Estimating population size with truncated sampling
Titel:
Estimating population size with truncated sampling
Auteur:
Blumenthal, Saul
Verschenen in:
Communications in statistics
Paginering:
Jaargang 6 (1977) nr. 4 pagina's 297-308
Jaar:
1977
Inhoud:
Let X1, X2,…,Xn be independent, indentically distributed random variables with density f(x,θ) with respect to a σ-finite measure μ. Let R be a measurable set in the sample space X. The value of X is observable if X ε (X-R) and not otherwise. The number J of observable X's is binomial, N, Q, Q = 1-P(X ε R). On the basis of J observations, it is desired to estimate N and θ. Estimators considered are conditional and unconditional maximum likelihood and modified maximum likelihood using a prior weight function to modify the likelihood before maximizing. Asymptotic expansions are developed for the Nˆ's of the form Nˆ = N + α√N + β + op(1), where α and β are random variables. All estimators have the same α, which has mean 0, variance σ2 (a function of θ) and is asymptotically normal. Hence all are asymptotically equivalent by the usual limit distributional theory. The β's differ and Eβ can be considered an “asymptotic bias”. Formulas are developed to compare the asymptotic biases of the various estimators. For a scale parameter family of absolutely continuous distributions with X = (0,∞) and R = (T,∞), special formuli are developed and a best estimator is found.
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