UC San Diego

This thesis is concerned with transformation-based statistical methods in different three areas. For problem of nonparametric regression, the transformed-based technique is called Model-free bootstrap method; for density nonparametric estimation we propose transformed flat-top series estimator; for Bayesian hypothesis testing, we employ a parametric transformation to help in evaluating the system upgrade of glucose monitoring device. First, we establish the theory of the asymptotic validity of Model-free method on construction of confidence interval under specified assumptions. The spirit of Model-free method is transforming non-i.i.d. data to i.i.d data and then bootstrapping the new i.i.d. data. We also conduct simulations to check the finite sample properties of Model-free estimators, compared to regular normal approximation and local bootstrap method. Next, we address the problem of nonparametric estimation of a smooth univariate density on compact support. If the density function has compact support and is non-zero at either boundary, regular kernel estimator will be seriously biased. A lot of bias correction methods were proposed to improve the bias on the boundary. In this chapter we propose the transformed flat-top series estimator, which keeps the same bias order as existing methods at boundary, and improves the bias in the interior region of the support to higher order. Theoretical analysis and simulations are provided, and the results are generally better than corresponding results of many other kernel density estimator with boundary correction. At last we propose modified Bayesian hypothesis method that can be assessed by type I&II errors and an alternative assessment of standard Bayesian hypothesis testing method, particularly suited to situations where modifications are made to continuous glucose monitoring (CGM) systems already approved. A parametric transformation of observations is the key to test the validation of this system upgrade. Simulations are conducted to assess the risks and benefits of the approach, which show that by careful planning and analysis prospective study sizes can be reduced and better decisions can be made on the effectiveness and safety of the modified systems