• Park, Hyo-Il (Department of Statistics, Cheongju University) ;
  • Na, Jong-Hwa (Department of Information and Statistics and Institute for Basic Science Research, Chungbuk National University)
  • Published : 2004.12.01


In this paper, we define the weighted U-empirical process for simple linear model and show the weak convergence to a Gaussian process under some conditions. Then we illustrate the usage of our result with examples. In the appendix, we derive the variance of the weighted U-empirical distribution function.


  1. BICKEL, P. J. AND DOKSUM, K. A. (1977). Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day, San Francisco
  2. H$\'A$JEK, J. (1968). 'Asymptotic normality of simple linear rank statistics under alternatives', The Annals of Mathematical Statistics, 39, 325-346
  3. KOUL, H. L. (1992). 'Weighted empiricals and linear models', IMS Lecture Notes Monograph Series, Vol. 21, Institute of Mathematical Statistics, Hayward, California
  4. MAJOR, P. (1994). 'Asymptotic distributions for weighted U-statistics', The Annals of Probability, 22, 1514-1535
  5. O'NEIL, K. A. AND REDNER, R. A. (1993). 'Asymptotic distributions of weighted U-statistics of degree 2', The Annals of Probability, 21, 1159-1169
  6. SEN, P. K. (1968). 'Estimates of the regression coefficient based on Kendall's tau', Journal of the American Statistical Association, 63, 1379-1389
  7. SERFLING, R. J. (1980). Approximation Theorems of Mathematical Statistics, John Wiley & Sons, New York
  8. SIEVERS, G. L. (1978). 'Weighted rank statistics for simple linear regression', Journal of the American Statistical Association, 73, 628-631
  9. SILVERMAN, B. W. (1983). 'Convergence of a class of empirical distribution functions of dependent random variables', The Annals of Probability, 11, 745-751