Abstract
The usual estimators of the regression under isotonicity are known to present the so-called spiking problem, that is, they are very sensitive at the tails. Three design-based strategies in order to alleviate this effect are discussed. The proposed strategies will provide uniform consistency on the (closed and bounded) working interval. Firstly, the usual isotonic regression with a suitable number of observations at the edges of the interval is considered. Secondly, a reallocation of part of the edge observations at some artificial adjacent points is suggested. Finally, a strategy based on constraining the isotonic regression to take values within some horizontal bands is investigated. Simulation studies illustrate the performance of the proposed estimators in practice.
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References
Álvarez EE, Yohai VJ (2012) M-estimators for isotonic regression. J Stat Plan Inference 142:2351–2368
Ayer M, Brunk HD, Ewing WT, Reid WT, Silverman E (1955) An empirical distribution function for sampling with incomplete information. Ann Math Stat 26:641–647
Barlow RE, Bartholomew D, Bremner JM, Brunk HD (1972) Statistical inference under order restrictions: the theory and applications of isotonic regression. Wiley, New York
Brunk HD (1970) Estimation of isotonic regression. In: Brunk HD, Puri ML (eds) Nonparametric techniques in statistical inference. Cambridge University Press, London
Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2006) Testing constancy for isotonic regressions. Scand J Stat 33:463–475
Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2007) A test for constancy of isotonic regressions using the \(L_{2}\)-norm. Stat Sin 17:713–724
Colubi A, Domínguez-Menchero JS, González-Rodríguez G (2017) New designs to consistently estimate the isotonic regression. Submitted
Cuesta-Albertos JA, Domínguez-Menchero JS, Matrán C (1995) Consistency of \(L_{p}\)-best monotone approximations. J Stat Plan Inference 47:295–318
de Leeuw J, Hornik K, Mair P (1973) Isotone optimization in R: pool-adjacent-violators algorithm (PAVA) and active set methods. J Stat Softw 32:1–24
Domínguez-Menchero JS, González-Rodríguez G (2007) Analyzing an extension of the isotonic regression problem. Metrika 66:19–30
Domínguez-Menchero JS, López-Palomo MJ (1997) On the estimation of monotone uniform approximations. Stat Probab Lett 35:355–362
Domínguez-Menchero JS, González-Rodríguez G, López-Palomo MJ (2005) An \(L_{2}\) point of view in testing monotone regression. Nonpar Stat 17:135–153
Hanson DL, Pledger G, Wright FT (1973) On consistency in monotonic regression. Ann Stat 1:401–421
Mukerjee H (1988) Monotone nonparametric regression. Ann Stat 16:741–775
Robertson T, Wright FT, Dykstra RL (1988) Order restricted statistical inference. Wiley, New York
Sampson AR, Singh H, Whitaker LR (2009) Simultaneous confidence bands for isotonic functions. J Stat Plan Inference 139:828–842
Wang Y, Huang J (2002) Limiting distribution for monotone median regression. J Stat Plan Inference 107:281–287
Wright FT (1981) The asymptotic behavior of monotone regression estimates. Ann Stat 9:443–448
Wu WB, Woodroofe M, Mentz G (2001) Isotonic regression: another look at the changepoint problem. Biometrika 88:793–804
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Colubi, A., Domínguez-Menchero, J.S., González-Rodríguez, G. (2018). The Spiking Problem in the Context of the Isotonic Regression. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_9
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DOI: https://doi.org/10.1007/978-3-319-73848-2_9
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