Maximum-likelihood-based confidence regions and hypothesis tests for selected statistical models.
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Riggs, Kent Edward.
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We apply maximum likelihood methods for statistical inference on parameters of interest for three different types of statistical models. The models are a seemingly unrelated regression model, a bivariate Poisson regression model, and a model of two inversely-related Poisson rate parameters with misclassified data. The seemingly unrelated regression (SUR) model promotes more efficient estimation as opposed to an ordinary least squares approach (OLS). However, the exact distribution of the SUR estimator is complex and does not yield easily-formed confidence regions of a coefficient parameter. Therefore, one can apply maximum likelihood (ML) asymptotic-based methods to construct a confidence region. Here, we invert a Wald statistic with a size-corrected critical value to construct a confidence region for a coefficient parameter vector. We compare the coverage and ellipsoidal volume properties of three SUR ML-related confidence regions and two SUR two-stage confidence region methods in a Monte Carlo simulation study, and demonstrated how they yield smaller confidence regions as compared to the OLS approach. The bivariate Poisson regression model allows for joint estimation of two Poisson counts as a function of possibly-different covariates. We derive the Wald, score, and likelihood ratio test statistics for testing a single coefficient parameter vector. Also, we derive a Wald statistic for testing the equality of two coefficient parameter vectors. In addition, we derive a confidence interval for the covariance parameter. We then study the size and power of the test statistics and the coverage properties of the confidence interval in a Monte Carlo simulation. Lastly, we consider confidence interval estimation for parameters in a Poisson count model where the rate parameters are inversely related and the data is subject to misclassification. We derive confidence intervals for a Poisson rate parameter by inverting the appropriate Wald, score, and likelihood ratio statistics. We also derive confidence intervals for a misclassification parameter by inverting the appropriate Wald, score, and likelihood ratio statistics. Another interesting parameter is the difference of the two complementary Poisson rate parameters, for which we derive a Wald-type confidence interval. We then examine the coverage and length properties of these confidence intervals via a Monte Carlo simulation study.