Estimation of the binominal distribution parameters using the method of moments and its asymptotic properties

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.