Universal Bounds for Mean Range and Extreme Observation

Consider any distribution $f(x)$ with standard deviation σ and let x₁, x₂ ⋯ xₙ denote the order statistics in a sample of size n from $f(x).$ Further let wₙ = xₙ - x₁ denote the sample range. Universal upper and lower bounds are derived for the ratio E(wₙ)/σ for any $f(x)$ for which aσ x bσ, where a and b are given constants. Universal upper bounds are given for E(xₙ)/σ for the case - ∞ < x < ∞. The upper bounds are obtained by adopting procedures of the calculus of variation on lines similar to those used by Plackett [3] and Moriguti [4]. The lower bounds are attained by singular distributions and require the use of special arguments.