Asymptotically efficient estimation of the index of regular variation

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.