Recent progress in the nonparametric estimation of monotone curves - With applications to bioassay and environmental risk assessment

Three recent nonparametric methodologies for estimating a monotone regression function F and its inverse F-1 are (1) the inverse kernel method DNP (Dette et al. (2005), Dette and Scheder (2010)), (2) the monotone spline (Kong and Eubank (2006)) and (3) the data adaptive method NAM (Bhattacharya and Lin (2010), (2011)), with roots in isotonic regression (Ayer et al. (1955), Bhattacharya and Kong (2007)). All three have asymptotically optimal error rates. In this article their finite sample performances are compared using extensive simulation from diverse models of interest, and by analysis of real data. Let there be m distinct values of the independent variable x among N observations y. The results show that if m is relatively small compared to N then generally the NAM performs best, while the DNP outperforms the other methods when m is O(N) unless there is a substantial clustering of the values of the independent variable x.

[1]  H. D. Brunk,et al.  AN EMPIRICAL DISTRIBUTION FUNCTION FOR SAMPLING WITH INCOMPLETE INFORMATION , 1955 .

[2]  H. Dette,et al.  A finite sample comparison of nonparametric estimates of the effective dose in quantal bioassay , 2010 .

[3]  Hari Mukerjee,et al.  Monotone Nonparametric Regression , 1988 .

[4]  Sangun Park,et al.  Parametric and nonparametric estimators of ED100α , 2006 .

[5]  Holger Dette,et al.  A Note on Nonparametric Estimation of the Effective Dose in Quantal Bioassay , 2005 .

[6]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[7]  Walter W Piegorsch,et al.  Nonparametric estimation of benchmark doses in environmental risk assessment , 2012, Environmetrics.

[8]  Hans-Georg Müller,et al.  Kernel and Probit Estimates in Quantal Bioassay , 1988 .

[9]  F. Nesslany,et al.  Study of oxidative DNA damage in TK6 human lymphoblastoid cells by use of the in vitro micronucleus test: Determination of No-Observed-Effect Levels. , 2009, Mutation research.

[10]  F. T. Wright.  Monotone Regression Estimates for Grouped Observations , 1982 .

[11]  R. L. Eubank,et al.  Monotone Smoothing with Application to Dose-Response Curve , 2006 .

[12]  R. Bhattacharya,et al.  Nonparametric benchmark analysis in risk assessment: a comparative study by simulation and data analysis , 2011, Sankhya. Series B. [Methodological.].

[13]  B. Efron.  Nonparametric standard errors and confidence intervals , 1981 .

[14]  Rabi Bhattacharya,et al.  An adaptive nonparametric method in benchmark analysis for bioassay and environmental studies. , 2010, Statistics & probability letters.

[15]  C. Kelly,et al.  Monotone smoothing with application to dose-response curves and the assessment of synergism. , 1990, Biometrics.

[16]  Rabi Bhattacharya,et al.  Consistency and asymptotic normality of the estimated effective doses in bioassay , 2007 .

[17]  Lizhen Lin.  Nonparametric Inference for Bioassay , 2012 .

[18]  R. Eubank.  Nonparametric Regression and Spline Smoothing , 1999 .