Patil, Prakash N.
Woody, Jonathan R.
Date of Degree
Dissertation - Open Access
Doctor of Philosophy
College of Arts and Sciences
Department of Mathematics and Statistics
The nonparametric confidence interval for an unknown function is quite a useful tool in statistical inferential procedures; and thus, there exists a wide body of literature on the topic. The primary issues are the smoothing parameter selection using an appropriate criterion and then the coverage probability and length of the associated confidence interval. Here our focus is on the interval length in general and, in particular, on the variability in the lengths of nonparametric intervals for probability density and hazard rate functions. We start with the analysis of a nonparametric confidence interval for a probability density function noting that the confidence interval length is directly proportional to the square root of a density function. That is variability of the length of the confidence interval is driven by the variance of the estimator used to estimate the square-root of the density function. Therefore we propose and use a kernel-based constant variance estimator of the square-root of a density function. The performance of confidence intervals so obtained is studied through simulations. The methodology is then extended to nonparametric confidence intervals for the hazard rate function. Changing direction somewhat, the second part of this thesis presents a statistical study of daily snow trends in the United States and Canada from 1960-2009. A storage model balance equation with periodic features is used to describe the daily snow depth process. Changepoint (inhomogeneities features) are permitted in the model in the form of mean level shifts. The results show that snow depths are mostly declining in the United States. In contrast, snow depths seem to be increasing in Canada, especially in north-western areas of the country. On the whole, more grids are estimated to have an increasing snow trend than a decreasing trend. The changepoint component in the model serves to lessen the overall magnitude of the trends in most locations.
Xu, Yang, "On Non-Parametric Confidence Intervals for Density and Hazard Rate Functions & Trends in Daily Snow Depths in the United States and Canada" (2016). Theses and Dissertations MSU. 3361.