Linear Estimation of the Normal Distribution Standard Deviation

Recently in a series of articles ([1], [2], [3], [6], [7], [8]) the estimation of the standard deviation of the normal distribution based on the sample standard deviation, s, has been thoroughly explored-both for unbiased estimation ([1], [2], [3], [6]) and minimum mean square error estimation ([7], [8]). If one is willing to retain the assumption of normality then it is well known that many simple highly efficient linear estimates of the standard deviation also exist (for example in [4] Chapter 9). The usual drawbacks of these linear estimates are that they are appropriate either to small samples (e.g., the sample range) or large samples (e.g., estimates using a few sample quantiles) or else they do apply to all sample sizes (e.g., the BLUE estimates) but involve elaborate weighting schemes for the ordered observations-thus rendering them chiefly only of academic interest. In the following we examine one linear estimate first proposed by Downton [5]. It does not suffer the above mentioned drawbacks and further it compares extremely well with the estimates based on s. Let X1,n ? ? Xn,n represent the ordered observations from a random sample of size n drawn from a normal distribution. Downton's unbiased estimate of the distribution's standard deviation ois