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

• Published in: Technometrics 1972-11, Vol.14 (4), p.841-846
• Main Author: Cohen, A. Clifford
• Format: Article
• Language: English
• Subjects: Biometrics; Data sampling; Estimation; Maximum likelihood estimators; Parameters; Poisson; Poisson process; Probability; Samples; Statistical variance; Surface areas; Truncated; Truncation
• ISSN: 0040-1706; 1537-2723
• DOI: 10.1080/00401706.1972.10488980

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 {x i , t i , c i ), i = 1, 2, n, with x i designating the number of occurrences, of interest during the ith interval of observation (or in the ith sample unit) of fixed, magnitude t i , subject to the restriction x i ≥ c i ≥ 0. Each x i is associated with its own, pair of constants t i and c i which are fixed prior to sampling. Accordingly, x i represents, an observation from a Poisson distribution with parameter λt i that is complete when, c i = 0, but which is truncated on the left at c i when c i ≥ 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.