Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error

This article considers estimation of constant and time-varying coefficients in nonlinear ordinary differential equation (ODE) models where analytic closed-form solutions are not available. The numerical solution-based nonlinear least squares (NLS) estimator is investigated in this study. A numerical algorithm such as the Runge--Kutta method is used to approximate the ODE solution. The asymptotic properties are established for the proposed estimators considering both numerical error and measurement error. The B-spline is used to approximate the time-varying coefficients, and the corresponding asymptotic theories in this case are investigated under the framework of the sieve approach. Our results show that if the maximum step size of the $p$-order numerical algorithm goes to zero at a rate faster than $n^{-1/(p\wedge4)}$, the numerical error is negligible compared to the measurement error. This result provides a theoretical guidance in selection of the step size for numerical evaluations of ODEs. Moreover, we have shown that the numerical solution-based NLS estimator and the sieve NLS estimator are strongly consistent. The sieve estimator of constant parameters is asymptotically normal with the same asymptotic co-variance as that of the case where the true ODE solution is exactly known, while the estimator of the time-varying parameter has the optimal convergence rate under some regularity conditions. The theoretical results are also developed for the case when the step size of the ODE numerical solver does not go to zero fast enough or the numerical error is comparable to the measurement error. We illustrate our approach with both simulation studies and clinical data on HIV viral dynamics.Comment: Published in at http://dx.doi.org/10.1214/09-AOS784 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Zap Q-Learning for Optimal Stopping Time Problems

The objective in this paper is to obtain fast converging reinforcement learning algorithms to approximate solutions to the problem of discounted cost optimal stopping in an irreducible, uniformly ergodic Markov chain, evolving on a compact subset of $\mathbb{R}^n$. We build on the dynamic programming approach taken by Tsitsikilis and Van Roy, wherein they propose a Q-learning algorithm to estimate the optimal state-action value function, which then defines an optimal stopping rule. We provide insights as to why the convergence rate of this algorithm can be slow, and propose a fast-converging alternative, the "Zap-Q-learning" algorithm, designed to achieve optimal rate of convergence. For the first time, we prove the convergence of the Zap-Q-learning algorithm under the assumption of linear function approximation setting. We use ODE analysis for the proof, and the optimal asymptotic variance property of the algorithm is reflected via fast convergence in a finance example