|Article title||BOUNDED QUANTITATIVELY EXPERIMENT BINOMIAL DISTRIBUTION PARAMETER ESTIMATION PROBLEM|
|Authors||R.B. Tregubov, M.V. Stremouhov|
|Section||SECTION II. COMPUTER ENGINEERING AND COMPUTER SCIENCE|
|Month, Year||02, 2015 @en|
|Abstract||Aim of this study is to find an approximation interval Clopper–Pearson provided out of the assumption that the frequency of occurrence of an event in n independent experiments (Bernoulli scheme) normal distribution. The practice shows that this assumption causes a serious error for the statistical analysis of rare events in a limited number of experiments. This object is achieved in the work by approximation the exact solution of the equations Clopper–Pearson with the help of sixth – degree polynomial. In turn, to obtain an accurate solution of the equations Clopper–Pearson used in the numerical method of bisection (method of bisection of the interval), implemented in an environment of mathematical modeling Mathcad. The value of the module of the proposed polynomial approximation (in a limited number n of independent experiments) error doesn’t exceed 5·10-3 . In turn, for the known approximation (in the same case) of the value of the module approximation error is much more, as evidence by the results of mathematical modeling Mathcad. The main results of the study are presented in tabular form coefficients of the approximating polynomials for different values of the confidence probability – β (0,9; 0,95; 0,99; 0,995 and 0,999) and number of tests – n (10; 20; 30… 100), the values of the polynominal to determine the left and right boundaries of the same parameter of the binominal distribution. A distinctive feature of the proposed method in the calculation of the boundaries of the confidence interval is that, firstly, the order of the approximating polynomial doesn’t depend on the namber of tests, and the coefficients of the boundary which is calculated; and secondly, eliminating the need to use the binomial distribution tables or approximating its beta distribution, F-distribution, normal distribution and Poisson distribution. The results can be applied for the analysis of probabilistic–temporal characteristics (loss probability PDUs transshipment, errors, not timely delivery etc.), communication systems for various purposes or their simulation models.|
|Keywords||Probability;relative frequency; point estimate; interval estimate; confidence interval; confidence probability; binomial distribution; Clopper–Pearson equation.|
|References||1. Thulin M. The cost of using exact confidence intervals for a binomial proportion, Electronic Journal of Statistics, 2014, Vol. 8, pp. 817-840.
2. Ivanov N.N., Strel'nikov V.P. Prognozirovanie ostatochnoy dolgovechnosti payanykh soedineniy [Prediction of residual life of solder joints], Matematichnі mashini і sistemi [Mathematical Machines and Systems], 2012, No. 3, pp. 162-165.
3. Kuznetsov A.G. Aleksandrovskaya L.N. Neparametricheskie metody "izmereniya" malykh riskov v zadachakh otsenki sootvetstviya trebovaniy k bezopasnosti avtomaticheskoy posadki samoletov normam letnoy godnosti [Nonparametric methods for measurement of small risks in the tasks of conformity assessment requirements for security of automatic landing aircraft airworthiness], Trudy Moskovskogo instituta elektromekhaniki i avtomatiki (MIEA) [Proceedings of the Moscow Institute of electromechanics and automation (MIEA)], 2011, Issue 3, pp. 2-11.
4. Gusev L.A. Ob interpretatsii nerazlichimosti v zadache interval'noy otsenki neizvestnoy veroyatnosti [About the interpretation of fuzzy in the problem of interval estimation of unknown probability], Avtomatika i telemekhanika [Avtomatika i Telemekhanika], 2010, Issue 8, pp. 38-48.
5. Gusev L.A. O nekotorykh svoystvakh doveritel'nykh intervalov dlya neizvestnykh veroyatnostey [On some properties of confidence intervals for unknown probabilities], Avtomatika i telemekhanika [Avtomatika i Telemekhanika], 2007, Issue 12, pp. 70-84.
6. Krishnamoorthy K., Peng J. Some properties of the exact and score methods for binomial proportion and sample size calculation, Communications in Statistics – Simulation and Computation, 2007, Vol. 36, pp. 1171-1186.
7. Venttsel' E.S. Teoriya veroyatnostey: Uchebnik dlya vuzov [Probability theory: the Textbook for high schools]. 7 th ed. Moscow: Vysshaya shkola, 2001, 575 p.
8. Kobzar' A.I. Prikladnaya matematicheskaya statistika. Dlya inzhenerov i nauchnykh rabotnikov [Applied mathematical statistics. For engineers and scientists]. Moscow: Fizmatlit, 2006, 816 p.
9. Agresti A. Score and pseudo-score confidence intervals for categorical data analysis, American Statistical Association. Statistics in Biopharmaceutical Research, 2011, Vol. 3, No. 2, pp. 163-172.
10. Brown L.D.,Cai T.T., DasGupta A. Confidence intervals for a binomial proportion and asymptotic expansions, The Annals of Statistics, 2002, Vol. 30, No. 1, pp. 160-201.
11. Reiczigel J. Confidence intervals for the binomial parameter: some new considerations, Statistics in Medicine, 2003, Vol. 22, pp. 611-621.
12. Boomsma A. Confidence intervals for a binomial proportion, University of Groningen. Department of statistics and measurement theory, 2005, pp. 1-9.
13. Agresti A., Coull B.A. Approximate is better than “exact” for interval estimation of binominal proportion // American Statistician. – 1998. – Vol. 52. – P. 119-125.
14. Agresti A., Caffo B. Simple and effective confidence intervals for proportions and differences of proportions result from adding two successes and two, American Statistician, 2000, Vol. 54, No. 4, pp. 280-288.
15. Robert C.P. The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. New York, Springer, 2007, 602 p.
16. Thulin M. On split sample and randomized confidence intervals for binomial proportions, Statistics & Probability Letters, 2014, Vol. 92, pp. 65-71.
17. Spravochnik po nadezhnosti. V 3 vol. Vol. 1 = Reliability handbook, Under ed. W.G. Ireson: translation from English, Under ed. B.R. Levina. Moscow: Mir, 1969, 340 p.
18. Ouen D.N. Sbornik statisticheskikh tablits [The collection of statistical tables]: translation from English. Moscow: VTs AN SSSR, 1966, 568 p.
19. Bol'shev L.N., Smirnov N.V. Tablitsy matematicheskoy statistiki [Tables of mathematical statistics]. Moscow: Nauka. Glavnaya redaktsiya fiziko-matematicheskoy literatury 1983, 416 p.
20. Yanko Ya. Matematiko-statisticheskie tablitsy [Mathematical-statistical tables]: translation from Czech. Moscow: Gosstatizdat, 1961, 244 p.
21. Venttsel' E.S., Ovcharov L.A. Teoriya veroyatnostey i ee inzhenernye prilozheniya. Ucheb. posobie dlya vtuzov [Probability theory and its engineering applications. Textbook for technical colleges]. 2 nd ed. Moscow: Vysshaya shkola, 2000, 480 p.