Statistical Analysis Using Local Properties of Smoothly Heteromorphic Stochastic Series

SUMMARY. A widening of the oonoept of stationarity leads to the concepts of smooth hetero-morphy and local homomorphy in stochastic series, obviating much of the need for the introduction of trend into structural specifications of statistical data. It is shown that formulae for sampling properties of local statistics in stationary stochastic series are applicable as they stand to series having these less restricted properties. 1. INTBODUCTION In a recent paper (1955a) the author established the principle that the sampling properties, and notably the standard errors, of statistics constructed from local comparisons of terms in stationary normal stochastic series were approximately deducible from the short term variational properties of the series themselves. In making practical use of such statistics, it is not necessary to know the mean, variance, or serial correlations of the series, all of which are dependent on the long-term variational properties of the series; it suffices to know the values of the serial and lag variation functions for short lags, i.e. the mean semi-squared differences of terms in the series separated by distances which are short. This principle is important because in many practical applications the stretches of series which constitute the data are muoh too short to provide accurate and unbiased estimates of long-term variational properties, whereas accurate and unbiased estimates of the lag varia-tion functions for short lags can be obtained even, say, by combining evidence from short scraps of series having the same variational properties but different means. In recent papers the author has proposed the application of this principle in a number of techniques, including the following: