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8 Estimation of parameters (The Method of Maximum Likelihood (Confidence…

- - - - \[l(\theta)=\sum_{i=1}^{n}\log[f(X_{i}|\theta)]\]
  - - - \[\hat{\theta}\pm z_{\alpha/2}/\sqrt{nI(\hat{\theta})}\]
    - - och antag detta känt: \(\Delta=\hat{\theta}-\theta_{0}\)
      - Låt \(\theta_0\) vara det sanna parametervärdet
      - \(\underline{\delta}=\alpha/2\,\textrm{kvantilen}\)
        \(\overline{\delta}=1-\alpha/2\,\textrm{kvantilen}\)
        av \(\Delta\)
      - \(P(\underline{\delta}\leq\hat{\theta}-\theta_{0}\leq\overline{\delta})=1-\alpha\)
        \(\Downarrow\)
        \(P(\hat{\theta}-\overline{\delta}\leq\theta_{0}\leq\hat{\theta}-\underline{\delta})=1-\alpha\)
      - Detta förutsatte att \(\Delta=\hat{\theta}-\theta_{0}\) var känt, vilket typiskt inte är fallet.
        Om \(\theta_0\) var känd skulle denna fördelning kunna approximeras med simulering: Många många samples av observationer kunde slumpmässigt genereras med det sanna värdet av \(\theta_0\); för varje sample kunde \(\Delta=\hat{\theta}-\theta_{0}\) räknas ut.
        Eftersom \(\theta_0\) inte är känd föreslår bootstrap-principen att vi använder \(\hat{\theta}\) i dess plats:
      - Generate many, many samples (say, B in all) from a distribution with value \(\hat{\theta}\); and for each sample construct an estimate of \(\theta\), say \(\theta_j^*\), j = 1, 2, ..., B. The distribution of \(\Delta=\hat{\theta}-\theta_{0}\) is then approximated by that of \(\theta^*-\hat{\theta}\), the quantiles of which are used to form an approximate confidence interval.
- - - - Let \(X_1, ..., X_n\) be i.i.d. with density function \(f(x|\theta)\).
        Let \(T=t(X_1, ..., X_n)\) be an unbiased estimate of \(\theta\). Then, under smoothness assumptions on \(f(x|\theta)\), \[\textrm{Var}(T)\geq\frac{1}{nI(\theta)}\]
- - - - In other words, given the value of \(T\), which is called a sufficient statistic, we can gain no more knowledge about \(\theta\) from knowing more about the probability distribution of \(X_1, ..., X_n\)
    - - \[f(x_{1},\ldots,x_{n}|\theta)={\color{OrangeRed}{g[T(x_{1},\ldots,x_{n}),\theta]}}{\color{DodgerBlue}{h(x_{1},\ldots,x_{n})}}\]
      - \(\)
    - - \(f(x|\theta)\).