br It is assumed that each sample j is generated

It is assumed that each sample , j=1, 2, …, is generated by a source with an explicitly unknown conditional distribution density (8). Let us represent the quantity dF(w) in the following form: where N(w) is the information capacity of the source w; is the total information capacity of all sources; g(w) is the weight taking into account the rest of the factors (weight selection is described below). If we take into account that the relative information capacity of a source can be estimated as , then we can obtain the following estimate for the quantity (11): If we take the unit weights g(w)=1, then the Formula (13) is reduced to the average value of the squared regression deviation over the whole aggregate sample , and we obtain the buy Caspase-3/7 Inhibitor coinciding with Eq. (12). The described scheme involving dividing the aggregate sample into groups is typical for the so-called variance analysis developed by Fisher [13] to take into account the influence of the systematic error of the group and the random error within the group. This scheme is convenient for describing the situation with heterogeneous information sources based on both the theoretical and the experimental methods for obtaining cross-sections. Within this scheme, it is also convenient to formulate and test statistical hypotheses involving clustering. Testing the condition under which the systematic error of a partial source exceeds certain bounds is of practical interest, in particular, for the case of electron-atom excitation we are examining. Let us assume that the parameters for approximating the cross-section have been found from minimizing the expectation of the squared deviation (13). Let us formulate for the chosen information source the following hypothesis about the magnitude of the conditional expectation (10) of the squared regression deviation:
It is known [11–13] that for testing this hypothesis, the statistic can be used, where is the sample mean of a random variable of the squared regression deviation δ=δ() of the information source ; is the respective squared sample variance. The Neyman–Pearson model bases the criterion for rejecting the hypothesis (14) at the regression level α on the critical region for the statistic (15). If this statistic falls into the critical region, then the hypothesis is rejected. In this case, the probability of rejecting a true hypothesis (type I error) does not exceed the quantity where φ(y) is the distribution of the statistic (15). Let us turn our attention to the distribution of the statistic (15). In principle, it can be obtained from the distributions of the conditional deviations that, as stated above, are unknown to us. The problem can be significantly simplified if we examine the case when the sample number is large enough. Then, according to the central limit theorem, the distribution of the numerator of the Expression (15) can be approximated by a normal distribution independent of the Distribution (17). The squared sample variance in the denominator of the Expression (15) tends to the variance. Thus, the Distribution (15) can be approximated by a standard normal distribution with the density
Let us note that in case of a normal distribution of the squared deviation quantities , the Statistic (15) has a Student\'s t-distribution with m= 1 degrees of freedom which at large values of m tends to a normal distribution. However, the same as for logarithmic deviations, the assumption that the distribution of the squared deviation quantities is normal may turn out to be unjustified. In our studies, this choice is made on the following principle. The data are taken from peer-reviewed journals with a unit weight =1. The other sources are taken with a zero weight, =0, i.e., they are not taken into account. Such a model is justified because the respective results have passed an expert test. Attempting to make a more clear-cut distinction between information sources inevitably introduces more subjectivity, and, due to the above-described considerations, applying statistical methods does not eliminate this subjectivity.