Bernoulli

  • Bernoulli
  • Volume 25, Number 2 (2019), 1536-1567.

New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains

Amir Sepehri

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper introduces two new families of non-parametric tests of goodness-of-fit on the compact classical groups. One of them is a family of tests for the eigenvalue distribution induced by the uniform distribution, which is consistent against all fixed alternatives. The other is a family of tests for the uniform distribution on the entire group, which is again consistent against all fixed alternatives. The construction of these tests heavily employs facts and techniques from the representation theory of compact groups. In particular, new Cauchy identities are derived and proved for the characters of compact classical groups, in order to accommodate the computation of the test statistic. We find the asymptotic distribution under the null and general alternatives. The tests are proved to be asymptotically admissible. Local power is derived and the global properties of the power function against local alternatives are explored.

The new tests are validated on two random walks for which the mixing-time is studied in the literature. The new tests, and several others, are applied to the Markov chain sampler proposed by Jones, Osipov and Rokhlin [Proc. Natl. Acad. Sci. 108 (2011) 15679–15686], providing strong evidence supporting the claim that the sampler mixes quickly.

Article information

Source
Bernoulli, Volume 25, Number 2 (2019), 1536-1567.

Dates
Received: June 2017
Revised: February 2018
First available in Project Euclid: 6 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.bj/1551862859

Digital Object Identifier
doi:10.3150/18-BEJ1029

Mathematical Reviews number (MathSciNet)
MR3920381

Zentralblatt MATH identifier
07049415

Keywords
Cauchy identity goodness-of-fit mixing-diagnostics for Markov Chains non-parametric hypothesis testing random rotation generators representation theory of compact groups spectral analysis

Citation

Sepehri, Amir. New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains. Bernoulli 25 (2019), no. 2, 1536--1567. doi:10.3150/18-BEJ1029. https://projecteuclid.org/euclid.bj/1551862859


