On the Estimation of Fano’s Dispersion Index using Auxiliary Information
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Keywords:
Keywords: Estimation of dispersion; Relative bias; Relative eficiency; Ratio and regression Type estimators; Fano factor; Simulation; Applications.Abstract
This paper addresses the problem of estimating the Fano’s dispersion index (also known as the Fano factor) in the presence of auxiliary information.
The dispersion index, defined as the ratio of the variance to the mean, is a key measure in quantifying over dispersion and arises in numerous applied
settings, including count data modeling in biology, epidemiology, and engineering. We begin by introducing the necessary notation and establishing
the theoretical framework, including the distributional assumptions and expected values underpinning the estimators. The core of the paper involves
a comparative study of three estimators: (i) the naïve estimator, which does not incorporate auxiliary information; (ii) a ratio-type estimator, and (iii)
a regression-type estimator, which utilize the known auxiliary variable. For each estimator, we derive expressions for the expected value, bias, and
mean squared error (MSE), facilitating a theoretical comparison of their performance. The analysis demonstrates the potential gains in efficiency
when auxiliary information is properly utilized. A comprehensive simulation study, carried out in SAS, evaluates the finite-sample properties of the
estimators across a range of scenarios. We conclude with an application to a real-world data set, highlighting the practical implications of the proposed
methods. The performance of the estimators is assessed via relative bias and relative efficiency, offering guidance for their use in applied statistical
practice.
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