Delattre et al. (2013) considered a system of stochastic differential equations (SDEs) in a random effects setup. Under the independent and identical (iid) situation, and assuming normal distribution of the random effects, they established weak consistency of the maximum likelihood estimators (M LEs) of the population parameters of the random effects. In this article, respecting the increasing importance and versatility of normal mixtures and their ability to approximate any standard distribution, we consider the random effects having mixture of normal distributions and prove asymptotic results associated with the MLEs in both independent and identical (iid) and independent but not identical (non-iid) situations. Besides, we consider iid and non-iid setups under the Bayesian paradigm and establish posterior consistency and asymptotic normality of the posterior distribution of the population parameters, even when the number of mixture components is unknown and treated as a random variable. Although ours is an independent work, we later noted that Delattre et al. (2016) also assumed the SDE setup with normal mixture distribution of the random effect parameters but considered only the iid case and proved only weak consistency of the M LE under an extra, strong assumption as opposed to strong consistency that we are able to prove without the extra assumption. Furthermore, they did not deal with asymptotic normality of M LE or the Bayesian asymptotics counterpart which we investigate in details. Ample simulation experiments and application to a real, stock market data set reveal the importance and usefulness of our methods even for small samples.

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关于具有混合正态分布的随机效应的随机微分方程中的经典渐近和贝叶斯渐近

德拉特等人。(2013) 考虑了随机效应设置中的随机微分方程 (SDE) 系统。在独立相同（iid）的情况下，假设随机效应呈正态分布，他们建立了随机效应总体参数的最大似然估计量（M LEs）的弱一致性。在本文中，考虑到正态混合物日益增加的重要性和多功能性及其逼近任何标准分布的能力，我们考虑了具有正态分布混合的随机效应，并证明了与独立和相同 (iid) 和独立的 MLE 相关的渐近结果但不完全相同（非 iid）的情况。除了，我们考虑贝叶斯范式下的 iid 和非 iid 设置，并建立总体参数后验分布的后验一致性和渐近正态性，即使混合成分的数量未知并被视为随机变量。虽然我们的工作是独立的，但我们后来注意到 Delattre 等人。(2016) 还假设 SDE 设置具有随机效应参数的正态混合分布，但只考虑了 iid 情况，并在额外的强假设下证明了 M LE 的弱一致性，而不是我们能够证明的强一致性额外的假设。此外，他们没有处理 M LE 的渐近正态性或我们详细研究的贝叶斯渐近对应物。大量的模拟实验和实际应用，