The estimation of the poisson parameter from a truncated distribution

Estimating the Parameter in a Conditional Poisson Distribution

has been considered by David and Johnson in a recent issue of this journal [1]. In particular, they derive the maximum likelihood estimate of X and its asymptotic variance, and discuss the efficiency of estimation by moments. Distribution (ii) has been studied by Mloore [2], who provides a simple estimate of X and shows the effect of truncation at different values s. The purpose of the present note is to provide a similar estimate of X for distribution (i), to show that it is highly efficient, and to estimate its sampling variance. 2. Suppose that a sample of size N from a truncated Poisson distribution of type (i) gives N, observations equal to r(r = 1, 2, 3,) To estimate an arbitrary function 6(X), use quantities of the form

The Truncated Poisson Distribution

1. Summary. A minimum variance unbiased estimator is provided for the parameter of a truncated Poisson distribution in the case of truncation on the left. In this connection the distribution is obtained for the sum of n independent identically distributed truncated Poisson random variables, and then well-known properties of sufficient statistics are employed to obtain the estimator. For the case of truncation away from the zero value results are expressed in terms of Stirling numbers of the second kind. The estimator has a particularly simple form and tables are available for its computation. For the general case results are expressed in terms of what we call generalized Stirling numbers. As a byproduct of the statistical considerations there arises an identity between generalized Stirling numbers which may be useful in the study of Difference Equations.

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Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution

A bibliography on life testing and related topics

Maximum likelihood estimators of the Poisson parameter applicable to both truncated and censored samples are derived in this paper. Singly and doubly truncated samples as well as singly and doubly censored samples are considered. The estimators obtained are presented in simple algebraic forms and their application to practical problems with the aid of standard Poisson tables is illustrated with numerical examples. Asymptotic variances of estimates for the different cases considered are obtained from second derivatives of the likelihood functions and are simplified to forms which permit ready evaluation. * Preliminary report presented before American Mathematical Society, Auburn, Alabama, November 23, 1951. published in Bulletin of the American Mathematical Society, 58 (1952), 60

The estimation of the poisson parameter from a truncated distribution

Citations

Estimating the Parameter in a Conditional Poisson Distribution

The Truncated Poisson Distribution

Minimum Variance Unbiased Estimation for the Truncated Poisson Distribution

A bibliography on life testing and related topics

Estimation of the Poisson Parameter from Truncated Samples and from Censored Samples

Related Papers (5)

The Truncated Poisson Distribution

The Truncated Poisson

Truncated Poisson Distributions

Estimation of the Poisson Parameter from Truncated Samples and from Censored Samples

Estimating the Parameter in a Conditional Poisson Distribution