×

Estimating a standard deviation with \(U\)-statistics of degree more than two: the normal case. (English) Zbl 1258.62022

Summary: We consider unbiased estimation of \(\sigma\) in a \(N(\mu, \sigma^2)\) population. Traditional unbiased stimators consist of appropriate multiples of both the sample standard deviation \(S\), that is \(T^{(1)}\), and Gini’s mean difference (GMD) that is \(T^{(2)}\). Both \(T^{(1)}, T^{(2)}\) depend upon \(U\)-statistics associated with symmetric kernels of degree two. We develop a new approach of constructing higher-order unbiased \(U\)-statistics \(T^{(3)}, T^{(4)}\) and \(T^{(5)}\) for \(\sigma\) based upon symmetric kernels with degrees three and four, respectively. From this investigation, we find that the new unbiased estimators \(T^{(3)}, T^{(4)}\), and \(T^{(5)}\) for \(\sigma\); (i) go practically head-to-head with the existing estimators \(T^{(1)}\) and \(T^{(2)}\), (ii) \(T^{(4)}\) beats \(T^{(2)}\) and (iii) \(T^{(3)}\) very nearly beats \(T^{(2)}\) whether \(n\) is small or moderately large. In other words, it is our belief that this new approach appears very promising.

MSC:

62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
65C60 Computational problems in statistics (MSC2010)
PDFBibTeX XMLCite