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Estimation, inference and model selection for jump regression models

Steffen Grønneberg, Gudmund Hermansen, Nils Lid Hjort

Abstract

We consider regression models with data of the type $y_i=m(x_i)+\varepsilon_i$, where the $m(x)$ curve is taken locally constant, with unknown levels and jump points. We investigate the large-sample properties of the minimum least squares estimators, finding in particular that jump point parameters and level parameters are estimated with respectively $n$-rate precision and $\sqrt{n}$-rate precision, where $n$ is sample size. Bayes solutions are investigated as well and found to be superior. We then construct jump information criteria, respectively AJIC and BJIC, for selecting the right number of jump points from data. This is done by following the line of arguments that lead to the Akaike and Bayesian information criteria AIC and BIC, but which here lead to different formulae due to the different type of large-sample approximations involved.

Estimation, inference and model selection for jump regression models

Abstract

We consider regression models with data of the type , where the curve is taken locally constant, with unknown levels and jump points. We investigate the large-sample properties of the minimum least squares estimators, finding in particular that jump point parameters and level parameters are estimated with respectively -rate precision and -rate precision, where is sample size. Bayes solutions are investigated as well and found to be superior. We then construct jump information criteria, respectively AJIC and BJIC, for selecting the right number of jump points from data. This is done by following the line of arguments that lead to the Akaike and Bayesian information criteria AIC and BIC, but which here lead to different formulae due to the different type of large-sample approximations involved.
Paper Structure (16 sections, 2 theorems, 119 equations, 3 figures, 2 tables)

This paper contains 16 sections, 2 theorems, 119 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The case of smooth regression models
  3. The model and its least false parameters
  4. The log-likelihood function and the $A_n$ process
  5. The Kullback--Leibler distance and the $B_n$ function
  6. The model robust AIC
  7. The BIC
  8. The two-window case: a single discontinuity
  9. The two-window case, outside model conditions
  10. The multi-windows model
  11. The AJIC
  12. The BJIC
  13. Bayes estimators
  14. Algorithms for computing estimates
  15. Illustrations
  16. ...and 1 more sections

Key Result

Lemma 1

There is process convergence where where $V_1$ and $V_2$ are independent zero-mean normals with variances $\sigma_0^2F(\gamma_0)$ and $\sigma_0^2\{1-F(\gamma_0)\}$ and also independent of $N^*(f(\gamma_0),s)$ and $W^*(f(\gamma_0),s)$, introduced above. The convergence takes place in the Skorokhod function space $D([-r,r]^3)$, for eac

Figures (3)

  • Figure 1.1: Illustration of the (\ref{['eq:intro2']}) model, with five windows and $n=250$ data points. The statistical task is to estimate the discontinuity points and the levels from the data, and also to infer the right number of windows in case this is not known a priori.
  • Figure 3.1: Ten simulated $M(s)$ curves of the type (\ref{['eq:Msfirsttime']}), with values $(0.5,2.0,3.0,1.0)$ for $(\sigma_0,a_0,b_0,\lambda)$. Here $\widehat{s}={\rm argmax}(M)$ is the limit distribution of $n(\widehat{\gamma}-\gamma_0)$.
  • Figure 10.1: A simulated illustration based on 1000 realisations from a true change point model with three break points. The AJIC winner is the model with three breaks, indicated by the vertical lines. The best smooth regression model, according to the model robust AIC, is a cubic polynomial regression model shown by dashed line above.

Theorems & Definitions (6)

  • Lemma 1
  • proof : Sketch of proof
  • Remark 1
  • Lemma 2
  • proof : Sketch of proof
  • Remark 2