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  • br It is reasonable to relate

    2018-10-24


    It is reasonable to relate the conditional expectation of the deviation for the regression (2) at the given Caspase-3/7 Inhibitor and source, namely, to the systematic error of the source, e.g., the procedure error of the method for obtaining the cross-sections for each energy. In this case the conditional expectation of the deviation for the regression (2) on all energies with the given source, taking the form can turn out to equal zero even with nonzero systematic errors (9) present, since errors for different energies can have unlike signs. This is why it is preferable to use the conditional expected squared deviation of regression for the given information source: in order to characterize the total procedure error. For an explicit representation of the contributions of single information sources, let us write, using the distribution (8), the expected squared deviation (5) in the form and through the conditional expectation (10) in the form
    This expression has the form of a weighted average of the conditional expectations (10) with the weights assigned to information sources dF(w). Let us note that the normalization condition is fulfilled. In our studies we selected the most frequently used parametric representation of the variable class of functions: where p is the set of parameters. We are going to discuss choosing the specific parametrization below. With the parametrization in mind, from now on, we will give the parameters p of the function σ as the variable quantity in the formulae instead of the symbol of the function itself. Consequently, the problem is reduced to searching for the minimum (7) of the quantity (5) depending on the parameters p, where . Above we have Caspase-3/7 Inhibitor laid out the general scheme for applying the statistical approach to a broad class of problems, including the one we are examining. It does not matter by whom the results have been obtained, whether it was experimentally or theoretically, and which method has been used. From a statistical viewpoint, it is only important how the random variable is distributed. This can be assessed only based on the samples. To find the Quantity (5), we should take the sample of the quantities , constructed based on the existing data on the studied problem:
    It is known that the sample average in the right-hand side of Eq. (12) converges in probability to the expectation with an increase in sample number, which is one of the forms of the law of large numbers [11,12]. The latter takes place provided that the sample is generated with a distribution densityf(v). It can be assumed, as a simplest model, that this condition is ensured by the nature of the aggregate information source. The consideration that the sample actually does not depend on the information users but results from the combined efforts of all the information makers can serve as a proof of this. Remarkably, if the above condition is satisfied, there is no need to find the form of the distribution function f unknown to us. Let us also note that such an estimate can be regarded as one for the integral by the Monte–Carlo method. Its accuracy, as follows from the central limit theorem [11,12] has the order , where D is the dispersion of the quantity δ2, and n is the sample number. The assumption that the sample of the quantities is generated with the density of the conditional distribution (8) for each information source separately can serve as a more rigorous model. In this case the formula (12) requires to be substantiated further. For this purpose, let us use the representation of the deviation variance in the form (11). Taking into account the structure of the expression (11), the sample can be divided into groups by information source. Let be a sample of the number I from information sources. For each selected source with the index i let us denote the random variable of the pairs of cross-sections and energies with the conditional distribution density (8) as , and the corresponding sample as , j=1, 2, …, . The total number of the aggregate sample is equal to . For the Quantity (11) we have the following estimate: