Sequential estimation of quantiles and adaptive sequential estimation of location and scale parameters
Navarro, Mercidita Tulay
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Permalink
https://hdl.handle.net/2142/22141
Description
Title
Sequential estimation of quantiles and adaptive sequential estimation of location and scale parameters
Author(s)
Navarro, Mercidita Tulay
Issue Date
1990
Doctoral Committee Chair(s)
Martinsek, Adam T.
Department of Study
Statistics
Discipline
Statistics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Statistics
Language
eng
Abstract
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 fixed-sample-size 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.
The problem of estimating the center of a symmetric distribution using the better of the sample mean and the sample median, and that of estimating scale using the better of the sample standard deviation and the sample interquartile range, are studied. Under certain conditions, the risk is minimized (asymptotically) by using the estimator that possesses the smaller asymptotic variance (in scale estimation, the smaller standardized asymptotic variance) and the sample size that minimizes the asymptotic risk of that estimator. Oftentimes, though, the asymptotic variances of the estimators are unknown. In this case, no single estimator and no fixed sample size would minimize the risk. Adaptive sequential procedures that allow one to choose not only the estimators but also the sample sizes in the two problems are investigated. Both interval and point estimation are considered.
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