# About the least-squares method, adapted to the Pearson-Jeffreys law of errors

2014;
: pp. 68 - 73
Received: January 17, 2014
Accepted: March 24, 2014
Authors:
1
International University of Economics and Humanities named after Academician S. Demianchuk

Introduction. K.F. Gausse creates the classical least-square method (LSM) based on the hypothesis of normality of the observation errors. However this hypothesis, as a rule, is uncapable, if the number of the numerous instrumentation is n>500. In this case the errors are described with the Pearson-Jeffreys symmetrical threeparametrical distribution, which as the Gausse law has the diagonal information matrix and one can name it as the universal distribution law of overall size errors in accordance with numerous research. The aim of this investigation is elaboration of evolutionary procedures of the modern update LSM adapted to the Pearson-Jeffreys law of errors.

Methods of solving this problem is based on the analytic theory adapted to errors of observation weighting function that we have developed. The basic result of the work is that this theory transforms robust estimation from heuristic attempts into true science. The scientific novelty of this investigation: the meaning of the residual errors analysis was shown firstly from the Fisher’s theory of estimation point of view and it allows to outline the weight functions zones of singularity when LSM using. Practical importance: the diagnostic technique of the results of LSM usage on the basis of analysis of statistical semi-invariant of residual errors was elaborated and well-grounded evolutional procedures for receiving of effective LSM estimations which do not change practically the data handling classical algorithms were created.

1. Alimov Ju., Shaevich A. B. Metodologicheskie osobennosti ocenivanija rezul'tatov kolichestvennogo himicheskogo analiza [Methodological features of estimation of quantitative chemical analysis]. Zhurnal analiticheskoj himii. [Journal of Analytical Chemistry]. 1988, t. XLIII, no.10 p. 1893-1917.
2. Granovskaja V. A., Siraj T. N. Metodologicheskie osobennosti ocenivanija rezul'tatov kolichestvennogo himicheskogo analiza [Processing methods of experimental data for measurements]. Leningrad, Energia Publ. 1990, p. 328.
3. Dzhun' I. V. Ob odnom obobshhenii fundamental'nogo principa metoda naimen'shih kvadratov v svjazi s jevoljuciej predstavlenij o zakone oshibok nabljudenij [A generalization of the fundamental principle of the method of least squares with the evolution of the laws of the observational errors]. Izvestija vuzov. Geodezija i ajerofotos#emka [Proceedings of the universities. Surveying and aerial photography]. 2013, № 6, p. 19–26.
4. Dzhun' I. V. Matematicheskaja obrabotka astronomicheskoj i kosmicheskoj informatsii pri negaussovykh oshibkakh nabljudenij. Avtoreferat Doct.Diss. [Mathematical Treatment of astronomical and space Information in non-Gaussian observation Errors. Author’s abstract]. Kiev, 1992. 46 p.
5. Novickij P. V. Otsenka pogreshnostej rezul'tatov izmereniyy [Estimation of Errors of measurement Results]. Leningrad, Energoatomizdat Publ., 1991, 304 p.
6. Frans A. Kniga Sjuzanny. Polnoe sobranie sochinenij [Suzanne's book. Complete Works.]. Moscow: Gostehizdat, Publ. 1957, t.1, p. 551–610.
7. Jel'jasberg P.E. Izmeritel'naja informatsija: skol'ko yeye nuzhno? Kak obrabatyvat' ? [The measuring Information: how it should be? How to handle?]. Moscow, Nauka Publ., 1983. 208 p.
8. Dzhun' I. V. About make use of Pearsons Distvibution of Type VII for the Approximation of observation’s Errors in Astrometry. Measurement Techniques: Springer Science + Business Media. Inc. – 1992, vol. 35, № 3, pp.298–304.
9. Dzhun' I. V. Pearsons Distribution of type VII of the Errors of Satellite Laser Ranging Data. Kinematics and Physics of Celestial Bodies,New York: Allerton Press, Inc., 1991, vol.7, №3, pp. 74–84.
10. Dzhun' I. V. Distribution of Errors in multiple large-volume observations. Measurement Techniques: Springer Sciene + Business Media. Inc., 2012, vol. 55, no.4, pp. 393–396.
11. Jeffreys H. Theory of Probabiliti. 3rd ed. Oxford, Clarendon Press, 1983, 459 p.
12. Hulme H. R., Syms L. S. T. The Law of Errors and the Combinations of Observations. Mon. Notic. Roy. Astron. Soc. 1939, 99, no. 8. – P. 642–658.
13. Kline Morris. Mathematics. The Loss of Certainty. New York. Oxford: Oxford University Press, 1980, 420 p.
14. Robust Statistics. The Approach Based on Influence Functions / F. R. Hampel, E. M. Ronchetti, P. J. Rousseeuw, W. A. Stahel. // John Wiley & Sons, Inc. 1986 – 488 p.