Annals of Statistics

Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives

Satoshi Kuriki and Akimichi Takemura

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Abstract

We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a spherical cone and a piecewise smooth cone.

Article information

Source
Ann. Statist., Volume 25, Number 6 (1997), 2368-2387.

Dates
First available in Project Euclid: 30 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1030741077

Digital Object Identifier
doi:10.1214/aos/1030741077

Mathematical Reviews number (MathSciNet)
MR1604465

Zentralblatt MATH identifier
0897.62055

Subjects
Primary: 62H10: Distribution of statistics 62H15: Hypothesis testing
Secondary: 52A39: Mixed volumes and related topics

Keywords
Multivariate one-sided alternative one-sided simultaneous confidence region mixed volume second fundamental form volume element internal angle external angle Gauss-Bonnet theorem Shapiro's conjecture

Citation

Takemura, Akimichi; Kuriki, Satoshi. Weights of $overline{\chi}{}\sp 2$ distribution for smooth or piecewise smooth cone alternatives. Ann. Statist. 25 (1997), no. 6, 2368--2387. doi:10.1214/aos/1030741077. https://projecteuclid.org/euclid.aos/1030741077


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References

  • Akkerboom, J. C. (1990). Testing Problems with Linear or Angular Inequality Constraints. Lecture Notes in Statist. 62. Springer, New York.
  • Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, London. Bohrer, R. and Francis, G. K. (1972a). Sharp one-sided confidence bounds for linear regression over intervals. Biometrika 59 99-107. Bohrer, R. and Francis, G. K. (1972b). Sharp one-sided confidence bounds over positive regions. Ann. Math. Statist. 43 1541-1548.
  • Johnstone, I. and Siegmund, D. (1989). On Hotelling's formula for the volume of tubes and Naiman's inequality. Ann. Statist. 17 184-194.
  • Kuriki, S. (1993). One-sided test for the equality of two covariance matrices. Ann. Statist. 21 1379-1384.
  • Lin, Y. and Lindsay, B. G. (1997). Projections on cones, chi-bar squared distributions, and Wey l's formula. Statist. Probab. Lett. 32 367-376.
  • McMullen, P. (1975). Non-linear angle-sum relations for poly hedral cones and poly topes. Math. Proc. Cambridge Philos. Soc. 78 247-261.
  • Naiman, D. Q. (1990). Volumes of tubular neighborhoods of spherical poly hedra and statistical inference. Ann. Statist. 18 685-716.
  • Pincus, R. (1975). Testing linear hy potheses under restricted alternatives. Math. Oper. Statist. 6 733-751.
  • Robertson, T., Wright, F. T. and Dy kstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, Chichester.
  • Santal ´o, L. A. (1976). Integral Geometry and Geometric Probability. Addison-Wesley, London. Schneider, R. (1993a). Convex Bodies: The Brunn-Minkowski Theory. Cambridge Univ. Press. Schneider, R. (1993b). Convex surfaces, curvature and surface area measures. In Handbook of Convex Geometry (P. M. Gruber and J. M. Wilks, eds.) Chapter 1.8. North-Holland, Amsterdam.
  • Shapiro, A. (1985). Asy mptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72 133-144.
  • Shapiro, A. (1987). A conjecture related to chi-bar-squared distributions. Amer. Math. Monthly 94 46-48.
  • Takemura, A. and Kuriki, S. (1995). Weights of ¯ 2 distribution for smooth or piecewise smooth cone alternatives. Discussion Paper 95-F-28, Faculty of Economics, Univ. Toky o.
  • Webster, R. (1994). Convexity. Oxford Univ. Press.
  • Wey l, H. (1939). On the volume of tubes. Amer. J. Math. 61 461-472.
  • Wy nn, H. P. (1975). Integrals for one-sided confidence bounds: a general result. Biometrika 62 393-396.