Pseudo-true parameter under misspecification

Let $\mathcal{X}$ be a set, let $\mathcal{A}$ be a sigma-algebra on it, let $\mathrm{\Theta }\subseteq {\mathbb{R}}^d$ be nonempty, and let $\mu$ be a fixed sigma-finite measure on $(\mathcal{X},\mathcal{A})$. Suppose $(p_{\theta }{)}_{\theta \in \mathrm{\Theta }}$ is a specified family of nonnegative measurable densities with $\int p_{\theta }d\mu =1$. Let $P$ be the actual data probability measure; it need not equal any model law $p_{\theta }\mu$ (Probability measures and probability spaces). Assume $p_{\theta }(X)>0$ $P$-almost surely and ${\mathrm{\ell }}_{\theta }(X):=\log p_{\theta }(X)$ is integrable under $P$ for every parameter considered. Expectation means integration against $P$ (Expectation of a nonnegative or integrable random variable).

The population log-likelihood criterion is $M_P(\theta ):={\mathbb{E}}_P[{\mathrm{\ell }}_{\theta }(X)]$. A pseudo-true parameter is any maximizer ${\theta }^{\ast }\in {\mathrm{argmax}}_{\theta \in \mathrm{\Theta }}{M}_P(\theta )$. It is unique when this argmax set is the singleton $\{{\theta }^{\ast }\}$. The model is misspecified relative to $P$ when $P\mathrm{\ne }{p}_{\theta }\mu$ for every $\theta \in \mathrm{\Theta }$. This definition does not assert existence, uniqueness, interiority, or separation of the maximizer; those are theorem hypotheses when needed.