## The Annals of Statistics

### Natural statistics for spectral samples

#### Abstract

Spectral sampling is associated with the group of unitary transformations acting on matrices in much the same way that simple random sampling is associated with the symmetric group acting on vectors. This parallel extends to symmetric functions, $k$-statistics and polykays. We construct spectral $k$-statistics as unbiased estimators of cumulants of trace powers of a suitable random matrix. Moreover we define normalized spectral polykays in such a way that when the sampling is from an infinite population they return products of free cumulants.

#### Article information

Source
Ann. Statist., Volume 41, Number 2 (2013), 982-1004.

Dates
First available in Project Euclid: 29 May 2013

https://projecteuclid.org/euclid.aos/1369836967

Digital Object Identifier
doi:10.1214/13-AOS1107

Mathematical Reviews number (MathSciNet)
MR3099128

Zentralblatt MATH identifier
1287.60009

#### Citation

Di Nardo, E.; McCullagh, P.; Senato, D. Natural statistics for spectral samples. Ann. Statist. 41 (2013), no. 2, 982--1004. doi:10.1214/13-AOS1107. https://projecteuclid.org/euclid.aos/1369836967

#### References

• [1] Capitaine, M. and Casalis, M. (2006). Cumulants for random matrices as convolutions on the symmetric group. Probab. Theory Related Fields 136 19–36.
• [2] Collins, B. and Śniady, P. (2006). Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264 773–795.
• [3] Di Nardo, E., Guarino, G. and Senato, D. (2008). A unifying framework for $k$-statistics, polykays and their multivariate generalizations. Bernoulli 14 440–468.
• [4] Di Nardo, E., Guarino, G. and Senato, D. (2009). A new method for fast computing unbiased estimators of cumulants. Stat. Comput. 19 155–165.
• [5] Di Nardo, E., Guarino, G. and Senato, D. (2011). A new algorithm for computing the multivariate Faà di Bruno’s formula. Appl. Math. Comput. 217 6286–6295.
• [6] Di Nardo, E. and Oliva, I. (2012). A new family of time–space harmonic polynomials with respect to Lévy processes. Ann. Mat. Pura Appl. DOI:10.1007/s10231-012-0252-3.
• [7] Di Nardo, E. and Senato, D. (2001). Umbral nature of the Poisson random variables. In Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota (H. Crapo and D. Senato, eds.) 245–266. Springer Italia, Milan.
• [8] Di Nardo, E. and Senato, D. (2006). An umbral setting for cumulants and factorial moments. European J. Combin. 27 394–413.
• [9] Fisher, R. A. (1929). Moments and product moments of sampling distributions. Proc. Lond. Math. Soc. (3) 30 199–238.
• [10] Graczyk, P., Letac, G. and Massam, H. (2003). The complex Wishart distribution and the symmetric group. Ann. Statist. 31 287–309.
• [11] McCullagh, P. (1984). Tensor notation and cumulants of polynomials. Biometrika 71 461–476.
• [12] McCullagh, P. (1987). Tensor Methods in Statistics. Chapman & Hall, London.
• [13] Nica, A. and Speicher, R. (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series 335. Cambridge Univ. Press, Cambridge.
• [14] Raviraj, P. and Sanavullah, M. Y. (2007). The modified 2D-Haar wavelet transformation in image compression. Middle East J. Scientific Research 2 73–78.
• [15] Rota, G.-C. (1964). On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2 340–368.
• [16] Rota, G. C. and Taylor, B. D. (1994). The classical umbral calculus. SIAM J. Math. Anal. 25 694–711.
• [17] Stanley, R. P. (1986). Enumerative Combinatorics. Vol. I. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA.
• [18] Strang, G. and Nguyen, T. (1996). Wavelets and Filter Banks. Wellesley–Cambridge Press, Wellesley, MA.
• [19] Stuart, A. and Ord, J. K. (1987). Kendall’s Advanced Theory of Statistics. Vol. 1, 5th ed. Oxford Univ. Press, New York.
• [20] Taylor, B. D. (1998). Difference equations via the classical umbral calculus. In Mathematical Essays in Honor of Gian-Carlo Rota (Cambridge, MA, 1996) (B. Sagan and R. Stanley, eds.). Progress in Mathematics 161 397–411. Birkhäuser, Boston, MA.
• [21] Tukey, J. W. (1950). Some sampling simplified. J. Amer. Statist. Assoc. 45 501–519.
• [22] Tukey, J. W. (1956). Keeping moment-like sampling computations simple. Ann. Math. Statist. 27 37–54.