A Bivariate Test for the Detection of a Systematic Change in Mean

Abstract

Abstract Let (xi, yi ), i = 1, …, n, be a sequence of observations such that yi = bi + cxi + ui , where bi and c are unknown parameters, and {ui } and {xi } are independent sequences of independent, identically distributed random variables. The likelihood ratio test is derived for the hypothesis that bi = b (i = 1, …, n), against the alternative that bi = b (i ≤ i 0) and bi = b + d (i > i 0) for some b, i 0, and d ≠ 0, assuming the ui 's are normal. Quantiles of the test statistic are computed by simulation, and the consistency of the test is proved. Some asymptotic properties of the test statistic are shown.