Optimal Aggregation of Clustered Sample Intervals for Applying the χ2 Test
https://doi.org/10.17586/0021-3454-2024-67-9-751-758
Abstract
The use of intervals of equal length or intervals of equal probability for using the χ2-type criterion is discussed. In this case, intervals of equal probability are predetermined by the distribution law being tested. When forming the initial sample based on real production data, it is often immediately grouped with predetermined and unchangeable grouping boundaries in production and may not satisfy the recommendations for applying χ2-type criteria. A method is proposed for constructing a set of optimal grouping intervals by combining some of the intervals available in the initial sample. An optimal set of such intervals is understood to be a set of intervals that has the least square deviation of weighted frequencies of hits from a discrete uniform distribution, which makes it possible not to change the set of intervals when changing the selected distribution law and to automatically solve the problem of choosing the optimal number of intervals. Some properties of such sets are listed, examples of situations arising during their construction are considered, and an example of forming such an optimal set is given.
About the Authors
P. М. Vinnik
D. F. Ustinov Baltic State Technical University VOENMEH
Russian Federation
Russian Federation
Petr M. Vinnik, Dr. Sci., Associate Professor, Head of the Department
Department of Higher Mathematics
St. Petersburg
Т. V. Vinnik
St. Petersburg State Institute of Technology
Russian Federation
Russian Federation
Tatyana V. Vinnik, PhD, Associate Professor
Department of Mathematics
St. Petersburg
Е. А. Еskova
D. F. Ustinov Baltic State Technical University VOENMEH
Russian Federation
Russian Federation
Ekaterina A. Eskova, Assistant
Department of Higher Mathematics
St. Petersburg
References
1. Mittag H.-J., Rinne H. Statistische Methoden der Qualitatssicherung, Munchen, Wien, 1993.
2. Kobzar' A.I. Prikladnaya matematicheskaya statistika. Dlya inzhenerov i nauchnykh rabotnikov (Applied Mathematical Statistics. For Engineers and Scientists), Moscow, 2006, 816 р. (in Russ.)
3. Lemeshko B.Yu., Postovalov S.N. Izvestiya vuzov. Fizika, 1995, no. 9, pp. 39–45. (in Russ.)
4. Novitskiy P.V., Zograf I.A. Otsenka pogreshnostey rezul'tatov izmereniy (Evaluation of Errors in Measurement Results), Leningrad, 1991, 304 р. (in Russ.)
5. Hald A. Statistical Theory with Engineering Applications, NY, Wiley, 1952, 783 p.
6. Mann H.B., Wald A. Ann. Math. Stat., 1942, vol. 13, рр. 306–317.
7. Williams C.A., jr., Journal of the American Statistical Association, 1950, no. 249(45), pp. 77–86.
8. Lemeshko B.Yu., Postovalov S.N. Industrial Laboratory. Diagnostics of Materials, 1998, no. 5(64), pp. 56–63. (in Russ.)
9. Lemeshko B.Yu., Chimitova E.V. Industrial Laboratory. Diagnostics of Materials, 2003, vol. 69, рр. 61–67. (in Russ.)
10. Kulldorff G. Contributions to the theory of estimation from grouped and partially grouped samples, NY, John Wiley, 1963, 144 p.
11. https://03.rosstat.gov.ru/storage/mediabank/05_tom1(1).pdf.
12. Vinnik P.M., Vinnik T.V., Eskova E.A. Bulletin of Education and Development of Science of the Russian Academy of Natural Sciences, 2022, no. 4, pp. 24–30, DOI: 10.26163/RAEN.2022.32.23.003. (in Russ.)
13. Andrews G.E. The theory of partitions, Addison-Wesley Pub. Co., 1976, 255 p.
14. Korf R. IJCAI'09, Proceedings of the 21<sup>st</sup> International Joint Conference on Artificial Intelligence Pasadena, California, USA, July 11-17, 2009, pp. 538–543.
15. Bardasov S.A. Bulletin of Tyumen State University, 2003, no. 5, pp. 217–219. (in Russ.)
Review
For citations:
Vinnik P.М., Vinnik Т.V., Еskova Е.А. Optimal Aggregation of Clustered Sample Intervals for Applying the χ2 Test. Journal of Instrument Engineering. 2024;67(9):751-758. (In Russ.) https://doi.org/10.17586/0021-3454-2024-67-9-751-758
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