Large Sample Properties And Applications Of Sample Quantiles Based On The Mid-distribution Function

In recent years, many scholars have pointed out that quantile functions areeffective and equivalent replacement of the distribution functions in modeling andanalysis statistical data. And in many cases, quantile functions are more effective thanthe distribution functions. Great progress has been made in the study of classicalsample quantiles whether in theoretical research or practical application. However,many drawbacks occur with the classical definition of sample quantiles. Samplequantiles based on mid-distribution functions introduced by Yanyuan Ma, Marc G.Genton and Emanuel Parzen (2011) make up for these drawbacks of the classicalquantiles to a certain extent. For example, the asymptotic distribution of samplequantiles in the classical definition is well-known to be normal for absolutelycontinuous distributions. However, a lot of Monte Carlo simulations show that this isno longer true for discrete distributions or samples with ties. The definition of samplequantiles based on mid-distribution functions resolves this issue to a certain extentand provides a unified framework for asymptotic properties of sample quantiles fromabsolutely continuous and from discrete distributions.Along the line of the proof of the Berry-Esseen bound of classical samplequantiles, and with the aid of the forced convergence criterion, this paper investigatefirstly the Berry-Esseen bound of sample quantiles based on mid-distributionfunctions for absolutely continuous distributions, and show that the rate ofconvergence in connection with the asymptotic normality of these sample quantiles isstill the typical rate1O n2.Next, the paper discusses the limit behaviors for deviation between thepopulation quantile pand the sample quantile pnbased on the mid-distributionfunctions. The moderate deviation and large deviation principles for pn pareestablished for absolutely continuous distribution.At last, inspired by the concept of sample quantiles based on the mid-distribution functions, this paper provides a correction method to solve differentquantile regression curves intersection problem in linear quantile regression analysis.The example analysis shows that the proposed correction method is effective.