A non-parametric analysis of transformations

Abstract A non-parametric estimator is proposed for the transformation model, where following Box and Cox (1964), we assume that the transform zi = z(yi, λ) of the original data {yi} satisfies a linear model. The estimator is based on the rank correlation of Kendall (1938) and is shown to be strongly consistent. We also find in a Monte Carlo study that the estimator attains greater efficiency in terms of mean squared error than the standard maximum likelihood estimator and the robust estimator of Carroll (1980) when the true error departs from normality. The size of this gain varies with the degree of the departure from normality as well as with the degree of non-linearity of the underlying transformation.

[1]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[2]  Norman R. Draper,et al.  On Distributions and Their Transformation to Normality , 1969 .

[3]  M. Kendall.  A NEW MEASURE OF RANK CORRELATION , 1938 .

[4]  D. F. Andrews,et al.  A note on the selection of data transformations , 1971 .

[5]  D. Ruppert,et al.  The Analysis of Transformed Data: Comment , 1984 .

[6]  David V. Hinkley,et al.  On power transformations to symmetry , 1975 .

[7]  Comment: Distinguishing between the Scale of the Estimand and the Transformation to Normality , 1984 .

[8]  J. Jurecková,et al.  Nonparametric Estimate of Regression Coefficients , 1971 .

[9]  Comment: The Transformation Controversy , 1984 .

[10]  Louis A. Jaeckel.  Estimating Regression Coefficients by Minimizing the Dispersion of the Residuals , 1972 .

[11]  The Analysis of Transformed Data: Comment , 1984 .

[12]  R. Serfling.  Approximation Theorems of Mathematical Statistics , 1980 .

[13]  W. Hoeffding.  A Class of Statistics with Asymptotically Normal Distribution , 1948 .

[14]  D. E. Shaw,et al.  Analyzing the results of a cloud-seeding experiment in tasmania , 1979 .

[15]  J. N. Adichie.  ESTIMATES OF REGRESSION PARAMETERS BASED ON RANK TESTS , 1967 .

[16]  P. Sen.  Estimates of the Regression Coefficient Based on Kendall's Tau , 1968 .

[17]  P. J. Huber.  Robust Statistical Procedures , 1977 .

[18]  D. Hinkley,et al.  The Analysis of Transformed Data , 1984 .

[19]  Kjell A. Doksum,et al.  An extension of partial likelihood methods for proportional hazard models to general transformation models , 1987 .

[20]  R. Carroll.  A Robust Method for Testing Transformations to Achieve Approximate Normality , 1980 .

[21]  Anthony C. Atkinson,et al.  Testing Transformations to Normality , 1973 .

[22]  On power transformation to symmetry , 1976 .

[23]  P. Bickel.  The Analysis of Transformed Data: Comment , 1984 .

[24]  P. Bickel,et al.  An Analysis of Transformations Revisited , 1981 .

[25]  J. Simpson,et al.  Rainfall Results, 1970-1975: Florida Area Cumulus Experiment , 1977, Science.