Nonparametric nonstationary density estimation including upper control limit methods for detecting change points

Nonstationary nonparametric densities occur naturally including applications such as monitoring the amount of toxins in the air and in monitoring internet streaming data. Progress has been made in estimating these densities, but there is little current work on monitoring them for changes. A new statistic is proposed which effectively monitors these nonstationary nonparametric densities through the use of transformed wavelet coefficients of the quantiles. This method is completely nonparametric, designed for no particular distributional assumptions; thus making it effective in a variety of conditions.;Existing methods for monitoring sequential data typically focus on using a single value upper control limit (UCL) based on a specified in control average run length (ARL) to detect changes in these nonstationary statistics. However, such a UCL is not designed to take into consideration the false alarm rate, the power associated with the test or the underlying distribution of the ARL. Additionally, if the monitoring statistic is known to be monotonic over time (which is typical in methods using maxima in their statistics, for example) the flat UCL does not adjust to this property. We propose several methods for creating UCLs that provide improved power and simultaneously adjust the false alarm rate to user-specified values. Our methods are constructive in nature, making no use of assumed distribution properties of the underlying monitoring statistic. We evaluate the different proposed UCLs through simulations to illustrate the improvements over current UCLs. The proposed method is evaluated with respect to profile monitoring scenarios and the proposed density statistic. The method is applicable for monitoring any monotonically nondecreasing nonstationary statistics.