Asymptotically efficient estimates of the index of regular variation.

Asymptotically efficient estimators of the index of regular variation are proposed and their convergence rates and other asymptotic properties are studied. Suppose that a distribution function $F\sb{\theta}(x)$ is regularly varying at $\infty$. Let $X\sb1,X\sb2,$ ...,$X\sb{n}$ be i.i.d. from $F\sb{\theta}(x)$ and $Z\sb1,Z\sb2,$ ... $,Z\sb{n}$ be the corresponding order statistics. We are interested in estimating $\theta$, the index of regular variation. It is known that if the slowly varying function $L\sb{\theta}(x)$ is a constant, Hill's estimator is a conditional MLE of $\theta$ based on the ($k\sb{n}$ + 1) largest order statistics. When $L\sb{\theta}(x)$ is not a constant, Hill's estimator is only a consistent estimator of $\theta$ if $k\sb{n}$ is such that $k\sb{n}$ goes to $\infty$ and $k\sb{n}$ is the order of $o(n)$ as $n$ goes to infinity. Hill's estimator is asymptotically normal only if $k\sb{n}$ goes to infinity slowly enough. In this dissertation we study the conditional MLE of $\theta$ when $L\sb{\theta}(x)$ is not a constant and satisfies certain conditions. We prove the Local Asymptotic Normality of $P\sbsp{\theta}{k\sb{n}}$, a probability measure whose density is the product of the conditional densities of the order statistics which are larger than $Z\sb{n-k}$ given $Z\sb{n-k}$ and a density of $Z\sb{n-k}$ with a parameter $\theta$. Based on this result we show that this conditional MLE is asymptotically normal whenever $k\sb{n}$ goes to $\infty$ and $k\sb{n}$ is $o(n)$ as $n$ goes to infinity and that it is asymptotically efficient in many senses. We also propose an iteration estimator of $\theta$ when only partial knowledge of $L\sb{\theta}(x)$ is known. This estimator is asymptotically normal, asymptotically unbiased and asymptotically efficient. We then derive the convergence rates to the normal distribution with mean zero and variance $\theta\sp2$ for the conditional MLE and our iteration estimator. It turns out that the conditional MLE and the iteration estimator possess the convergence rates of 1/$\sqrt{k\sb{n}}$ if $k\sb{n}$ goes to infinity at a proper rate which depends on the functional form of $L\sb{\theta}(x)$.PhDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103334/1/9308477.pdfDescription of 9308477.pdf : Restricted to UM users only