Geodesic
Estimation of the binominal distribution parameters using the method of moments and its asymptotic properties
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 2, pp. 221-230

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The problem of estimating the parameters $m$ and $p$ of the binomial distribution for a sample having the fixed volume $n$ with the help of the method of moments is considered in this paper. Using the delta method, the joint asymptotic normality of the estimates is established and the parameters of the limit distribution are calculated. The moment estimates of the parameters $m$ and $p$ do not have averages and variance. An explanation is offered for the asymptotic normality parameters in terms of characteristics of the accuracy properties of the estimates. On the basis of the data of statistical modelling, the accuracy properties of the estimates by the delta-method and their modifications which do not have initial defects of the estimates (the values of the estimates of $p$ are below zero and those of $m$ are smaller than the greatest value in the sample) are explored. An example of estimating the parameters $m$ and $p$ according to the observations of the number of responses in the experiment with nervous synapse ($m$ is the number of vesicles with acetylcholine in the vicinity of the synapse, $p$ is the probability of acetylcholine release by each vesicle) is provided.
Mots-clés : binomial distribution
Keywords: estimation of parameters, method of moments, delta method, asymptotic normality, accuracy properties of estimates. 
A. N. Safiullina. Estimation of the binominal distribution parameters using the method of moments and its asymptotic properties. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 2, pp. 221-230. http://geodesic.mathdoc.fr/item/UZKU_2016_158_2_a5/
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[1] DasGupta A., Rubin H., “Estimation of binomial parameters when both $n,p$ are unknown”, J. Stat. Plann. Inference, 130:1–2 (2005), 391–404 | DOI | MR | Zbl

[2] Nicholls J., Martin R., Wallace B., Fuchs P., From Neuron to Brain, Editorial URSS, Moscow, 2003, 672 pp. (In Russian)

[3] Haldane J. B. S., “The fitting of binomial distributions”, Ann. Eugen., 11:1 (1941), 179–181 | DOI | MR | Zbl

[4] Binet F. E., “The fitting of the positive binomial distribution when both parameters are estimated from the sample”, Ann. Eugen., 17:1 (1954), 117–119 | DOI | MR

[5] Blumenthal S., Dahiya R. C., “Estimating the binomial parameter $n$”, J. Am. Stat. Assoc., 76:376 (1981), 903–909 | MR | Zbl

[6] Olkin I., Petkau A. J., Zidek J. V., “A comparison of n estimators for the binomial distribution”, J. Am. Stat. Assoc., 76:375 (1981), 637–642 | MR | Zbl

[7] Hoel P. G., “Discriminating between binomial distribution”, Ann. Math. Stat., 18:4 (1947), 556–564 | DOI | MR | Zbl

[8] Hall P., “On the erratic behavior of estimators of $N$ in the binomial $N,p$ distribution”, J. Am. Stat. Assoc., 89:425 (1994), 344–352 | MR | Zbl

[9] Draper N., Guttman I., “Bayesian estimation of the binomial parameter”, Technometrics, 13:3 (1971), 667–673 | DOI | Zbl

[10] Raftery A. E. Inference for the binomial $N$ parameter: A hierarchical Bayes approach, Biometrika, 75:2 (1988), 223–228 | DOI | Zbl

[11] Carroll R. J., Lombard F., “A note on $N$ estimators for the binomial distribution”, J. Am. Stat. Assoc., 80:390 (1985), 423–426 | MR

[12] Kramer H., Mathematical Methods of Statistics, GIIL, Moscow, 1948, 631 pp. (In Russian)

[13] van der Vaart A. W., Asymptotic statistics, Cambridge Univ. Press, Cambridge, 1998, 443 pp. | MR | Zbl