Fulfilment of the law of large numbers in case of variance determinations

Hajagos, B. and Steiner, Ferenc (2001) Fulfilment of the law of large numbers in case of variance determinations. Acta Geodaetica et Geophysica Hungarica, 36 (2). pp. 163-174. ISSN 1217-8977

Text ageod.36.2001.2.4.pdf Restricted to Repository staff only Download (393kB)

Abstract

As the variance (the square of the minimum L2-norm, i.e., the square of the scatter) is one of the basic characteristics of the conventional statistics, it is of practical importance to know the errors of its determination for different parent distribution types. This statement is outstandingly valid for the geostatistics because the (h) variogram (called also as semi-variogram) is defined as the half variance of some quantity-difference (e.g. difference of ore concentrations) in function of the h dis- tance of the measuring points and this g (h)-curve plays a basic role in the classical geostatistics. If the scatter (s VAR) is chosen to characterize the determination uncertainties of the variance (denoted the latter by VAR), this can be easily calculate as the quotient A AVAR=p n(if the number n of the elements in the sample is large enough); for the so-called asymptotic scatter A VAR is known a simple formula (containing the fourth moment). The present paper shows that the AVAR has finite value unfortunately only for about a quarter of distribution types occurring in the earth sciences, it must be especially accentuate that A VAR has infinite value for that distribution type which most frequent occurs in the geostatistics. It is proven by the present paper that the law of large numbers is always fulfilled (i.e., the error always decreases if n increases) for the error-determinations if the semi-intersextile range is accepted (instead of the scatter); the single (quite natural) condition is the existence of the theoretical variance for the parent distribution. __

Actions (login required)

Edit Item