function $F$ is the weighted sum of squared residuals, $$\begin{aligned}
    \label{eq:wls}
        F(\boldsymbol{\theta}) =
            \left[\mathbf{s}_n - \boldsymbol{\sigma}(\theta)\right]^T
            \mathbf{V}
            \left[\mathbf{s}_n - \boldsymbol{\sigma}(\theta)\right],\end{aligned}$$ and $\mathbf{s}_n = \text{vech}(\mathbf{S}_n)$ is the half-vectorized observed covariance matrix obtained from an i.i.d. sample of size $n$, with $\boldsymbol{\sigma}(\boldsymbol{\theta}) = \text{vech}[\boldsymbol{\Sigma}(\boldsymbol{\theta})]$, of length $p^*$. Throughout, we will assume that both the observed and the population covariance matrices are positive-definite. (This assumption is violated, for example, in the case of high-dimensional data with $q>n$.)

Setting $$\mathbf{V} := 2^{-1} \mathbf{D}^T (\hat{\boldsymbol{\Sigma}}^{-1} \otimes \hat{\boldsymbol{\Sigma}}^{-1}) \mathbf{D},$$ where $\mathbf{D}$ is the “duplication matrix”, yields normal-theory maximum likelihood estimation [@neudecker_linear_1991]. Here $\hat{\boldsymbol{\Sigma}}$ is a consistent estimate of the population covariance matrix $\text{Var}(\mathbf{y})$. Some procedures set $\hat{\boldsymbol{\Sigma}} := \boldsymbol{\Sigma}(\hat{\boldsymbol{\theta}})$ iteratively during estimation, while others use $\hat{\boldsymbol{\Sigma}} := \mathbf{S}_n$. More generally, we will assume $\mathbf{V}$ to be any positive-definite matrix of estimation weights.

Estimation, nonconvergence, and “inadmissible” estimates
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Sample parameter estimates $\hat{\boldsymbol{\theta}}_n$ are generally found through an optimization procedure with objective given in Equation \[eq:objective\]. The gradient of $F$ with respect to the parameters plays a key role in convergence of any such procedure. For example, in gradient descent optimization, the updating step at iteration $t+1$ is $$\hat{\boldsymbol{\theta}}_{t+1} \leftarrow \hat{\boldsymbol{\theta}}_t - \boldsymbol{A}_t \cdot \boldsymbol{g}(\hat{\boldsymbol{\theta}}_{t}),$$ where $\boldsymbol{g}(\hat{\boldsymbol{\theta}}_{t})$ 