A robust sequential fixed-width confidence interval for count data based on Bhattacharyya-Hellinger distance estimator

Abstract

ABSTRACTFor count data having an unknown mass function g0, we use the minimum Bhattacharyya-Hellinger distance (MBHD) estimator and a stopping rule to construct a sequential fixed-width confidence interval for a functional T(g0) = θ0, where fθ0 is the best-fitting parametric model achieving the MBHD between g0 and any member of a parametric class of mass functions. We establish the asymptotic consistency and efficiency properties of the sequential confidence interval and the expected sample size, respectively, as the half-width d → 0. When the count data come from a gross-error contamination model gα, L = (1 − α)fθ + αδ{L} for α ∈ (0, 1), where a parametric model fθ is mixed with a point mass δ{L} located at a value L, we reparameterize L = Ld such that Ld → ∞ as d → 0 and theoretically show that the expected sample size is affected by α, whereas the coverage probability of the sequential confidence interval depends on the rate of [T(gα, Ld) − θ]/d, as d → 0. Our reparameterization fully exploits the MBHD estimator's inherent ability to progressively ignore increasing values of L, providing an asymptotically consistent sequential fixed-width interval estimator of θ. When fθ is Poisson (θ), simulations are conducted to corroborate our theoretical results and to contrast the performance of the MBHD with that of the maximum likelihood estimator (MLE) of θ. A real data on Death Notice, modeled as a negative binomial, are analyzed to compare and contrast the performance of our sequential MBHD procedure with that of the MLE.