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Mostly common definition of Maximum Likelihood Estimator in certain parametric problem $(\mathcal{X},\mathcal{B}(\mathbb{R}^n),\mathcal{P}=\{\mathbb{P}_\theta\mid\theta\in\Theta\})$is what follows: $$ \hat\theta_{MLE}(\mathbf{x})\equiv\underset{\theta\in\Theta}{\operatorname{argmax}}L_n(\theta\mid\mathbf{x}),\quad\forall\mathbf{x}\in\mathcal{X} $$ It means that for all n-dimensional data points $\mathbf{x}$ we have unique solution, which maximize Likelihood function. Then, during the process of maximizing Likelihood is considered as non-random function of $\theta$, with data given. I am interested under what assumption we can treat the Likelihood as random in process of deriving estimator- i look for more formal, measure-theoretic approach to this problem. I think that the likelihood should be measurable function, differentiable in $\theta\quad a.s\quad\mathcal{P}$. Am I right? What else?

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    $\begingroup$ There is no need for the likelihood to be either measurable nor differentiable in $\theta.$ Insisting on that would introduce unnecessary restrictions and rule out the applicability of this theory in some significant situations. $\endgroup$
    – whuber
    Mar 13, 2020 at 16:37
  • $\begingroup$ The random function $L_n(\theta\mid\mathbf{x})$ should be measurable in $\theta$ (at the very least, to talk about existence of argmax you need more structure) $\mathcal{P}$-a.s. The argmax $\hat\theta_{MLE}$ should be a measurable function of data $\mathbf{x}$, as any estimator should be. There is no requirement that $L_n(\theta\mid\mathbf{x})$ be jointly measurable in $\theta$ and $\mathbf{x}$. $\endgroup$
    – Michael
    Mar 13, 2020 at 20:59

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Let $\left(\mathcal X, \mathcal A, \left(\mathbb P_\vartheta\right)_{\vartheta \in \Theta}\right)$ be a statistical model consisting of

  • a set $\mathcal X$ (the sample space),
  • a $\sigma$-algebra $\mathcal A$ on $\mathcal X$,
  • a family of probability measures $\left(\mathbb P_\vartheta\right)_{\vartheta \in \Theta}$ on $\mathcal A$ with
  • index set (the parameter space) $\Theta$ of cardinality bigger than one.

If, for all $\vartheta \in \Theta$, $\mathbb P_\vartheta$ is absolutely continuous w.r.t. a $\sigma$-finite measure $\mu$ on $\mathcal A$, then the likelihood function $\mathcal L$ is defined as the Radon–Nikodym derivative of $\mathbb P_\vartheta$ w.r.t. $\mu$: $$ \mathcal L(\vartheta, x) \mathrel{:=} \frac{\mathrm d \mathbb P_\vartheta}{\mathrm d \mu}(x), \; \forall \, \vartheta \in \Theta, x \in \mathcal X, $$ and often viewed as a function in $\vartheta$.

Thus, the (random) likelihood process is the random process of the corresponding Radon–Nikodym derivative.

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