A Statistical Index Representing a Gross Feature of Several Climatic Time Series by Hard Limiting

Abstract

An index of gross feature of several climatic time series is given by a total of its hard limiting series obtained by level-crossing to the original time series. Under admitting a weak concordance between the paired hard limiting series, a distinguished cold period "Maunder minimum" or a warm period can be interpreted by the above index series.
For a pair of random variable at any two time points, an unbiased estimator of correlation is given by a certain function of level, under the condition that a pair of random variable follows to a bi-variate normal distribution with known mean values and variances. An optimum level for the estimation of correlation is discussed by the criterion of minimum variance under the statistical optimality condition; the variance of estimator is explicitly given by a function of such a bi-variate correlation and level of crossing.
Autocorrelation of index series can be also expressible by an integral form which is a function of such a correlation and level of crossing.