# Réponse finale obtenue du bot:
response = « Here’s a proposed conjecture:
**Conjecture:** « The maximum likelihood estimator of a stochastic process is asymptotically optimal in the sense that it achieves the minimum mean squared error among all possible estimators, regardless of the underlying distribution. »
This conjecture combines elements from descriptive statistics (estimation), inferential statistics (hypothesis testing and confidence intervals), and stochastic models (stochastic processes). It proposes a fundamental property of maximum likelihood estimation, which is widely used in statistical theory.
The conjecture can be formalized as follows:
Let $(X_n)_{n\geq 1}$ be a sequence of i.i.d. random variables with an underlying distribution $P$. Let $\theta$ be a parameter of interest, and let $\hat{\theta}_n$ be the maximum likelihood estimator of $\theta$ based on the first $n$ observations.
Conjecture: For any other estimator $\tilde{\theta}_n$ (e.g., Bayes estimator, method-of-moments estimator), we have:
$$\lim_{n\to\infty} E[ (\hat{\theta}_n – \theta)^2 ] \leq \lim_{n\to\infty} E[ (\tilde{\theta}_n – \theta)^2 ]$$
This conjecture has far-reaching implications in statistical theory, as it would imply that maximum likelihood estimation is asymptotically optimal for a wide range of stochastic models.
What do you think? Is this conjecture worth exploring further? »