## 3 Estimation Procedure Based on StEM
### The Estimation Procedure  
The EM algorithm introduced by [17] is a classical approach to estimate parameters of models with non-observed or incomplete data. Let us briefly cover the principle. Denote \(\mathbf{z}\) as the vector of non-observed data, \((\mathbf{y},\mathbf{z})\) the complete data, and \(L_{e}(\mathbf{y},\mathbf{z};\mathbf{\theta})\) the log-likelihood of the complete data; the EM algorithm maximizes the \(Q(\mathbf{\theta}|\mathbf{\theta}^{\prime})=E(L_{e}(\mathbf{y},\mathbf{z};\mathbf{\theta})|\mathbf{y} ;\mathbf{\theta}^{\prime})\) function in two steps. At the \(k^{th}\) iteration, the E-step is the evaluation of \(Q^{(k)}(\mathbf{\theta})=Q(\mathbf{\theta}|\mathbf{\theta}^{(k-1)})\), where the M-step updates \(\mathbf{\theta}^{(k-1)}\) by maximizing \(Q^{(k)}(\mathbf{\theta})\). For cases where the E-step has no analytic form, ref. [18] introduces the MCEM algorithm, which calculates the conditional expectations at the E-step via many simulations within each iteration and hence is quite computationally intensive. The choice of replicate size is the central issue in guaranteeing convergence. Ref. [10] introduces a stochastic version of the EM algorithm, namely the StEM, which replaces the E-step with a single imputation of the complete data and then averages the last batch of \(M\) estimates in the Markov Chain iterative sequence to obtain the point estimate of the parameters. The imputed data \(\mathbf{z}^{(k)}\) at the \(k^{th}\) iteration are a random draw from the conditional distribution of the missing data given the observed data and the estimated parameter values at the \((k-1)^{th}\) iteration, \(f(\mathbf{z}^{(k)}|\mathbf{y},\boldsymbol{\theta}^{(k-1)})\). As \(\mathbf{z}^{(k)}\) only depends on \(\mathbf{z}^{(k-1)}\), \(\{\mathbf{z}^{(k)}\}_{k\geq 1}\) is a Markov chain. Assuming that \(\mathbf{z}^{(k)}\) take values in a compact space and the kernel of the Markov chain is positive continuous for a Lebesgue measure, the Markov chain is ergodic, and that ensures the existence of a unique stationary distribution [19; 20]. In extending the StEM algorithm for the ZIE-based nonlinear random change point model, the imputation step is a crucial part of the process. At the \(k^{th}\) iteration, we aim to draw the missing data \((\mathbf{y}_{cen,i}^{(k)},\mathbf{a}_{i}^{(k)},\mathbf{b}_{i}^{(k)},\tau_{i}^{ (k)})\), where direct sampling from the joint conditional distribution is often intractable. To address this, we employ a Metropolis-within-Gibbs sampler, wherein each component is updated conditionally. For variables with tractable full conditional distributions (e.g., censored outcomes), we use standard Gibbs updates. For components lacking closed-form conditionals, we embed Metropolis-Hastings steps within the Gibbs framework to sample from the appropriate target distributions. Unlike the EM or MCEM algorithms, this procedure does not require monotonic increases in the likelihood, but instead ensures that the Markov chain explores the parameter space in a way consistent with the joint posterior distribution [21; 22]. As an example, after initializing \(\theta^{(0)}\), and \((\mathbf{y}_{cen,i}^{(0)},\mathbf{a}_{i}^{(0)},\mathbf{b}_{i}^{(0)},\tau_{i}^ {(0)})\), we update \(\mathbf{y}_{cen,i}^{(k)}\), \(k=1,\ldots,\) as follows:  
1.