Summary
The class (1) of transformations of a binomial variablek is studied. The famous De Moivre-Laplace local and integral limit theorems are generalized for the asymptotically standardized transformations (2). The transformation (4) respectively (3) is singled out as the only one with an error of order 1/n. Besides it is shown, that the error term of ordern −1/2 of the identical transformation, the angular transformation, and the logarithmic transformation are in proportion to 1/6, −1/12, and −1/3. Results on the influence of a correction of unbiasedness are mentioned in the last section. Such corrections allow a slight improvement of our new transformation.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Literatur
Anscombe, F. J.: Discussion of the analysis of variance with various binomial transformations by R. Fisher. Biometrics10, 141–144 (1954).
Bernstein, S.: Retour au problème de l'évaluation de la formule limite de Laplace [Russisch]. Izvestija Akad. Nauk SSSR, Ser. mat.7, 3–16 (1943).
Blom, G.: Transformations of the binomial, negative binomial, Poisson and x2-distributions. Biometrika41, 302–316 (1954);43, 235 (1956).
Bol'shev, L. N.: Asymptotically Pearson's transformations [Russisch]. Teor. Verojatn. Primen.8, 129–155(1963).
Curtiss, J. H.: On transformations used in the analysis of variance. Ann. math. Statistics14, 107–122 (1943).
Dyke, G.V., Patterson, H.D.: Analysis of factorial arrangements when the data are proportions. Biometrics8, 1–12 (1952).
Feller, W.: On the normal approximation to the binomial distribution. Ann. math. Statistics16, 319–329 (1945).
—: An introduction to probability theory and its applications. Vol. I, 2. Aufl. New York: Wiley 1957.
Fisher, R. A.: On the dominance ratio. Proc. roy. Soc. Edinburgh42, 321–341 (1921–1922).
—: The distribution of generatios for rare mutations. Proc. roy. Soc. Edinburgh, Sect. A50, 205–220 (1930).
Gebhardt, F.: Some numerical comparisons of several approximations to the binomial distribution. To appear in: J. Amer. statist. Assoc. December 1969.
Govindarajulu, Zakkula: Normal approximation to the classical discrete distributions. Sankhyā, Ser. A27, 143–167 (1965). (Abgedruckt aus: Classical and contagious distributions. Proc. International Symp. McGill Univ., Montreal, Canada, August 1963, ed. by Ganapati P. Patil.)
Kalinin, V. M.: A theorem on summation and its application to special functions [Russisch]. Doklady Akad. Nauk. SSSR175, 1004–1007 (1967).
—: Convergent and asymptotic expansions for probability distributions [Russisch, englische Zusammenfassung]. Teor. Verojatn. Primen.12, 24–38 (1967 a).
Meshalkin, L. D.: On the approximation of polynomial distributions by infinitely divisible laws [Russisch], Teor. Verojatn. Primen.5, 114–124 (1960) und englische übersetzung in: Theor. Probab. Appl.5, 106–114 (1960).
Molenaar, W.: How to poison Poisson (when approximating binomial tails). Statistica Neerlandica23, 19–40 (1969).
Peizer, D. B., Pratt, J. W.: A normal approximation for binomial,F, Beta, and other common, related tail probabilities, I. J. Amer. statist. Assoc.63, 1416–1456 (1968).
Pratt, J. W.: A normal approximation for binomial,F, beta, and other common related tail probabilities, II. J. Amer. statist. Assoc.63, 1457–1483 (1968).
Prohorov, Yu. V.: Asymptotic behavior of the binomial distribution [Russisch]. Uspehi mat. Nauk8, 135–142 (1953).
Raff, M. S.: On approximating the point binomial. J. Amer. statist. Assoc.51, 293–303 (1956).
Rowe, Ch. H.: A proof of the asymptotic series for logγ (z) and logγ (z+ a). Ann. of Math., II. Ser.32, 10–16 (1931).
Stange, K.: Die zeichnerische Ermittlung von Plänen für Gut-Schlecht-Prüfung in einem geeigneten Funktionsnetz. Qualitätskontrolle8, 85–91 (1963).
Tsaregradskii, I. P.: On the uniform approximation of the binomial distribution [Russisch]. Teor. Verojatn. Primen.3, 470–474 (1958).
Whittaker, E. T., Watson, G. N.: A course of modern analysis. 4. Aufl. Cambridge Univ. Press 1927 (Nachdruck 1952).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borges, R. Eine Approximation der Binomialverteilung durch die Normalverteilung der Ordnung 1/n . Z. Wahrscheinlichkeitstheorie verw Gebiete 14, 189–199 (1970). https://doi.org/10.1007/BF01111416
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01111416