Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Sequential Shape Modification for Monotone Convex Function: L2 Monotonization and Uniform Convexifiation

  • Lim, Jo-Han (Department of Statistics, Seoul National University) ;
  • Lee, Sung-Im (Department of Statistics, Dankook University)
  • Published : 2008.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

This paper studies two sequential procedures to estimate a monotone convex function using $L_2$ monotonization and uniform convexification; one, denoted by FMSC, monotonizes the data first and then, convexifis the monotone estimate; the other, denoted by FCSM, first convexifies the data and then monotonizes the convex estimate. We show that two shape modifiers are not commutable and so does FMSC and FCSM. We compare them numerically in uniform error(UE) and integrated mean squared error(IMSE). The results show that FMSC has smaller uniform error(UE) and integrated mean squared error(IMSE) than those of FCSC.

Keywords

References

  1. Barlow, R. E. (1972). Statistical Inference under Order Restrictions, John Wiley & Sons, New York
  2. Brunk, H. D. (1955). Maximum likelihood estimates of monotone parameters, The Annals of the Mathematical Statistics, 26, 607-616 https://doi.org/10.1214/aoms/1177728420
  3. Brunk, H. D. (1958). On the estimation of parameters restricted by inequalities, The Annals of the Mathematical Statistics, 29, 437-454 https://doi.org/10.1214/aoms/1177706621
  4. Davis, L. and Gather, U. (1993). The identification of multiple outliers, Journal of the American Statistical Association, 88, 782-801 https://doi.org/10.2307/2290763
  5. Kim, S. J. and Lim, J. (2006). Uniform approximation and estimation of shape restricted functions, Preprint available upon request
  6. Lee, S., Lim, J., Kim, S. J. and Joo, Y. (2008). Estimating monotone convex functions via sequential shape modification, To appear in Journal of Statistical Compuation and Simulation
  7. Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference, John Wiley & Sons, New York