Linear Estimation of the Location and Scale Parameters of the Cauchy Distribution Based on Sample Quantiles

Abstract

Abstract The asymptotically best linear estimate of the location or scale parameter, assuming one is known, based on a few, say k, order statistics selected from a large sample is considered. Through an analytic approach, some spacings (a spacing is a set of proportions 0 <λ1 <λ2 < … <λk 1) are proposed such that the asymptotically best linear estimates based on the sample quantiles x[nλ1] x[n λ2] < … x [nλk] determined by them have high efficiencies compared with those based on most other choices of spacings. Coefficients of the sample quantiles in these estimates are computed for k = 1(1)10. These estimates yield more than 92 percent asymptotic relative efficiencies (in the Cramér-Rao sense) for k≥4. It is also found that the asymptotically best linear estimate of the parameters, when both of them are unknown, determined by the spacings {i/(k+1), i = 1, 2, …, k} have joint asymptotic relative efficiencies ≥65 percent for all k >2.