Export citation

References

  • [1] Abramowitz, M. and Stegun, I.A. (1966). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. 55. New York: Dover Publications.
  • [2] Ajne, B. (1968). A simple test for uniformity of a circular distribution. Biometrika 55 343–354.
  • [3] Andrews, G.E. (1998). The Theory of Partitions. Cambridge Mathematical Library. Cambridge: Cambridge Univ. Press.
  • [4] Arias-Castro, E., Pelletier, B. and Saligrama, V. (2018). Remember the curse of dimensionality: The case of goodness-of-fit testing in arbitrary dimension. J. Nonparametr. Stat. 30 448–471.
  • [5] Baringhaus, L. (1991). Testing for spherical symmetry of a multivariate distribution. Ann. Statist. 19 899–917.
  • [6] Beran, R. (1975). Tail probabilities of noncentral quadratic forms. Ann. Statist. 3 969–974.
  • [7] Beran, R.J. (1968). Testing for uniformity on a compact homogeneous space. J. Appl. Probab. 5 177–195.
  • [8] Birnbaum, A. (1955). Characterizations of complete classes of tests of some multiparametric hypotheses, with applications to likelihood ratio tests. Ann. Math. Stat. 26 21–36.
  • [9] Bump, D. (2004). Lie Groups. Graduate Texts in Mathematics 225. New York: Springer.
  • [10] Coram, M. and Diaconis, P. (2003). New tests of the correspondence between unitary eigenvalues and the zeros of Riemann’s zeta function. J. Phys. A 36 2883–2906.
  • [11] Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 155–178.
  • [12] Diaconis, P. and Mallows, C. (1986). On the trace of random orthogonal matrices. Unpublished Manuscript. Results Summarized in Diaconis (1990).
  • [13] Diaconis, P. and Shahshahani, M. (1986). Products of random matrices as they arise in the study of random walks on groups. In Random Matrices and Their Applications (Brunswick, Maine, 1984). Contemp. Math. 50 183–195. Providence, RI: Amer. Math. Soc.
  • [14] Diaconis, P. and Shahshahani, M. (1987). The subgroup algorithm for generating uniform random variables. Probab. Engrg. Inform. Sci. 1 15–32.
  • [15] Downs, T.D. (1972). Orientation statistics. Biometrika 59 665–676.
  • [16] Giné, E.M. (1975). Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Statist. 3 1243–1266.
  • [17] Giné, E.M. (1975). The addition formula for the eigenfunctions of the Laplacian. Adv. Math. 18 102–107.
  • [18] Goodman, R. and Wallach, N.R. (2009). Symmetry, Representations, and Invariants. Graduate Texts in Mathematics 255. Dordrecht: Springer.
  • [19] Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 97–109.
  • [20] Hermans, M. and Rasson, J.-P. (1985). A new Sobolev test for uniformity on the circle. Biometrika 72 698–702.
  • [21] Janssen, A. (1995). Principal component decomposition of non-parametric tests. Probab. Theory Related Fields 101 193–209.
  • [22] Jones, P.W., Osipov, A. and Rokhlin, V. (2011). Randomized approximate nearest neighbors algorithm. Proc. Natl. Acad. Sci. 108 15679–15686.
  • [23] Jupp, P.E. and Spurr, B.D. (1983). Sobolev tests for symmetry of directional data. Ann. Statist. 11 1225–1231.
  • [24] Jupp, P.E. and Spurr, B.D. (1985). Sobolev tests for independence of directions. Ann. Statist. 13 1140–1155.
  • [25] Kac, M. (1959). Probability and Related Topics in Physical Sciences 1. American Mathematical Soc.
  • [26] Kerkyacharian, G., Nickl, R. and Picard, D. (2012). Concentration inequalities and confidence bands for needlet density estimators on compact homogeneous manifolds. Probab. Theory Related Fields 153 363–404.
  • [27] Lehmann, E.L. and Romano, J.P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer Texts in Statistics. New York: Springer.
  • [28] Liberty, E., Woolfe, F., Martinsson, P.-G., Rokhlin, V. and Tygert, M. (2007). Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. USA 104 20167–20172.
  • [29] Mardia, K.V. and Jupp, P.E. (2000). Directional Statistics. Wiley Series in Probability and Statistics. Chichester: Wiley.
  • [30] Mehta, M.L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier/Academic Press.
  • [31] Oliveira, R.I. (2009). On the convergence to equilibrium of Kac’s random walk on matrices. Ann. Appl. Probab. 19 1200–1231.
  • [32] Pak, I. and Sidenko, S. (2007). Convergence of Kac’s random walk. Preprint. Available at http://www-math.mit.edu/~pak/research.html.
  • [33] Pillai, N.S. and Smith, A. (2016). On the mixing time of Kac’s walk and other high-dimensional Gibbs samplers with constraints. Preprint. Available at ArXiv:1605.08122.
  • [34] Porod, U. (1996). The cut-off phenomenon for random reflections. Ann. Probab. 24 74–96.
  • [35] Prentice, M.J. (1978). On invariant tests of uniformity for directions and orientations. Ann. Statist. 6 169–176.
  • [36] Rayleigh, L. (1880). XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase. Philos. Mag. 10 73–78.
  • [37] Rokhlin, V. and Tygert, M. (2008). A fast randomized algorithm for overdetermined linear least-squares regression. Proc. Natl. Acad. Sci. USA 105 13212–13217.
  • [38] Römisch, W. (2005). Delta method, infinite dimensional. Encyclopedia of Statistical Sciences.
  • [39] Rosenthal, J.S. (1994). Random rotations: Characters and random walks on $\operatorname{SO}(N)$. Ann. Probab. 22 398–423.
  • [40] Sengupta, A. and Pal, C. (2001). On optimal tests for isotropy against the symmetric wrapped stable-circular uniform mixture family. J. Appl. Stat. 28 129–143.
  • [41] Sepehri, A. (2019). Supplement to “New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains.” DOI:10.3150/18-BEJ1029SUPP.
  • [42] Strasser, H. (1985). Mathematical Theory of Statistics: Statistical Experiments and Asymptotic Decision Theory. De Gruyter Studies in Mathematics 7. Berlin: de Gruyter.
  • [43] Thomson, B.S., Bruckner, J.B. and Bruckner, A.M. (2008). Elementary Real Analysis. Available at ClassicalRealAnalysis.com.
  • [44] van der Vaart, A.W. and Wellner, J.A. (1996). Weak convergence. In Weak Convergence and Empirical Processes 16–28. New York: Springer.
  • [45] Watson, G.S. (1961). Goodness-of-fit tests on a circle. Biometrika 48 109–114.
  • [46] Watson, G.S. (1962). Goodness-of-fit tests on a circle. II. Biometrika 49 57–63.
  • [47] Watson, G.S. (1967). Another test for the uniformity of a circular distribution. Biometrika 54 675–677.
  • [48] Wellner, J.A. (1979). Permutation tests for directional data. Ann. Statist. 7 929–943.
  • [49] Weyl, H. (1939). The Classical Groups. Their Invariants and Representations. Princeton, NJ: Princeton Univ. Press.

Supplemental materials

  • Supplement to “New tests of uniformity on the compact classical groups as diagnostics for weak-$^{*}$ mixing of Markov chains”. We provide additional supporting material including background and proofs from representation theory, proofs of some of the results, introduction to Le Cam’s theory, further derivation and analysis of local properties, as well as a test based on the trace. The motivating example is also reviewed.