Estimation of quality of a small sampling biometric data using a more efficient form of the chi-square test

Aim. The purpose is to increase the power of the Pearson’s chi-square test so that this test will become efficient on small test samplings. . It is necessary to reduce the scope of a test sample from 200 examples to 20 examples while maintaining the probability of errors of the first and the second kind. Selection of 20 examples of biometric images is considered by users to be a comfortable level of effort. The need to select more examples is perceived by users negatively.

Methods. The article offers one more (the second) form of the Pearson test that is much less sensitive to the scope of data in a test sampling. It is shown that a traditional form of the chi-square test is more sensitive to the scope of a test sampling than the Cramer-von Mises test. The offered (second) form of the chi-square test is less sensitive to the scope of a test sampling than a classical form of the chi-square test and less sensitive than the Cramer-von Mises test as well. This effect is achieved by the transition from the space of frequency of occurrence of events and probabilities of a group of similar events occurring in the space of more accurately evaluated junior statistical moments (mean and standard deviation). The fractal dimension of the new synthetic form of chi-square test coincides with the fractal dimension of the classical form of the chi-square test.

Results. The offered second variant of the chisquare test is presumably one of the most powerful of all existing statistical tests. The analytical description of correlation of standard deviations of a classical form of the chi-square test and a new form of the chi-square test is given. The standard deviation of the second form of the chi-square test decreases by half on retention of a statistical expectation on samplings of the same scope. The latter is equivalent to a four-time reduction of the requirements to the scope of a test sampling within the interval from 16 to 20 examples. Power gain as the result of the application of a new test is growing with the growth of a test sampling scope.

Conclusions. When creating a classical chi-square test in 1900, Pearson was guided by limited computing opportunities of the existing computer facilities, and he had to rely on the analytical relations that he found. Today the situation has changed and there are no more restrictions in relation to the engaged computing resources. However we continue to rely on those created with computing resources of 1900 by inertia. Probably, we should try to consider modern opportunities of computer facilities and to build more powerful options of statistical tests. Even if new tests will require a search of large number of possible states (they will have big tables calculated in advance instead of analytical relations), it is not a constraining factor today. When data is insufficient (in biometrics, in medicine, in economy) a computing complexity of statistical tests does not play a special role if the result of estimations is more accurate.

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