Precision Statistics: Fractional Number of Degrees of Freedom Chi-Square Criterion for Small Samples of Biometric Data

The Aim of this paper is to increase the power of statistical tests through their joint application to reduce the requirement for the size of the test sample. Methods. It is proposed to combine classical statistical tests, i.e. chi square, Cram r-von Mises and Shapiro-Wilk by means of using equivalent artificial neurons. Each neuron compares the input statistics with a precomputed threshold and has two output states. That allows obtaining three bits of binary output code of a network of three artificial neurons. Results. It is shown that each of such criteria on small samples of biometric data produces high values of errors of the first and second kind in the process of normality hypothesis testing. Neural network integration of three tests under consideration enables a significant reduction of the probabilities of errors of the first and second kind. The paper sets forth the results of neural network integration of pairs, as well as triples of statistical tests under consideration. Conclusions. Expected probabilities of errors of the first and second kind are predicted for neural network integrations of 10 and 30 classical statistical tests for small samples that contain 21 tests. An important element of the prediction process is the symmetrization of the problem, when the probabilities of errors of the first and second kind are made identical and averaged out. Coefficient modules of pair correlation of output states are averaged out as well by means of artificial neuron adders. Only in this case the connection between the number of integrated tests and the expected probabilities of errors of the first and second kind becomes linear in logarithmic coordinates.

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The Neural Network Analysis of Normality of Small Samples of Biometric Data through Using the Chi-Square Test and Anderson–Darling Criteria

Engineering Technologies and Systems ◽

10.15507/2658-4123.029.201902.205-217 ◽

2019 ◽

pp. 205-217

Author(s):

Vladimir I. Volchikhin ◽

Aleksandr I. Ivanov ◽

Alexander V. Bezyaev ◽

Evgeniy N. Kupriyanov

Keyword(s):

Neural Network ◽

Sample Size ◽

Single Layer ◽

Test Sample ◽

Small Samples ◽

Chi Square ◽

Biometric Data ◽

The Neural Network ◽

Anderson Darling ◽

Statistical Criteria

Introduction. The aim of the work is to reduce the requirements to test sample size when testing the hypothesis of normality. Materials and Methods. A neural network generalization of three well-known statistical criteria is used: the chi-square criterion, the Anderson–Darling criterion in ordinary form, and the Anderson–Darling criterion in logarithmic form. Results. The neural network combining of the chi-square criterion and the Anderson–Darling criterion reduces the sample size requirements by about 40 %. Adding a third neuron that reproduces the logarithmic version of the Andersоn–Darling test leads to a small decrease in the probability of errors by 2 %. The article deals with single-layer and multilayer neural networks, summarizing many currently known statistical criteria. Discussion and Conclusion. An assumption has been made that an artificial neuron can be assigned to each of the known statistical criteria. It is necessary to change the attitude to the synthesis of new statistical criteria that previously prevailed in the 20th century. There is no current need for striving to create statistical criteria for high power. It is much more advantageous trying to ensure that the data of newly synthesized statistical criteria are low correlated with many of the criteria already created.

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Statistical Justification of Pearson's Criterion for Testing a Complex Hypothesis on the Uniform Distribution

Mechanical Engineering and Computer Science ◽

10.24108/0418.0001392 ◽

2018 ◽

pp. 45-53

Author(s):

T. V. Oblakova

Keyword(s):

