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Estimation of quantities determined as implicit functions of unknown parameters

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Abstract

Consider a family of distribution functions \({\{F(x, \theta),\,\theta \in \Theta\}}\) . Suppose that there exists an estimator of the unknown parameter vector θ based on given data set. Then it is readily to obtain an estimator of any quantity given as an explicit function g(θ). Particularly, it is the case when the maximum likelihood estimator of θ is available. However, often some quantities of interest can not be expressed as an explicit function, rather it is determined as an implicit function of θ. The present article studies this problem. Sufficient conditions are given for deriving estimators of these quantities. The results are then applied to estimate change point of failure rate function, and change point of mean residual life function.

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Authors and Affiliations

  1. Department of Statistics, Florida International University, University Park, Miami, FL, 33199, USA

    Jie Mi

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  1. Jie Mi
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Correspondence to Jie Mi.

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Mi, J. Estimation of quantities determined as implicit functions of unknown parameters. Metrika 71, 353–359 (2010). https://doi.org/10.1007/s00184-009-0235-6

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  • DOI: https://doi.org/10.1007/s00184-009-0235-6

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