Nonparametric Estimation of Sample Percentage Point Standard Deviation

Abstract

The available data consists of a random sample $x(1) < \cdots < x(n)$ from a reasonably well-behaved continuous statistical population. The problem is to estimate the standard deviation of a specified $x(r)$ that is not in the tails of the sample. The estimates examined are of the form $$a\lbrack x(r + i) - x(r - i)\rbrack$$ and the explicit problem consists of determining suitable values for $a$ and $i.$ The solution $$a = (\frac{1}{2})(n + 1)^{-3/10}\{\lbrack r/(n + 1)\rbrack\lbrack 1 - r/(n + 1)\rbrack\}^{1/2}, i \doteq (n + 1)^{4/5}$$ appears to be satisfactory. Then the expected value of the estimate equals the standard deviation of $x(r)$ plus $O(n^{-9/10});$ also the standard deviation of this estimate is $O(n^{-9/10}).$ That is, the fixed and random errors for this point estimate are of the same order of magnitude with respect to $n.$ Solutions can be obtained which decrease the order of one of these types of error. However, these solutions increase the order of the other type of error, so that the overall error magnitude exceeds $O(n^{-9/10}).$