Optimal spacing of the selected sample quantiles for the joint estimation of the location and scale parameters of a symmetric distribution

Abstract

We are considering the ABLUE’s – asymptotic best linear unbiased estimators – of the location parameter μ and the scale parameter σ of the population jointly based on a set of selected k sample quantiles, when the population distribution has the density of the form g(x)= 1 σ f x−μ σ , −∞<μ<∞, σ>0, where the standardized function f( u) being of a known functional form. A set of selected sample quantiles with a designated spacing λ : λ 1<λ 2<⋯<λ k, or in terms of u=( x− μ)/ σ u : −∞<u 1<u 2<⋯<u k<+∞, where λ i=∫ −∞ u i f(t) dt, i=1,2,…,k are given by x(n 1)<x(n 2)<⋯<x(n k), where n i= nλ i if nλ i is an integer , i=1,2,…,k, [nλ i]+1 otherwise. Asymptotic distribution of the k sample quantiles when n is very large is given by h(x(n 1),x(n 2),…,x(n k);μ,σ)=(2 πσ 2) −k/2[λ 1(λ 2−λ 1)⋯(λ k−λ k−1)(1−λ k)] −1/2n k/2 exp(−nS/2σ 2), where S= ∑ i=1 k λ i+1−λ i−1 (λ i+1−λ i)(λ i−λ i−1) f i 2(x(n i)−μ−u iσ) 2−2 ∑ i=2 k f if i−1 λ i−λ i−1 (x(n i)−μ−u iσ)(x(n i−1)−μ−u i−1σ), f i=f(u i), i=0,1,…,k,k+1, f 0=f k+1=0, λ 0=0, λ k+1=1. The relative efficiency of the joint estimation is given by η(μ,σ)= 1 κ Δ Δ=K 1K 2−K 3 2, where K 1= ∑ i=1 k+1 (f i−f i−1) 2 λ i−λ i−1 , K 2= ∑ i=1 k+1 (u if i−u i−1f i−1) 2 λ i−λ i−1 , K 3= ∑ i=1 k+1 (f i−f i−1) (u if i−u i−1f i−1) λ i−λ i−1 and κ being independent of the spacing λ . The optimal spacing is the spacing which maximizes the relative efficiency η( μ, σ). We will prove the following rather remarkable theorem. Theorem. The optimal spacing for the joint estimation is symmetric, i.e. λ i+λ k−i+1=1, or u i+u k−i+1=0, i=1,2,…,k, if the standardized density f( u) of the population is differentiable infinitely many times and symmetric f(−u)=f(u), f′(−u)=−f′(u).