Conditional estimation of average on the basis of weighting data

Abstract

LetF(x,y) be a distribution function of a two dimensional random variable (X,Y). We assume that a distribution functionF _x(x) of the random variableX is known. The variableX will be called an auxiliary variable. Our purpose is estimation of the expected valuem=E(Y) on the basis of two-dimensional simple sample denoted by:U=[(X ₁, Y₁)…(X_n, Y_n)]=[X Y]. LetX=[X ₁…X _n]andY=[Y ₁…Y _n].This sample is drawn from a distribution determined by the functionF(x,y). LetX _(k)be the k-th (k=1, …,n) order statistic determined on the basis of the sampleX. The sampleU is truncated by means of this order statistic into two sub-samples:$U_{k,1} = \left[ {\left( {X_{\left( 1 \right)} ,Y_{\left( {t_1 } \right)} } \right)...\left( {X_{\left( {k - 1} \right)} ,Y_{\left( {t_{k - 1} } \right)} } \right)} \right]$% MathType!End!2!1! and$U_{k,2} = \left[ {\left( {X_{\left( k \right)} ,Y_{\left( {t_k } \right)} } \right)...\left( {X_{\left( n \right)} ,Y_{\left( {t_n } \right)} } \right)} \right]$% MathType!End!2!1!.Let$\bar Y_{U_{1,k} }$% MathType!End!2!1! and$\bar Y_{U_{2,k} }$% MathType!End!2!1! be the sample means from the sub-samplesU _k,1 andU _k,2, respectively. The linear combination$\bar Y_U \left( k \right)$% MathType!End!2!1! of these means is the conditional estimator of the expected valuem. The coefficients of this linear combination depend on the distribution function of auxiliary variable in the pointx _(k).We can show that this statistic is conditionally as well as unconditionally unbiased estimator of the averagem. The variance of this estimator is derived.

The variance of the statistic$\bar Y_U \left( k \right)$% MathType!End!2!1! is compared with the variance of the order sample mean. The generalization of the conditional estimation of the mean is considered, too.