Small nonparametric tolerance regions for directional data

We present a natural approach, based on minimum volume sets, for constructing non­ parametric tolerance regions for directional data. The tolerance regions have desirable features like invariance and are asymptotically minimal under certain conditions. We es­ tablish the asymptotic correctness of our tolerance regions by using the theory of empirical processes and generalized quantiles. The results are obtained under minimal conditions. In case of circular data, the finite sample properties of the tolerance arcs are studied through simulations. The method is also applied to a real data example.

[1]  A. Di Bucchianico,et al.  Smallest nonparametric tolerance regions , 2001 .

[2]  Donald Fraser,et al.  Nonparametric Estimation IV , 1951 .

[3]  John Aitchison,et al.  Statistical Prediction Analysis , 1975 .

[4]  H. Ackermann.  Multivariate Non‐parametric Tolerance Regions: A New Construction Technique , 1983 .

[5]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[6]  R. Dudley.  Central Limit Theorems for Empirical Measures , 1978 .

[7]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[8]  D. Mason,et al.  Generalized quantile processes , 1992 .

[9]  I. R. Dunsmore,et al.  Statistical Prediction Analysis: Diagnosis , 1975 .

[10]  Abraham Wald,et al.  An Extension of Wilks' Method for Setting Tolerance Limits , 1943 .

[11]  W. Polonik.  Minimum volume sets and generalized quantile processes , 1997 .

[12]  J. K. Ord,et al.  Statistical Tolerance Regions: Classical and Bayesian , 1971 .

[13]  S. S. Wilks.  Determination of Sample Sizes for Setting Tolerance Limits , 1941 .

[14]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[15]  J. Tukey.  Non-Parametric Estimation II. Statistically Equivalent Blocks and Tolerance Regions--The Continuous Case , 1947 .

[16]  K. Mardia.  Statistics of Directional Data , 1972 .

[17]  B. Heller.  Circular Statistics in Biology, Edward Batschelet. Academic Press, London & New York (1981), 371, Price $69.50 , 1983 .

[18]  John W. Tukey,et al.  Nonparametric Estimation, III. Statistically Equivalent Blocks and Multivariate Tolerance Regions--The Discontinuous Case , 1948 .

[19]  F. Ruymgaart.  Strong uniform convergence of density estimators on spheres , 1989 .

[20]  D. Nolan.  The excess-mass ellipsoid , 1991 .

[21]  K. Alexander,et al.  Probability Inequalities for Empirical Processes and a Law of the Iterated Logarithm , 1984 .

[22]  Small nonparametric tolerance regions , 1998 .