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Clear All Modify Search  # Fulfilment of the law of large numbers in case of variance determinations

Acta Geodaetica et Geophysica Hungarica
Authors: B. Hajagos and Ferenc Steiner

As the variance (the square of the minimum L 2-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 VAR= Ön (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. __

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# Practical - but mysterious - behaviour of the most frequent values

Acta Geodaetica et Geophysica Hungarica
Authors: Ferenc Steiner and B. Hajagos

The paper proves the practically advantageous fact that for the determination errors of the most frequent value calculations the simple asymptotic rule is valid for the whole sample-size domain 1 ? n ³ 4.

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# Reliability of the estimations of the correlation coeficcient

Acta Geodaetica et Geophysica Hungarica
Authors: Ferenc Steiner and B. Hajagos

On the basis upon n corresponding value-pairs (x i; y i), i = 1, …, n, the closeness of correspondence between the random variables x and h is customarily characterized by the classical correlation coefficient r (see Eq. (2) in the present paper), equally in the geosciences and in the everyday life. It is shown in the present paper the lack of the robustness of Eq. (2) (r has even no meaning for circa 40% of the types occurring in the geosciences), and the lack of the resistance (one single outlying value-pair can distort the r-value in an incredible degree). The modern correlation coeffcient r rob (see Eq. (9) in this paper) is completely resistant against outliers, and in the same time also robust: Eq. (9) is applicable even if x and h are of Cauchy type, very far lying from the Gaussian distribution and even from the most frequently occurring so-called statistical distribution (see Eq. 6). For the Cauchy distribution neither the scatter (variance) nor the expected value exist therefore for this distribution-type even the classical theoretical value (see Eq. 3) does not exist: the calculation of r according to Eq. (2) gives in this case an "estimation" of a not existing quantity. In the paper are presented the results of a time consuming series of Monte Carlo calculations made equally for the statistical and Gaussian distributions and for n = 10;   30 and   100; the errors characterized by the semi-interquartile and semi- intersextile ranges of the modern rrob (Eq. 9) were calculated and tabulated for r t = 0; 0.1; 0. 2; … 0. 7 and 0. 8. An approximate method is also given (see the simple Eqs 16 and 17) to determine that value of n which assures a prescribed accuracy of the modern r rob.

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