MATHEMATICAL ENGINEERING TECHNICAL REPORTS The Tube Method for the Moment Index in Projection Pursuit

Abstract The projection pursuit index defined by a sum of squares of the third and the fourth sample cumulants is known as the moment index proposed by Jones and Sibson [1987. What is projection pursuit? J. Roy. Statist. Soc. Ser. A 150, 1–36]. The limiting distribution of the maximum of the moment index under the null hypothesis that the population is multivariate normal is shown to be the maximum of a Gaussian random field with a finite Karhunen–Loeve expansion. An approximate formula for tail probability of the maximum, which corresponds to the p -value, is given by virtue of the tube method through determining Weyl's invariants of all degrees and the critical radius of the index manifold of the Gaussian random field.

[1]  Akimichi Takemura,et al.  Tail probabilities of the maxima of multilinear forms and their applications , 2001 .

[2]  Erkki Oja,et al.  Independent Component Analysis , 2001 .

[3]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[4]  Robin Sibson,et al.  What is projection pursuit , 1987 .

[5]  T. W. Anderson,et al.  Tests for randomness of directions against equatorial and bimodal alternatives , 1972 .

[6]  Jiayang Sun.  Tail probabilities of the maxima of Gaussian random fields , 1993 .

[7]  Robert J. Adler,et al.  Euler characteristics for Gaussian fields on manifolds , 2003 .

[8]  S. Klinke,et al.  Exploratory Projection Pursuit , 1995 .

[9]  R. Adler.  RANDOM FIELDS AND THEIR GEOMETRY , 2003 .

[10]  R. Adler,et al.  The Geometry of Random Fields , 1982 .

[11]  Satoshi Kuriki,et al.  Asymptotic distribution of inequality-restricted canonical correlation with application to tests for independence in ordered contingency tables , 2005 .

[12]  D. Siegmund,et al.  On Hotelling's Approach to Testing for a Nonlinear Parameter in Regression , 1989 .

[13]  K. Worsley,et al.  Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics , 1995, Advances in Applied Probability.

[14]  Akimichi Takemura,et al.  ON THE EQUIVALENCE OF THE TUBE AND EULER CHARACTERISTIC METHODS FOR THE DISTRIBUTION OF THE MAXIMUM OF GAUSSIAN FIELDS OVER PIECEWISE SMOOTH DOMAINS , 2002 .

[15]  Michael B. Marcus,et al.  Central limit theorems for C(S)-valued random variables , 1975 .

[16]  A. Gorin.  ON THE VOLUME OF TUBES , 1983 .

[17]  Iain M. Johnstone,et al.  Hotelling's Theorem on the Volume of Tubes: Some Illustrations in Simultaneous Inference and Data Analysis , 1990 .

[18]  K. Worsley.  Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images , 1995 .

[19]  A. Takemura,et al.  Maximum of Gaussian field on piecewise smooth domain: Equivalence of tube method and Euler characteristic method , 1999 .

[20]  R. Adler.  On excursion sets, tube formulas and maxima of random fields , 2000 .

[21]  R. Adler.  The Geometry of Random Fields , 2009 .

[22]  H. Hotelling.  Tubes and Spheres in n-Spaces, and a Class of Statistical Problems , 1939 .

[23]  Jiayang Sun,et al.  Significance levels in exploratory projection pursuit , 1991 .

[24]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[25]  A. Takemura,et al.  Tail probabilities of the limiting null distributions of the Anderson-Stephens statistics , 2000 .

[26]  Akimichi Takemura,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Validity of the expected Euler characteristic heuristic , 2003 .

[27]  J. Schmee.  An Introduction to Multivariate Statistical Analysis , 1986 .