properties.

Large Sample Properties of Parameter Estimates {#LSP}
==============================================

Because unconditional MLEs ($\widehat\theta$) retain more information and have smaller MSEs than conditional MLEs ($\widehat\theta^c$), henceforth only unconditional MLEs are considered and they are now referred to simply as MLEs. To describe local asymptotic characteristics, the approach considered for sample size recalculation in @Tarima2019 is adopted. Let $f(x|\theta,\eta)$ depend on a parameter of interest, $\theta$, and a nuisance parameter, $\eta$. The objective is to test a null $H_0:\theta=0$ versus local alternatives $H_{Ak}: \theta = h_{k}/\sqrt{n_1}$. The $k$th stage-specific estimates $(\widehat\theta_k,\widehat\eta_k)$ of $(\theta,\eta)$ and their statistical models are called *regular* if, without the possibility of early stopping, $$\label{ctlholds}
\sqrt{n_k}\left(\begin{array}{c}
    \widehat\theta_k - \theta \\
    \widehat\eta_k - \eta
\end{array}
\right) \overset{d}{\to} N\left[
\left(\begin{array}{c}
    0\\
    0
\end{array}
\right),
\left(\begin{array}{cc}
    \sigma_{\theta\theta} & \sigma_{\theta\eta}\\
    \sigma_{\theta\eta} &  \sigma_{\eta\eta}
\end{array}
\right)
\right],$$ where the limiting covariance matrix is positive definite with finite elements. Then, $$\label{proMLEdist}
\xi_k:= \sqrt{n_k}\left(\widehat\theta_k - \theta\right) \sigma_{\theta}^{-1} \overset{d}{\to} N\left(0, 1 \right), \qquad k=1,\ldots,K,$$ where $\sigma_{\theta}^2 := \sigma_{\theta\theta}$. Assumption was described in @Tarima2019 to encompass specific assumptions for

-   independent, identically distributed observations by Cramér \[e.g., for example, @Ferguson1996\],

-   independent not identically distributed observations \[e.g., @philippou1973asymptotic\],

-   dependent observations \[e.g., @crowder1976maximum\],

-   and densities whose support 