Estimation in a Poisson Process Based on Combinations of Complete and Truncated Samples

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Abstract
Results presented here concern maximum likelihood estimation of a common Poisson, parameter λ in a process where available sample data consists of the independent, observations {xi , ti , ci ), i = 1, 2, n, with xi designating the number of occurrences, of interest during the ith interval of observation (or in the ith sample unit) of fixed, magnitude ti , subject to the restriction xi ≥ ci ≥ 0. Each xi is associated with its own, pair of constants ti and ci which are fixed prior to sampling. Accordingly, xi represents, an observation from a Poisson distribution with parameter λti that is complete when, ci = 0, but which is truncated on the left at ci when ci ≥ 1. Based on a sample of this, type, the likelihood estimating equation for λ is found to be Where The estimate, , is readily found by interpolating linearly between approximations obtained using trial and error procedures. Of course, the Newton-Raphson or various other standard iterative procedures are also applicable. The asymptotic variance of is obtained from the second derivative of the likelihood function. An illustrative example is included.

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