**Sankhya:
The Indian Journal of Statistics**

2006, Volume 68, Pt. 2, 326--356

**Higher Order Cumulants of Random Vectors and Applications to Statistical Inference and Time Series**

S. Rao Jammalamadaka, University of California, Santa Barbara, USA

T. Subba Rao, University of Manchester, Manchester, UK

Gy\"{o}rgy Terdik, University of Debrecen, Debrecen, Hungary

SUMMARY. This paper provides a unified and comprehensive approach for deriving expressions for higher-order cumulants of random vectors. The approach is based on expanding the characteristic functions and cumulant generating functions in terms of the Kronecker products of differential operators. The use of this methodology is then illustrated in three diverse and novel contexts, namely: (i) in obtaining a lower bound (Bhattacharya bound) for the variance-covariance matrix of a vector of unbiased estimators where the density depends on several parameters, (ii) in studying the asymptotic theory of multivariate statistics when the population is not necessarily Gaussian and finally, (iii) in obtaining higher order cumulant spectra in the study of multivariate nonlinear time series models. Our objective here is to derive such expressions for the higher-order cumulants of random vectors using only elementary calculus of several variables and to highlight some important and novel applications in statistics.

*AMS (2000) subject classification. *62E17, 62E20; secondary 62H10, 60E05.

*Key words and phrases. *Cumulants for random vectors, cumulants for likelihood functions, Bhattacharya
lower bound, Taylor series expansion, multivariate time series.