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# Bias-corrected estimator and confidence intervals based on the Monte Carlo method

Hakoyama, H. and Iwasa, Y.

To estimate parameters without bias and to determine confidence intervals of the parameter in general cases are difficult. Even the maximum likelihood (ML) estimators may have significant bias if the sample-size is small. In this paper, we develop a simple method of removing large bias in parameter estimation and calculating confidence intervals based on the Monte Carlo sampling. The method is applicable to any parameteric models for which (1) the computer simulation can be performed and (2) a biased (but correctable) estimator theta(hat)(bias) can be constructed. Let E[theta(hat)|theta] be the expected value of the estimator theta(hat)(s(*)) calculated for data s(*) that is generated by the model with parameter theta. For a given estimator with a large bias, theta(hat)(bias), we can calculate bias-corrected estimator, theta(hat)(bc), which satisfies the following relationship; E[theta(hat)(bias)(s(*))|theta(hat)(bc)(s)]=theta(hat)(bias)(s), where s is the observed data. We can find the bias corrected estimate theta(hat)(bc)(s) and its confidence intervals by trial and error, using the Monte Carlo sampling repeatedly. We can prove that theta(hat)(bc) is the unbiased estimator if theta and theta(hat)(bias) are linearly related. To illustrate the use of this method, we apply it to a stochastic differential equation model for a logistically growing population with environmental and demographic stochasticities. An approximate maximum likelihood (AML) estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity sigma(e)(2)) has a significant bias, especially if the time series data of population size is short. However we can remove the bias very effectively by the Monte Carlo sampling.

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