The likelihood ratio test for a change-point in simple linear regression.

*(English)*Zbl 0676.62027The purpose of this paper is to indicate that the use of the likelihood ratio statistic for change point problems in simple linear regression models does not present the apparently intractable analytic difficulties which have hindered its application in the past.

Given \(x_ 1,...,x_ m\), suppose that \(y_ i\) \((i=1,...,m)\) are independent and normally distributed with common variance \(\sigma^ 2\). For some j (the change point) the expectation of \(y_ i\) equals \(\alpha_ 0+\beta_ 0x_ i\), if \(i\leq j\) and equals \(\alpha_ 1+\beta_ 1x_ i\), if \(i>j\). The paper is concerned with the likelihood ratio test of the hypothesis of no change, \(H_ 0: \beta_ 0=\beta_ 1\) and \(\alpha_ 0=\alpha_ 1\) against one of the alternatives, \(H_ 1:\beta_ 0=\beta_ 1=\beta\) and there exists a j \((1\leq j<m)\) such that \(\alpha_ 0\neq \alpha_ 1\) or \(H_ 2:\) there exists a j \((1\leq j<m)\) so that \(\beta_ 0\neq \beta_ 1\) or \(\alpha_ 0\neq \alpha_ 1\). Furthermore, when \(H_ 1\) holds, a confidence region for j is obtained.

Approximations for the significance level of the corresponding likelihood ratio test are given under general assumptions about the empirical distribution of the \(x_ i\). The approximations are compared with the results of simulations; the agreement was very good. Some open problems are listed in the final chapter.

Given \(x_ 1,...,x_ m\), suppose that \(y_ i\) \((i=1,...,m)\) are independent and normally distributed with common variance \(\sigma^ 2\). For some j (the change point) the expectation of \(y_ i\) equals \(\alpha_ 0+\beta_ 0x_ i\), if \(i\leq j\) and equals \(\alpha_ 1+\beta_ 1x_ i\), if \(i>j\). The paper is concerned with the likelihood ratio test of the hypothesis of no change, \(H_ 0: \beta_ 0=\beta_ 1\) and \(\alpha_ 0=\alpha_ 1\) against one of the alternatives, \(H_ 1:\beta_ 0=\beta_ 1=\beta\) and there exists a j \((1\leq j<m)\) such that \(\alpha_ 0\neq \alpha_ 1\) or \(H_ 2:\) there exists a j \((1\leq j<m)\) so that \(\beta_ 0\neq \beta_ 1\) or \(\alpha_ 0\neq \alpha_ 1\). Furthermore, when \(H_ 1\) holds, a confidence region for j is obtained.

Approximations for the significance level of the corresponding likelihood ratio test are given under general assumptions about the empirical distribution of the \(x_ i\). The approximations are compared with the results of simulations; the agreement was very good. Some open problems are listed in the final chapter.

Reviewer: D.Rasch