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Mathematical Statistics (Definition)

The application of probability theory to statistics, dealing with data analysis and inference.
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The statement of the theorem

Let (Ω,F,P)(\Omega, \mathcal{F}, P) be a complete probability space. Consider a statistical model defined by a parameter θ∈Θ⊂Rk\theta \in \Theta \subset \mathbb{R}^k, where Θ\Theta is the parameter space. We assume that an observed sample X=(X1,…,Xn)X = (X_1, \dots, X_n) is drawn independently and identically distributed (i.i.d.) according to a probability measure PθP_{\theta} parameterized by θ\theta. The core of Mathematical Statistics is the rigorous development of inference procedures. Specifically, given the likelihood function L(θ∣X)=∏i=1nf(Xi∣θ)L(\theta | X) = \prod_{i=1}^{n} f(X_i | \theta), the field provides the theoretical foundation for: \n\n1. **Estimation:** Constructing an estimator θ^:X→Θ\hat{\theta}: \mathcal{X} \to \Theta that minimizes the expected risk R(θ,θ^)=E[L(θ,θ^)]R(\theta, \hat{\theta}) = E[L(\theta, \hat{\theta})]. For instance, the Maximum Likelihood Estimator (MLE) θ^MLE\hat{\theta}_{MLE} is defined by maximizing the log-likelihood function: \nθ^MLE=arg⁡max⁡θ∈Θlog⁡L(θ∣X)=arg⁡max⁡θ∈Θ∑i=1nlog⁡f(Xi∣θ)\hat{\theta}_{MLE} = \arg \max_{\theta \in \Theta} \log L(\theta | X) = \arg \max_{\theta \in \Theta} \sum_{i=1}^{n} \log f(X_i | \theta) \n\n2. **Hypothesis Testing:** Formulating a test statistic T(X)T(X) and a rejection region R\mathcal{R} such that the p-value p=P(T(X)≥tobs∣H0)p = P(T(X) \ge t_{obs} | H_0) allows for a decision regarding the null hypothesis H0:θ∈Θ0H_0: \theta \in \Theta_0. This involves establishing asymptotic distributions (e.g., n(θ^−θ0)→dN(0,I−1(θ0))\sqrt{n}(\hat{\theta} - \theta_0) \xrightarrow{d} \mathcal{N}(0, I^{-1}(\theta_0))) and controlling the Type I error rate α\alpha.