Uniform Distribution ◽

Degrees Of Freedom ◽

Likelihood Function ◽

Random Number Generator ◽

Maximum Likelihood Estimates ◽

Chi Square ◽

Main Hypothesis ◽

Pearson Criterion ◽

Complex Hypothesis ◽

Uniform Law

The paper is studying the justification of the Pearson criterion for checking the hypothesis on the uniform distribution of the general totality. If the distribution parameters are unknown, then estimates of the theoretical frequencies are used [1, 2, 3]. In this case the quantile of the chi-square distribution with the number of degrees of freedom, reduced by the number of parameters evaluated, is used to determine the upper threshold of the main hypothesis acceptance [7]. However, in the case of a uniform law, the application of Pearson's criterion does not extend to complex hypotheses, since the likelihood function does not allow differentiation with respect to parameters, which is used in the proof of the theorem mentioned [7, 10, 11].A statistical experiment is proposed in order to study the distribution of Pearson statistics for samples from a uniform law. The essence of the experiment is that at first a statistically significant number of one-type samples from a given uniform distribution is modeled, then for each sample Pearson statistics are calculated, and then the law of distribution of the totality of these statistics is studied. Modeling and processing of samples were performed in the Mathcad 15 package using the built-in random number generator and array processing facilities.In all the experiments carried out, the hypothesis that the Pearson statistics conform to the chi-square law was unambiguously accepted (confidence level 0.95). It is also statistically proved that the number of degrees of freedom in the case of a complex hypothesis need not be corrected. That is, the maximum likelihood estimates of the uniform law parameters implicitly used in calculating Pearson statistics do not affect the number of degrees of freedom, which is thus determined by the number of grouping intervals only.

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PRECISION STATISTICS: NEUROET NETWORKING OF CHI-SQUARE TEST AND SHAPIRO–WILK TEST IN THE ANALYSIS OF SMALL SELECTIONS OF BIOMETRIC DATA

Reliability and Quality of Complex Systems ◽

10.21685/2307-4205-2019-2-4 ◽

2019 ◽

Cited By ~ 1

Author(s):

A. I. Ivanov ◽

◽

S. E. Vjatchanin ◽

E. A. Malygina ◽

V. S. Lukin ◽

...

Keyword(s):

Chi Square ◽

Biometric Data ◽

Chi Square Test

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Crossover Interference in Arabidopsis

Genetics ◽

10.1093/genetics/160.4.1631 ◽

2002 ◽

Vol 160 (4) ◽

pp. 1631-1639 ◽

Cited By ~ 6

Author(s):

G P Copenhaver ◽

E A Housworth ◽

F W Stahl

Keyword(s):

Arabidopsis Thaliana ◽

Degrees Of Freedom ◽

Double Strand Breaks ◽

Crossing Over ◽

Chi Square ◽

Strand Breaks ◽

Crossover Interference

AbstractThe crossover distribution in meiotic tetrads of Arabidopsis thaliana differs from those previously described for Drosophila and Neurospora. Whereas a chi-square distribution with an even number of degrees of freedom provides a good fit for the latter organisms, the fit for Arabidopsis was substantially improved by assuming an additional set of crossovers sprinkled, at random, among those distributed as per chi square. This result is compatible with the view that Arabidopsis has two pathways for meiotic crossing over, only one of which is subject to interference. The results further suggest that Arabidopsis meiosis has >10 times as many double-strand breaks as crossovers.

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Confronting Theories of Firm Growth in Light of Degrees-of-Freedom Analysis

Organizações & Sociedade ◽

10.1590/1984-9230745 ◽

2015 ◽

Vol 22 (74) ◽

pp. 385-404

Author(s):

Sérgio Fernando Loureiro Rezende ◽

Ricardo Salera ◽

José Márcio de Castro

Keyword(s):

Degrees Of Freedom ◽

Firm Growth ◽

Dynamic Capabilities ◽

Statistical Tests ◽

Explanatory Power ◽

Stage Theory ◽

Chi Square ◽

Growth Optimum ◽

Chi Square Test ◽

Do So

This article aims to confront four theories of firm growth – Optimum Firm Size, Stage Theory of Growth, The Theory of the Growth of the Firm and Dynamic Capabilities – with empirical data derived from a backward-looking longitudinal qualitative case of the growth trajectory of a Brazilian capital goods firm. To do so, we employed Degree of Freedom-Analysis for data analysis. This technique aims to test the empirical strengths of competing theories using statistical tests, in particular Chi-square test. Our results suggest that none of the four theories fully explained the growth of the firm we chose as empirical case. Nevertheless, Dynamic Capabilities was regarded as providing a more satisfactory explanatory power.

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