Extrapolated empirical likelihood as a solution to the convex-hull-violation problem

Empirical likelihood (EL) breaks down when the hypothesised mean falls outside the convex hull of the sample. We propose extrapolated EL (ExEL) – two splicing schemes that extend the log-EL ratio beyond the hull while leaving it unchanged on a user-chosen interior region. The first scheme, ExEL1, continues EL past a data-driven cut-off using its local quadratic (Taylor) expansion. The second scheme, ExEL2, smoothly splices EL to its globalWald quadratic approximation via a convex bridge. Both methods extend naturally to multiple dimensions by radial reduction. In simulations with small samples – where convex-hull violations are common – ExEL remains well-behaved and distinguishes mild from severe violations. It also has attractive inferential properties, delivering accurate coverage probabilities with bootstrap calibration.