Nonlinear measurement error analysis for system monitoring

In many physical and biological systems, underlying variables satisfy restrictions, but some or all of the variables are measured with error. The restrictions are often nonlinear in variables and may contain unknown parameters. Some of the restrictions may fail to hold at certain time points due to some system anomaly. While a good estimate of the measurement error covariance matrix is often available, systematic measurement biases may be present due to calibration or human errors. Statistical analysis of such systems is considered using a nonlinear errors-in-variables approach;The case considered first deals with a system in a stable condition, where the restrictions do not contain any unknown parameters, and measurement error biases have been corrected. For estimating the true underlying values of the variables, the maximum likelihood estimator is shown to have a sizable bias due to the nonlinearity of the restrictions. An alternative estimator is proposed modifying the restrictions for the non-linearity. Small-error asymptotic theory is used to show that the bias-adjusted estimator has smaller bias and smaller mean squared error than the maximum likelihood estimator up to a certain order of approximation. For checking the possible violation of some restrictions, the Wald, Lagrange multiplier, and likelihood ratio tests are introduced and compared based on the asymptotic theory and a simulation study;For the full-scale system which is sufficiently dynamic, statistical inferences for the unknown parameters in the restrictions and possible systematic measurement biases are investigated. If the restrictions are linear in variables, the systematic measurement biases are non-estimable. However, the measurement biases can be estimated if the restrictions are sufficiently nonlinear, even with unknown parameters. A condition for the measurement bias estimability and test procedures for the systematic measurement biases are presented. For estimating the restriction parameters and measurement biases, an estimator adjusting for the nonlinearity bias is proposed based on an idea slightly different from the one used for the simple case. Testing for the possible violation of some restriction at some time point is also discussed