1Universidad de Guadalajara, CUCEI, Departamento de Matemáticas, Guadalajara, México. Profesor titular. Email: humberto.gutierrez@cucei.udg.mx
2Universidad de Guadalajara, CUCEI, Departamento de Matemáticas, Guadalajara, México. Profesor titular. Email: jmgarciat@prodigy.net.mx
El interés de este trabajo se centra en el problema de probar la aleatoriedad de los resultados de los juegos de números de d dígitos. Es usual que este problema se aborde con pruebas aproximadas del tipo χ2 y otras pruebas de independencia de resultados sucesivos. Pero estas pruebas, tienen entre otras limitantes, el hecho de que requieren muestras grandes. Como alternativa, en este trabajo se detalla una prueba bayesiana basada en el modelo multinomial. Además, para monitorear los resultados de este tipo de juego de azar y detectar en forma oportuna patrones y resultados no aleatorios, se propone la utilización de una carta de control geométrica. Se hace un breve estudio Monte Carlo para comprender mejor las características de la carta propuesta. Como caso práctico se analizan los resultados de 500 sorteos de la lotería mexicana Tris y se detectan problemas de falta de aleatoriedad, tanto con la prueba bayesiana como con la carta de control.
Palabras clave: carta de control, distribución geométrica, distribución multinomial, distribución Dirichlet, lotería, prueba ji-cuadrada.
this work is centered on testing the randomness of d-digit number game. It is usual that this problem is studied by the χ 2 test and other tests for independence of successive draws. However, these tests require of large sample sizes. As an alternative, it is proposed a Bayesian methodology based on the multinomial model. This methodology does not depend on asymptotic results. Besides, for monitoring the results of this type of game, it is proposed a geometric control chart. Monte Carlo study is carried out to analyze this chart. As a practical case, the data of 500 draws of mexican lottery Tris were analyzed, and problems of lack of randomness are detected.
Key words: Control chart, Dirichlet distribution, Geometric distribution, Lottery Chi-square test, Multinomial distribution.
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Referencias
]]>1. Bernardo, J. M. & Smith, A. F. M. (1994), Bayesian Theory, John Wiley, Chichester. [ Links ]
2. Box, G. E. P. & Pierce, D. A. (1970), 'Distribution of Residua Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models', Journal of the American Statistical Association 65, 1509-1526. [ Links ]
3. Cai, Y. & Krishnamoorthy, K. (2006), 'Exact Size and Power Properties of Five Tests for Multinomial Proportions', Communications in Statistics-Simulation and Computation 35, 149-160. [ Links ]
4. Clotfelter, C. T. & Cook, P. J. (2008), 'The ''Gambler's Fallacy'' in Lottery Play', Management Science 39(12), 1521-1525. [ Links ]
5. Coronel-Brizio, H., Hernández-Montoya, A., Rapallo, F. & Scalas, E. (2008), 'Statistical Auditing and Randomness Test of Lotto k/N-Type Games', Physica A 387(25), 6385-6390. [ Links ]
6. Finkelstein, M. (1995), 'Estimating the Frequency Distribution of the Numbers bet on the California Lottery', Applied Mathematics and Computation 69(2-3), 195-207. [ Links ]
7. Genest, C., Lockhart, R. A. & Stephens, M. A. (2002), '\chi^2 and the lottery', Journal of the Royal Statistical Society: Series D (The Statistician) 51(2), 243-257. [ Links ]
8. Goodman, L. A. (1965), 'On Simultaneous Confidence Intervals for Multinomial Proportions', Technometrics 7, 247-254. [ Links ]
9. Gutierrez, H. (2006), 'Cartas de control bayesianas para atributos y el tamaño de subgrupo grande en la carta p', Revista Colombiana de Estadistica 29(2), 163-180. [ Links ]
