Abstract
Sasabuchi et al. (Biometrika 70(2):465–472, 1983) introduces a multivariate version of the well-known univariate isotonic regression which plays a key role in the field of statistical inference under order restrictions. His proposed algorithm for computing the multivariate isotonic regression, however, is guaranteed to converge only under special conditions (Sasabuchi et al., J Stat Comput Simul 73(9):619–641, 2003). In this paper, a more general framework for multivariate isotonic regression is given and an algorithm based on Dykstra’s method is used to compute the multivariate isotonic regression. Two numerical examples are given to illustrate the algorithm and to compare the result with the one published by Fernando and Kulatunga (Comput Stat Data Anal 52:702–712, 2007).
Similar content being viewed by others
References
Barlow RE, Bartholomew DJ, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions. Wiley & Sons, New York
Boyle JP, Dykstra RL (1985) A method for finding projections onto the intersection of convex sets in Hilbert spaces. In: Advances in order restricted statistical inference, Lecture Notes in Statistics, vol 37, Springer, Berlin, pp 28–47
Brunk HD, Ewing GM, Utz WR (1957) Minimizing integrals in certain classes of monotone functions. Pac J Math 7: 833–847
Fernando WTPS, Kulatunga DDS (2007) On the computation and some applications of multivariate isotonic regression. Comput Stat Data Anal 52: 702–712
Hansohm J (2007) Algorithms and error estimations for monotone regression on partially preordered sets. J Multivariate Anal 98: 1043–1050
Hu XM, Hansohm J (2008) Merge and chop in the computation for isotonic regression. J Stat Plan Inference 138(10):3099–3106
Perkins C (2002) A convergence analysis of Dykstra’s algorithm for polyhedral sets. SIAM J Numer Anal 40(2): 792–804
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley & Sons, New York
Sasabuchi S, Inutsuka M, Kulatunga DDS (1983) A multivariate version of isotonic regression. Biometrika 70(2): 465–472
Sasabuchi S, Inutsuka M, Kulatunga DDS (1992) An algorithm for computing multivariate isotonic regression. Hiroshima Math J 22(3): 551–560
Sasabuchi S, Miura T, Oda H (2003) Estimation and test of several multivariate normal means under an order restriction when the dimension is larger than two. J Stat Comput Simul 73(9): 619–641
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hansohm, J., Hu, X. A convergent algorithm for a generalized multivariate isotonic regression problem. Stat Papers 53, 107–115 (2012). https://doi.org/10.1007/s00362-010-0317-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-010-0317-6