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

ABSTRACT

For 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 f_θ0 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 g_{α, L} = (1 − α)f_θ + αδ_{L} for α ∈ (0, 1), where a parametric model f_θ is mixed with a point mass δ_{L} located at a value L, we reparameterize L = L_d such that L_d → ∞ 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(g_{α, 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 f_θ 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.

KEYWORDS:

• Asymptotic consistency
• asymptotic efficiency
• fixed-width confidence interval
• gross-error contamination
• maximum likelihood
• minimum Hellinger distance
• robustness
• stopping rule

MATHEMATICS SUBJECT CLASSIFICATION:

• 62L10
• 62G20
• 62G35

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Erratum

Acknowledgments

The authors thank the Editor, Dr. Mukhopadhyay, for handling the article swiftly, pointing out missing references, raising interesting questions, and making many suggestions for improvement.