10. Gutiérrez, H. & de la Vara, R. (2009), Control estadístico de calidad y seis sigma, Segunda edn, McGraw-Hill, México. [ Links ]
11. Hou, C. D., Chiang, J. & Tai, J. J. (2003), 'A Family of Simultaneous Confidence Intervals for Multinomial Proportions', Computational Statistics & Data Analysis 43, 29-45. [ Links ]
12. Joe, H. (1987), 'An Ordering of Dependence for Distribution of k-tuples, with Applications to Lotto Games', The Canadian Journal of Statistics 15(3), 227-238. [ Links ]
13. Joe, H. (1990), 'A Winning Strategy for Lotto Games?', The Canadian Journal of Statistics 18(3), 233-244. [ Links ]
14. Joe, H. (1993), 'Tests of Uniformity for Sets of Lotto Numbers', Statistics & Probability Letters 16(3), 181-188. [ Links ]
15. Johnson, R. & Klotz, J. (1993), 'Estimating hot Numbers and Testing Uniformity for the Lotto', Journal of the American Statistical Association 88(422), 662-668. [ Links ]
16. Koning, R. H. & Vermaat, M. B. (2002), A Probabilistic Analysis of the Dutch Lotto, University of Groningen. Research report. [ Links ]
17. Ljung, G. M. & Box, G. E. P. (1978), 'On a Measure of Lack of Fit in Time Series Models', Biometrika 65(2), 297-303. [ Links ]
18. Nass, C. A. G. (1959), 'The 2-test for Small Expectations in Contingency Tables, with Special Feference to Accidents and Absenteeism', Biometrika 46, 365-385. [ Links ]
19. Percy, D. F. (2006), Bayesian Methods for Testing the Randomness of Lottery Draws, University of Salford, Centre for Operational Research and Applied Statistics. Research report. [ Links ]
20. Quesenberry, C. P. & Hurst, D. C. (1964), 'Large-Sample Simultaneous Confidence Intervals for Multinomial Proportions', Technometrics 6, 191-195. [ Links ]
21. Royal Statistical Society, (2000), Reports on the Randomness of U. K. Lottery Games. Obtenidos en septiembre de 2007. *http://www.natlotcomm.gov.uk [ Links ]
22. Royal Statistical Society, (2002), Reports on the Randomness of U. K. Lottery Games. Obtenidos en septiembre de 2007. *http://www.natlotcomm.gov.uk [ Links ]
23. Schwartz, D. G. (2003), Suburban Xanadu: The Casino Resort on the Las Vegas Strip and Beyond, Routledge, New York. [ Links ]
24. Schwartz, D. G. (2006), Roll the Bones: The History of Gambling, Gotham Books, New York. [ Links ]
25. Sison, C. P. & Glaz, J. (1995), 'Simultaneous Confidence Intervals and Sample size Determination for Multinomial Proportions', Journal of the American Statistical Association 90(429), 366-369. [ Links ]
26. Stern, H. & Cover, T. M. (1989), 'Maximum Entropy and the Lottery', Journal of the American Statistical Association 84, 980-985. [ Links ]
27. Teo, C. P. & Leong, S. M. (2002), 'Managing Risk in a Four-Digit Number Game', SIAM REVIEW 44(4), 601-615. [ Links ]
28. University of Salford, (2004), Randomness of the Lotto Draws: Summary of Findings, Centre for the Study of Gambling. Obtenido en septiembre de 2007. *http://www.natlotcomm.gov.uk [ Links ]
29. University of Salford, (2005a), Randomness of the Lotto Lucky Dip, Centre for the Study of Gambling. Obtenido en septiembre de 2007. *http://www.natlotcomm.gov.uk [ Links ]
30. University of Salford, (2005b), Randomness of thunderball draws, Centre for the Study of Gambling. Obtenido en septiembre de 2007. *http://www.natlotcomm.gov.uk [ Links ]
31. Yang, Z., Xie, M., Kuralmani, V. & Tsui, K. L. (2002), 'On the Performance of Geometric Charts with Estimated Control Limits', Journal of Quality Technology 34(4), 448-459. [ Links ]
Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:
@ARTICLE{RCEv33n2a01,
AUTHOR = {Gutiérrez-Pulido, Humberto and García, Juan Manuel},
TITLE = {{Verificación y monitoreo de la aleatoriedad de los juegos de números de d dígitos}},
JOURNAL = {Revista Colombiana de Estadística},
YEAR = {2010},
volume = {33},
number = {2},
pages = {167-190} ]]>
}