamma\) and \(\sigma^ 2\) are given.
Mathematics Subject Classification ID: 62J02
Mathematics Subject Classification ID: 62F05
Mathematics Subject Classification ID: 62F25
zbMATH DE Number: 4003295
zbMATH Keywords: Edgeworth expansion
zbMATH Keywords: median unbiased estimators
zbMATH Keywords: least-squares estimator
zbMATH Keywords: Second order asymptotically efficient tests
zbMATH Keywords: confidence regions
MaRDI profile type: Publication
cites work: Q4181803
cites work: Q3903870
cites work: On Efficiency of First and Second Order
cites work: Q4097679
cites work: Simple consistent estimation in nonlinear regression by data transformation and design of experiments
cites work: Asymptotic results on nonlinear approximation of regression functions and weighted least squares
cites work: Confidence bands with a given maximal size for linear regression functions
cites work: The roles of transformations and besponse functions chosen by maximum likelihood
cites work: On unification of the asymptotic theory of nonlinear econometric models
cites work: Q4199381
cites work: An Asymptotic Expansion for the Distribution of a Statistic Admitting an Asymptotic Expansion
cites work: An Asymptotic Expansion for the Distribution of Sums of a Special Form with an Application to Minimum-Contrast Estimates
cites work: Asymptotic theory of nonlinear least squares estimation
cites work: Jackknifing in Nonlinear Regression
cites work: Testing a Subset of the Parameters of a Nonlinear Regression Model
cites work: Q4139492
cites work: Accounting for intrinsic nonlinearity in nonlinear regression parameter inference regions
cites work: Non-linear time series regression
cites work: An asymptotic expansion of the distribution of least squares estimators in the nonlinear regression model
cites work: Asymptotic Properties of Non-Linear Least Squares Estimators
cites work: The Consistency of Nonlinear Regressions
cites work: An asymptotic expansion for the distribution of asymptotic maximum likelihood estimators of vector parameters
cites work: The Fitting of a Generalization of the Logistic Curve
cites work: Q4085018
cites work: On the measurability and consistency of minimum contrast estimates
cites work: Asymptotic expansions related to minimum contrast estimators
cites work: Q4127170
cites work: Q3862863
cites work: On the numerical accuracy of approximations based on edgeworth expansions
cites work: A note on the ban property of the residual sum of squares in regression models
cites work: Asymptotic results for estimation and testing variances in regression models
cites work: Q3040344
cites work: Testing hypotheses in nonlinear regressions
cites work: Q3935276
cites work: Differential geometry of curved exponential families. Curvatures and information loss
cites work: Polynomial approximation of nonlinear regression functions
cites work: The choice of approximative models in nonlinear regression
cites work: A note to the paper of ivanov and zwanzig on the asymptotic expansion of the least squares estimator
description: scientific article; zbMATH DE number 4003295