Measures of uncertainty are investigated for estimates and predictions using nonlinear mixed-effects models including state–space models in particular. These nonlinear mixed-effects models include fixed parameters and random effects. Maximum likelihood estimation of the parameters and conditional mean predictors of random effects are commonly used to estimate important quantities for a wide spectrum of applications. These quantities of interest may be functions of the parameters and random effects. In this case, software packages such as TMB and glmmTMB use a generalized delta method to provide standard errors and statistical inference. In the frequentist framework, it is clarified that these packages actually provide estimates of mean squared errors (MSE’s) based on a multivariate normal approximation of the distribution of the random effects given data. It is further shown that the MSE’s are not the variance of estimates due to repeated sampling of the data and the random effects. Equations are provided for that variance, including orders of approximations. In many cases the MSE’s will be more appropriate to use for statistical inference, but not always, and this is demonstrated for a simple random-walk state–space model example.

使用非线性混合效应模型（尤其是状态空间模型）来研究不确定性的度量，以进行估计和预测。这些非线性混合效应模型包括固定参数和随机效应。参数的最大似然估计和随机效应的条件平均预测变量通常用于估计广泛应用中的重要数量。这些感兴趣的量可以是参数和随机效应的函数。在这种情况下，诸如TMB和glmmTMB之类的软件包使用广义增量方法来提供标准错误和统计推断。在常客框架中，需要说明的是，这些软件包实际上是根据给定数据的随机效应分布的多元正态近似来提供均方误差（MSE）的估计值。进一步表明，由于数据的重复采样和随机效应，MSE不是估计值的方差。提供了针对该方差的方程式，包括近似阶数。在许多情况下，MSE将更适合用于统计推断，但并非总是如此，这在一个简单的随机游走状态空间模型示例中得到了证明。