文档介绍：ESTIMATION OF THE NUMBER OF JUMPS OF THE JUMP REGRESSIONFUNCTIONSPeihua QiuDepartment of StatisticsUniversity of Wisconsin - Madison1210 West Dayton StreetMadison, WI 53706Key Words and Phrases: Jump regression functions; Kernel smoothers; . con-sistent; Rate of convergenceABSTRACTThis paper suggests an estimator of the number of jumps of the jump regressionfunctions. The estimator is based on the di erence between right and left one-sided kernel smoothers. It is proved to be . consistent. Some results about itsrate of convergence are also IntroductionRegression analysis is one of the most mature branches in statistics. For a longtime, however, its main theory is about the continuous regression functions (c. and Smith (1981), Hardle (1990), etc. ). Recently, discontinuous regressionfunctions have gotten more and more attention from statisticians all over the , we think, is mainly due to their great application background (e. g. Wahba(1986) used the discontinuous regression model to explore the equi-temperaturesurfaces of the high sky and the deep ocean).By now, we have found that jump regression functions are discussed in twostatistical elds. One is the change-point eld, in which statisticians discuss thejump regression functions by means of the change-point methods (see Chen (1988),Medvedev and Kazachenok (1986), etc. ). The other is the partial spline method,which merges the jump regression functions into the so-called partial linear regres-sion models and deals with them as a whole (see Wahba (1986), Shiau (1987), etc.). There is mon point in these two methods that they both merge jumpregression models into other more general models. Doing so inevitably loses muchof the special features of the jump regression and Owen (1986) may be the rst to analyse the discontinuousregression functions directly from their own features. In that paper, they provideda split linear smoothing algorithm which is based on three s