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

Abstract. This paper provides a uni\u85ed and comprehensive approach that is useful in deriving expres-sions for higher order cumulants of random vectors. 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 multivariable statistics when the population is not necessarily Gaussian and \u85nally, (iii) in the study of multivariate nonlinear time series models and in obtaining higher order cumulant spectra. The approach depends on expanding the characteristic functions and cumulant generating functions in terms of the Kronecker products of di¤erential operators. Our objective here is to derive such expressions using only elementary calculus of several variables and also to highlight some important applications in statistics