The Annals of Statistics

Asymptotically efficient estimation of the index of regular variation

Xiaoying Wei

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Abstract

We propose a conditional MLE of the index of regular variation when the functional form of a slowly varying function is assumed known in the tail, and we study its asymptotic properties. We prove asymptotic normality of $P_{\theta}^{k_n}$, a probability measure whose density is the product of the joint conditional density of the $k_n$ largest order statistics from $F_{\theta} (x)$ given $Z_{n-k}$, the $$(n-k)$th order statistic, and a density of $Z_{n-k}$ with parameter $\theta$. Based on this result, we show that this conditional MLE is asymptotically normal and asymptotically efficient in many senses whenever $k_n$ is $o(n)$. We also propose an iterative estimator of $\theta$ given only partial knowledge of $L_{\theta}(x)$. This estimator is asymptotically normal, asymptotically unbiased and asymptotically efficient.

Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2036-2058.

Dates
First available in Project Euclid: 15 October 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1034713646

Digital Object Identifier
doi:10.1214/aos/1034713646

Mathematical Reviews number (MathSciNet)
MR1389864

Zentralblatt MATH identifier
0854.62022

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 62G20: Asymptotic properties

Keywords
LAN asymptotic efficient estimator the index of regular variation

Citation

Wei, Xiaoying. Asymptotically efficient estimation of the index of regular variation. Ann. Statist. 23 (1995), no. 6, 2036--2058. doi:10.1214/aos/1034713646. https://projecteuclid.org/euclid.aos/1034713646


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