The content of this dissertation is divided into two main topics: 1) nonlinear profilemonitoring and 2) an improved approximate distribution for the T^2 statistic based on thesuccessive differences covariance matrix estimator. (Part 1) In an increasing number of cases the quality of a product or process cannot adequately berepresented by the distribution of a univariate quality variable or the multivariate distributionof a vector of quality variables. Rather, a series of measurements are taken across some continuum,such as time or space, to create a profile. The profile determines the product quality atthat sampling period. We propose Phase I methods to analyze profiles in a baseline dataset wherethe profiles can be modeled through either a parametric nonlinear regression function or anonparametric regression function. We illustrate our methods using data from Walker and Wright(2002) and from dose-response data from DuPont Crop Protection. (Part 2) Although the T^2 statistic based on the successive differences estimator has been shown to beeffective in detecting a shift in the mean vector (Sullivan and Woodall (1996) and Vargas (2003)),the exact distribution of this statistic is unknown. An accurate upper control limit (UCL) for theT^2 chart based on this statistic depends on knowing its distribution. Two approximatedistributions have been proposed in the literature. We demonstrate the inadequacy of these twoapproximations and derive useful properties of this statistic. We give an improved approximatedistribution and recommendations for its use.