Estimating the Parameters of Degradation Models when Error Terms are Autocorrelated

Abstract

The need of autocorrelation models for degradation data comes from the facts that the degradation measurements are often correlated, since such measurements are taken over time. Time series can exhibit autocorrelation caused by modeling error or cyclic changes in ambient conditions in the measurement errors or in degradation process itself. Generally, autocorrelation becomes stronger when the times between measurements are relativelyshort and becomes less noticeable when the times between process are longer. In this paper, we assume that the error terms are autocorrelated and have an autoregressive of order one, AR(1). This case is a more general case of the assumption that the error terms are identically and independently normally distributed. Since when the error terms are uncorrelated over the time, the estimate of the parameter of AR(1) is approximately zero.If the parameter of AR(1) is unknown, one can estimate it from the data set. Using two real data sets, the model parameters are estimated and compared with the case when the error terms are independent and identically distributed. Such computations are available by using procedures AUTOREG and model in SAS. Computations show that an AR(1) can be used as a useful tool to remove the autocorrelation between the residuals.