The article deals with the problem of a statistical analysis of time series connected with the estimation of mutual variogram, a measure of spatial correlation. G. Matheron [1] has coined the term variogram, although earlier appearances of this function can be found in the scientific literature [2, 3]. The problem of estimating the mutual variogram and examination the statistical properties of this statistics has been considered in [1, 4, 5]. We present the limiting expressions of the first two moments and the higher order cumulants of the mutual variogram estimate of the second-order-stationary stochastic process with discrete time. These expressions are then used to prove the theorem concerning the asymptotic distribution of the mutual variogram estimate. The approach is similar to the approach taken in the time series literature, and the reader is referred to D. Brillinger [6] for theorems regarding the asymptotic distribution of the spectral density estimate of a time series.
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In the paper, a sequential confidence set based on an estimation process of a multivariate parameter is constructed. Under the assumption that the estimation process scaled by an increasing positive process has an asymptotic distribution it is proved that the sequential confidence set is asymptotically consistent and asymptotic- ally efficient. The results are applied to the sequential confidence sets based on maximum likelihood estimators of a multivariate parameter in the iid case and in the exponential class of processes.
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