Estimation of a support curve via order statistics (Q5954057)

An i.i.d. sample \((X_i,Y_i)\), \(i=1,\dots,n\) is observed with unknown pdf \(f(x,y)\) in \({\mathbb{R}}^d\). It is supposed that the support of \(f\) is of the form \(\{(x,y):y<g(x)\}\), where the function \(g\) (frontier or support curve) is unknown and has to be estimated. The authors consider the cases where \(f(x,g(x,y))\) can be 0 or \(\infty\) in \((0,\infty)\) for some points \(x\). They propose to build the estimator in two stages. At the first stage some estimators like the maximum estimator \(\hat g(x)=\max\{Y_j:|X_j-x|\leq h\}\) (\(h\) being a bandwidth) or the Dekker-de Haan (DH) estimator are used to ``improve'' the data taking \(\tilde Y_i=\hat g(x)\) if \(|X_i-x|\leq h\) and \(\tilde Y_j=Y_j\) otherwise. Then a Data Envelopment Analysis Estimator (DEAE) is applied to the improved points \((X_i,\tilde Y_i)\). (The DEAE is the lowest concave and increasing function covering all the sample points). It is shown that the obtained estimator is better than the DEAE as \(n\to\infty\) with probability 1. Convergence rates of the maximum and DH estimators are obtained. Results of simulation studies are presented.