Letter to the Editor: Hyndman, R.J. & Wand, M.P. (1997) Nonparametric autocovariance function estimation

Hyndman & Wand (1997 Theorem 1 and Appendix) established expressions for the conditional bias and, under normality, for the variance of γˆj , an estimator of the conditional autocovariance function γj at lag j. Although they were aware of the relevance of the Hadamard matrix product, they did not fully explore its richness. In Lemma 1 they considered the variance of Ay BsAy, where the random vector y obeyed a normal law with mean m and variance V , and matrices A and Bs were square and constant. They used a procedure based on generalized cumulants. They also considered the case of an unspecified distribution. They showed that each element of D(Ay BsAy) depended on a typical term involving fourth-order moments, for which it was ‘not easy to express this term in matrix notation’. It is, however, possible to get D(Ay BsAy) for normal and unspecified distributions by using algebraic procedures based upon the Hadamard and Kronecker matrix products