Real-time detection of a change-point in a linear expectile model

Real-time changepoint detection in a nonlinear expectile model

“…Next, similarly as in the proof of Lemma 1 in Ciuperca (2022), under Assumptions (A1) and (A2), using the last relation above, we have that for all constants C 1 , C 4 > 0, there exists a constant C > 0 such that…”

“…If the regression function f is linear in β ∈ , then the asymptotic behaviour in the proposition reduces to a special case of Proposition 1 from Ciuperca (2022). Similarly, if the regression function f in (1) is nonlinear in β ∈ , but the random error terms follow some normal distribution N (0, σ 2 ) with σ 2 < ∞, the asymptotic behaviour in Proposition 1 gives the results of Theorem 2.1 in Seber and Wild (2003).…”

“…The test statistic in Theorem 2 is based on the expectile estimator β m of the true parameter vector β 0 ∈ calculated from the historical data. However, the limit process is the same as for the expectile estimator in the linear model considered in Ciuperca (2022), or the quantile estimator proposed in Zhou et al (2015). On the other hand, the test statistic is different from that proposed by Ciuperca (2013) or Horváth et al (2004) where the authors rather considered the CUSUM type statistic based on the least squares residuals of the linear model or the nonlinear model respectively.…”