Sequential estimation of quantiles and adaptive sequential estimation of location and scale parameters
Suppose that $X\sb1,X\sb2,\cdots$ are independent observations from a distribution F, and that one wishes to estimate the $p\sp{\rm th}$ quantile $\xi\sb{p}(0 0).$ If $f(\xi\sb{p})$ is known, one may use the best fixed sample size (i.e., in the sense of minimum risk). If $f(\xi\sb{p})$ is unknown, then the best fixed sample size is also unknown. For this case, a stopping rule T = $T\sb{A}$ is proposed. It is shown that, under certain smoothness conditions on F and a growth condition on the delay, the sequential procedure derived is asymptotically risk efficient, i.e., it performs asymptotically as well as the best fixedsamplesize procedure. In the proof, asymptotic moment bounds for the remainder terms in Bahadur's representations of sample quantiles, and certain central order statistics, are derived and used to verify uniform integrability of $$\left\{\left(A\sp{1/4}\vert \ \xi\sb{pT}  \xi\sb{p}\vert\right)\sp2, A \geq 1\right\}.$$Results are extended to a problem that utilizes a more general loss function, $L\sb{n}\prime = A\vert \ \xi\sb{pn}  \xi\sb{p}\vert\sp{r} + n\ (A > 0, r > 0),$ and to estimation of a linear combination of two quantiles.
Year of publication: 
1990


Authors:  Navarro, Mercidita Tulay 
Other Persons:  Martinsek, A. T. (contributor) 
Subject:  Statistics 
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