On improved confidence intervals for parameters of discrete distributions
Qiansheng Zhu, Matthew A. Bognar, Joseph E. Cavanaugh, Jian Huang, Aixin Tan, Joseph B. Lang · 2020
Constructing confidence intervals for parameters in discrete distributions is an important problem in statistics. This dissertation focuses on improving confidence intervals in the following two settings. In the first setting, we are interested in constructing test-inversion approximate confidence intervals for estimands in contingency tables subject to equality constraints. Recommended test statistics include the difference in G2 statistic and nested versions of a family of power-divergence statistics. By imposing constraints, interval lengths are expected to decrease, although some bias might be introduced. Efficient and robust computational algorithms are proposed. The computational approach herein broadens the applicability in both estimands and constraints: (i) Compared with existing methods where only likelihood explicit estimands are applicable, our algorithms can also handle likelihood implicit estimands, where the log-likelihood cannot be reparameterized in terms of the estimand of interest and a collection of nuisance parameters; (ii) Only mild conditions on equality constraints are required, and it is unnecessary to re-express the constraints as a generalized linear model. A large-scale simulation study highlights the advantages of using likelihood-ratio intervals rather than bootstrap and Wald intervals, especially when table counts are small and/or the true estimand is close to the boundary. In addition, appropriate loss functions are proposed to investigate efficiency gain upon imposing constraints. Examples are presented to illustrate the appropriateness of imposing constraints and the utility of test-inversion intervals. In the second setting, suppose one observation is believed to be a realization of a discrete random variable with a distribution indexed by an unknown parameter, and our goal is to construct a confidence interval for this parameter. One example is that we have a single observation that comes from a Poisson variate with unknown mean. Another example would be a realization of a negative-binomial random variable with the desired number of successes known, but with the success rate in each independent Bernoulli trial unknown. We study properties and performance of various exact procedures, and propose new approximate procedures, with investigations on coverage and length. Generally, as exact procedures are conservative while approximate ones usually suffer from undercoverage issues, Mean-Minimum (MM) confidence procedures provide a compromise between the two. For an MM procedure, the minimum coverage probability is allowed to fall a little bit down the nominal level so that a narrower interval can be obtained, while the "mean" coverage probability is close to the nominal level. Because of the possible unboundedness of the parameter space, we propose a data-driven weighting criterion for assessing the "mean" coverage probability, and we construct MM intervals in both the Poisson and the negative-binomial settings. Graphical and numerical summaries of coverage and length reveal that MM intervals have better coverage properties than currently available approximate intervals, and they are narrower than exact intervals. Finally, we make recommendations on gold-standard procedures in various scenarios.