## Nonparametric change-point estimation for observations following a random walk with drift

##### Abstract

This research follows a three paper format. Abstracts of each paper are:
Paper 1: Nonparametric Estimator for the Time of a Step Change in the Trend of Random Walk Models with Drift
This is the first out of three concatenated research endeavors in change-point analysis for trended observations in a time series following a random walk model with drift (RWWD) where the random variable is not restricted to normality and might be heterocedastic over time. In particular, this research proposes a nonparametric change-point estimator based on clustering techniques and the median test. When a time series follows two different RWWD schemes, the estimator is used to identify the moment in time were the drift of the characteristic function changes. It employs the p-value function of the median test to determine the two partitions that give the most statistical evidence of a difference between sets. Direct application is found in the management of living systems, a model is proposed, and a numerical example is presented to demonstrate feasibility.
Paper 2: Nonparametric Estimator for the Time of a Step Change in the Variation of Random Walk Models with Drift
This is the second out of three concatenated research endeavors on change-point analysis for trended observations in a time series following a random walk model with drift (RWWD) where the random variable is not restricted to normality and might be heterocedastic over time. Specifically, this research employs the previously developed estimator for changes in median trend to handle changes in the variation of the trend by using an artifice employed in the construction of Conover’s test for variation. When a time series follows two different schemes of RWWD models, the estimator is used to identify the moment in time when the variation of the series changes one step away from the original scheme. It converts the problem to estimate changes in variation into a problem to estimate a change in the median. It employs the p-value function of the median test to separate the time series in two partitions where there exist the most statistical evidence of a difference between sets. A direct application is found in volatility analysis of living systems such as macroeconomic or financial systems which are known to adjust to RWWD schemes in many cases. A numerical example is presented to demonstrate methodological feasibility.
Paper 3: Performance of Estimators for Change-Point in Median and Variance Based on the P-Value Function of the Median Test
This is the third out of three concatenated efforts of research in change-point analysis for trended observations in a time series following a random walk model with drift (RWWD) where the random variable is not restricted to normality and might be heterocedastic over time. This paper evaluates the performance of the two p-value-based estimators that used the maximum evidence of the median test to determine the moment a time series changes its median trend and variance. Previous research developed the corresponding procedures for the change-point estimation, however, the bias and variance of those was yet to be measured for different scenarios. To evaluate the effectiveness of the estimators, the p-value-based estimators were compared with the maximum likelihood estimators for the mean and variance of normal processes. Shift size, sample size, location of the change-point, and the underlying distribution of the observations were the factors under study.