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

Abstract

Let ( x i , y i ), i = 1, …, n , be a sequence of observations such that y i = b i + cx i + u i , where b i and c are unknown parameters, and { u i } and { x i } are independent sequences of independent, identically distributed random variables. The likelihood ratio test is derived for the hypothesis that b i = b ( i = 1, …, n ), against the alternative that b i = b ( i ≤ i 0 ) and b i = b + d ( i > i 0 ) for some b, i 0 , and d ≠ 0, assuming the u i '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.