Change-point estimator in gradually changing sequences (Q2713592)

The following change-point problem is studied: Suppose that observations \(Y_i,\;i =1,\dots , n\), satisfy the regression model NEWLINE\[NEWLINEY_i = \tilde \alpha _0 + \tilde \alpha _1(i/n) + \dots + \tilde \alpha _p(i/n)^p + \beta[((i-k^\ast)/n)_+]^m + e_i,NEWLINE\]NEWLINE where \(e_i\) are i.i.d. random errors with zero mean and unit variance, \(p \geq 0,\;m\geq 1 \) are known integers, \( \tilde \alpha _0, \tilde \alpha _1,\dots , \tilde \alpha _p \) and \(\beta \) are unknown regression coefficients, \(\beta = \beta _n \) may depend on the sample size \(n\), and \(k^\ast \) is an unknown change point. The symbol \(a_+\) stands for \(\max \{a, 0\}.\)NEWLINENEWLINENEWLINEThe change point \(k^\ast \) is estimated by means of the least-squares method and then the asymptotic distribution of this estimator is established under mild moment conditions on the random errors \(e_i\) and under some assumptions on the limit behaviour of \(\beta _n.\)