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.

Key Words:

• Change in mean
• Test consistency
• Bivariate slippage
• Monte Carlo method
• Slippage