Estimation and prediction for large models with saturated output observation and general input condition

This paper considers the estimation and prediction problems for large models with saturated output observations. Here large models are referred to models with a large or infinite number of unknown parameters. The investigation of such models appears to be necessary even for finite dimensional linear stochastic systems when the output observations are saturated or binary-valued, since the regressors used in the traditional parameter estimation algorithms are not available due to partial observations of the output signals. We will propose a two-step projected recursive estimation algorithm and analyze the global convergence and the convergence rate under quite weak excitation conditions on the input signals, which do not exclude strongly correlated feedback signals. Moreover, the accuracy of prediction is also established by analyzing the asymptotic upper bound of the accumulated regret without resorting to any excitation conditions. This paper can be regarded as an extension of the recent results established for finite dimensional stochastic regression models with saturated output observations, but the new results can also be used to solve finite dimensional estimation problems that can hardly be solved without using large models. One of the key techniques used in the theoretical analysis for large models in the current paper is the theory of double array martingales developed by one of the authors. A case study based on judicial empirical data is also provided.