Improved M-estimates in Convex Minimization – an Easy Empirical Likelihood Approach

Let m(z, θ) be a criterion function convex in parameter θ for every z. For a random sample Z1, . . . , Zn, the M-estimate θ̃ of θ minimizes the criterion function ∑n j=1 n −1m(Zj , θ). Suppose side information is available given by E(u(Z1)) = 0 for some square-integrable function u. In this article, we are concerned with the use of side information and propose to estimate θ by θ̂ which minimizes the criterion function ∑n j=1 πnjm(Zj , θ) with πnj = n (1+ ζ n u(Zj)) −1 for some random variable ζn determined by u(Zj)’s. We show θ̂ is asymptotically normal and more efficient than θ̃. As applications of the results, we construct efficient estimates of quantitles, parameters in quantitle regression and in the Cox proportational hazard (PH) regression. A simulation study and real data application are performed to illustrate the use of side information in the Cox PH model to improve the efficiency of maximum partial likelihood estimates.