The Annals of Statistics

Smallest nonparametric tolerance regions

Alessandro Di Bucchianico, John H. Einmahl, and Nino A. Mushkudiani

Full-text: Open access

Abstract

We present a new, natural way to construct nonparametric multivariate tolerance regions. Unlike the classical nonparametric tolerance intervals, where the endpoints are determined by beforehand chosen order statistics, we take the shortest interval that contains a certain number of observations. We extend this idea to higher dimensions by replacing the class of intervals by a general class of indexing sets, which specializes to the classes of ellipsoids, hyperrectangles or convex sets.The asymptotic behavior of our tolerance regions is derived using empirical process theory, in particular the concept of generalized quantiles. Finite sample properties of our tolerance regions are investigated through a simulation study. Real data examples are also presented.

Article information

Source
Ann. Statist. Volume 29, Number 5 (2001), 1320-1343.

Dates
First available in Project Euclid: 8 February 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1013203456

Digital Object Identifier
doi:10.1214/aos/1013203456

Mathematical Reviews number (MathSciNet)
MR1873333

Zentralblatt MATH identifier
1043.62045

Subjects
Primary: 62G15: Tolerance and confidence regions 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions 60F05: Central limit and other weak theorems

Keywords
Nonparametric tolerance region prediction region empirical process asymptotic normality minimum volume set

Citation

Di Bucchianico, Alessandro; Einmahl, John H.; Mushkudiani, Nino A. Smallest nonparametric tolerance regions. Ann. Statist. 29 (2001), no. 5, 1320--1343. doi:10.1214/aos/1013203456. http://projecteuclid.org/euclid.aos/1013203456.


Export citation

References

  • [1] Ackermann, H. (1983). Multivariate nonparametric tolerance regions: a new construction technique. Biometrical J. 25 351-359.
  • [2] Agull ´o, J. (1996). Exact iterative computation of the multivariate minimum volume ellipsoid estimator with a branch and bound algorithm. In COMPSTAT '96 (A. Prat, ed.) 175- 180. Physica, Heidelberg.
  • [3] Aitchison, J., and Dunsmore, I. R. (1975). Statistical Prediction Analysis. Cambridge Univ. Press.
  • [4] Alexander, K. S. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12 1041-1067.
  • [5] Azzalini, A. (1981). A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68 326-328.
  • [6] Beirlant, J. and Einmahl, J. H. J. (1995). Asymptotic confidence intervals for the length of the shortt under random censoring. Statist. Neerlandica 49 1-8.
  • [7] Bolthausen, E. (1978). Weak convergence of an empirical process indexed by the closed convex subsets of I2. Z. Wahrsch. Verw. Gebiete 43 173-181.
  • [8] Carroll, R. J. and Ruppert, D. (1991). Prediction and tolerance intervals with transformation and/or weighting. Technometrics 33 197-210.
  • [9] Chatterjee, S. K. and Patra, N. K. (1980). Asymptotically minimal multivariate tolerance sets. Calcutta Statist. Assoc. Bull. 29 73-93.
  • [10] Davies, L. (1992). The asymptotics of Rousseeuw's minimum volume ellipsoid estimator. Ann. Statist. 20 1828-1843.
  • [11] Dudley, R. M. (1978). Central limit theorems for empirical measures. Ann. Probab. 6 899- 929.
  • [12] Dudley, R. M. (1982). Empirical and Poisson processes on classes of sets or functions too large for central limit theorems. Z. Wahrsch. Verw. Gebiete 61 355-368.
  • [13] Einmahl, J. H. J. and Mason, D. M. (1992). Generalized quantile processes. Ann. Statist. 20 1062-1078.
  • [14] Eppstein, D. (1992). New algorithms for minimum area k-gons. In Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms 83-88. ACM, New York.
  • [15] Eppstein, D., Overmars, M. Rote, G. and Woeginger, G. (1992). Finding minimum area k-gons. Discrete Comput. Geom. 7 45-58.
  • [16] Fraser, D. A. S. (1953). Nonparametric tolerance regions. Ann. Math. Statist. 24 44-55.
  • [17] Gaenssler, P. (1983). Empirical Processes. IMS, Hayward, CA.
  • [18] Gr ¨ubel, R. (1988). The length of the shorth. Ann. Statist. 16 619-628.
  • [19] Guttman, I. (1970). Statistical Tolerance Regions: Classical and Bayesian. Griffin, London.
  • [20] J´ilek, M. (1981). A bibliography of statistical tolerance regions. Math. Operations. Statist. Ser. Statist. 12 441-456.
  • [21] J´ilek, M. and Ackermann, H. (1989). A bibliography of statistical tolerance regions. II. Statistics 20 165-172.
  • [22] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405-414.
  • [23] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: descriptive statistics, graphics and inference (with discussion). Ann. Statist. 27 783- 858.
  • [24] Mushkudiani, N. (2000). Statistical applications of generalized quantiles: nonparametric tolerance regions and P-P plots. Eindhoven Univ. Technology.
  • [25] Mushkudiani, N. (2002). Small nonparametric tolerance regions for directional data. J. Statist. Plann. Inference 100 67-80.
  • [26] Nolan, D. (1991). The excess-mass ellipsoid. J. Multivariate Anal. 39 348-371.
  • [27] Polonik, W. (1997). Minimum volume sets and generalized quantile processes. Stochastic Process. Appl. 69 1-24.
  • [28] Rousseeuw, P. and Leroy, A. (1988). A robust scale estimator based on the shortest half. Statist. Neerlandica 42 103-116.
  • [29] Rousseeuw, P. and van Zomeren, B. C. (1991). Robust distances: simulations and cutoff values. In Directions in Robust Statistics and Diagnostics II (W. Stahel and S. Weisberg, eds.) 195-203. Springer, New York.
  • [30] Schneider, R. (1993). Convex Bodies: the Brunn-Minkowski Theory. Cambridge Univ. Press.
  • [31] Shephard, G. C. and Webster, R. J. (1965). Metric sets of convex bodies. Mathematika 12 73-88.
  • [32] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • [33] Silverman, B. W. and Titterington, D. M. (1980). Minimum covering ellipses. SIAM J. Sci. Statist. Comput. 1 401-409.
  • [34] Tukey, J. W. (1947). Nonparametric estimation, II. Statistical equivalent blocks and tolerance regions the continuous case. Ann. Math. Stat. 18 529-539.
  • [35] Tukey, J. W. (1948). Nonparametric estimation, III. Statistical equivalent blocks and multivariate tolerance regions the discontinuous case. Ann. Math. Stat. 19 30-39.
  • [36] van der Vaart, A. W. (1994). Weak convergence of smoothed empirical processes. Scand. J. Statist. 4 501-504.
  • [37] van der Vaart, A. W. (1996). New Donsker classes. Ann. Probab. 24 2128-2140.
  • [38] Wald, A. (1943). An extension of Wilks' method for setting tolerance limits. Ann. Math. Statist. 14 45-55.
  • [39] Webster, R. (1994). Convexity. Oxford Univ. Press.
  • [40] Wilks, S. S. (1941). Determination of sample sizes for setting tolerance limits. Ann. Math. Statist. 12 91-96.