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Mean-square-error-robustness of linear estimates in the exponential model. (English) Zbl 0603.62040

The model \(M_ 1\) is defined as the exponential distribution with unknown mean \(\lambda\). The purpose of the paper is to find a robust estimator of \(\lambda\) when \(M_ 1\) is violated in such a way that observations are drawn from a distribution in an extension of \(M_ 1\). Two such extensions are considered, namely the family of exponential power distribution \(M_{p_ 1,p_ 2}\) with pdf \[ f_{\lambda,p}(x)=\exp \{-(x/\lambda)^ p\}/\{\lambda \Gamma (1+1/p)\},\quad x>0,\quad p_ 1\leq p\leq p_ 2,\quad and\quad 0<p_ 1\leq 1\leq p_ 2\leq 2.16, \] and the family of gamma distributions \(M^*_{p_ 1,p_ 2}\) with pdf \[ f^*_{\lambda,p}(x)=\{x^{p-1} \exp (-x/\lambda)\}/\{\lambda^ p \Gamma (p)\}\quad x>0,\quad p_ 1\leq p\leq p_ 2,\quad and\quad 0<p_ 1\leq 1\leq p_ 2<\infty. \] The estimators considered belong to the class \(\tau^+\) of linear combinations of order statistics which are unbiased under the original model \(M_ 1\). The criterion used for selecting estimators is ’mean- square-error-robustness’, defined for an estimator T as \[ v_ T(\lambda)=\sup_{p}E[(T-\lambda)^ 2]-\inf_{p}E[(T-\lambda)^ 2]. \] An estimator \(T_ 0\in \tau^+\) is called the uniformly most v- robust estimator (UMVRE) if \(v_{T_ 0}(\lambda)\leq v_ T(\lambda)\) for each \(\lambda >0\) and every \(T\in \tau^+\). Sufficient conditions for the existence of an UMVRE under the extensions \(M_{p_ 1,p_ 2}\) and \(M^*_{p_ 1,p_ 2}\) are proved. It is also proved that the sample mean is not the UMVRE. For the extension \(M_{p_ 1,p_ 2}\) the case with two observations is completely solved.
The results are very interesting and encouraging for further research in several ways. One important question that is left unsolved concerns the performance of the UMVRE relative to an estimator where \(\lambda\) and the nuisance parameter p are jointly estimated, e.g. in a maximum likelihood approach.
Reviewer: H.Nyquist

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62F10 Point estimation
62G30 Order statistics; empirical distribution functions
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