Three essays on testing hypotheses with irregular conditions

This dissertation consists of three chapters coping with nonstandard models with one or multiple following properties under the null hypothesis: (i) when models are not identified, (ii) when models are not differentiable, and (iii) when the maximum expected value of models is attained on the boundary of the parameter space. The main goal of the dissertation is in investigating how to implement hypothesis tests under these circumstances, and get the asymptotic critical values of relevant test statistics.; Chapter I considers regime switching models, and investigates how to test the number of regimes. The null model turns out to have the identification problem along with the boundary parameter problem. Under this circumstance, the log-likelihood (LR) statistic converges to the functional of a Gaussian stochastic process, and tail lower bounds for the asymptotic distribution of the LR statistic are suggested as substitutes for the asymptotic critical values.; Chapter II further extends the regime switching testing framework of Chapter I and suggests how to estimate the most parsimonious regime switching model by a sequential testing procedure. In attaining this goal, bootstrap techniques for stochastic processes are adopted as a way to get the asymptotic critical value for the LR statistic.; Chapter III considers models that are only partially differentiable and have the boundary parameter problem, and shows that slightly modified standard Wald, Lagrange Multiplier, and LR statistics converge to the functionals of Gaussian stochastic processes for which critical values can be conveniently approximated.