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+基本方法就是$EX=\dfrac{1}{n}\sum\limits_{i=1}^nX_i$。 + +\subsubsection{一阶矩} + +\subsubsection{二阶矩} + +\textbf{例题:}设$X_i$为来自区间$[-a,a]$上均匀分布的总体$X$的简单随机样本,求$a$的矩估计量。 + +解:首先矩估计就是$E(X^k)=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^k$。 + +又对于均匀分布$X_i\sim U(-a,a)$,$EX=\dfrac{a+b}{2}=0$,$DX=\dfrac{(b-a)^2}{12}=\dfrac{a^2}{3}$。 + +所以$EX$不含有$a$,使用二阶矩$EX^2=DX+E^2X=\dfrac{a^2}{3}=\dfrac{1}{n}\sum\limits_{i=1}^nX_i^2$。 + +解得$a=\sqrt{\dfrac{3}{n}\sum\limits_{i=1}^nX_i^2}$。 + +\subsection{最大似然估计} + +步骤:写出概率函数或密度函数;写出似然函数(代入观测值$x_i$并连乘);两边取对数;求导数并令为0。 + +\textbf{例题:}设随机变量$X$在区间$[0,\theta]$上服从均匀分布,$X_1,X_2,\cdots,X_n$是来自$X$的简单随机样本,求$\theta$的最大似然估计量$\hat{\theta}$ + +解:$X\sim U(0,\theta)$,$f(x)=\left\{\begin{array}{ll} + \dfrac{1}{\theta}, & 0\max x_i$) + +\textbf{例题:}设$X_1,X_2,\cdots X_n$是来自总体$X$的简单随机样本,$X$的概率密度函数$f(x)=\dfrac{1}{2\lambda}e^{-\frac{\vert x\vert}{\lambda}}$,$x\in R$,$\lambda>0$,求$\lambda$的最大似然估计量$\hat{\lambda}$。 + +解:$\because f(x)=\dfrac{1}{2\lambda}e^{-\frac{\vert x\vert}{\lambda}}$,$\therefore L(\lambda)=\prod\limits_{i=1}^n\dfrac{1}{2\lambda}e^{-\frac{\vert x\vert}{\lambda}}=\left(\dfrac{1}{2\lambda}\right)^ne^{-\frac{1}{\lambda}\sum\limits_{i=1}^n\vert x_i\vert}$。 + +$\ln L(\lambda)=-n\ln2-n\ln\lambda-\dfrac{1}{\lambda}\sum\limits_{i=1}^n\vert x_i\vert$,$\dfrac{\textrm{d}\ln L(\lambda)}{\textrm{d}\lambda}=-\dfrac{n}{\lambda}+\dfrac{1}{\lambda^2}\sum\limits_{i=1}^n\vert x_i\vert$。 + +令$\dfrac{\textrm{d}\ln L(\lambda)}{\textrm{d}\lambda}=0$,则$\dfrac{n}{\lambda}=\dfrac{1}{\lambda^2}\sum\limits_{i=1}^n\vert x_i\vert$,解得$\lambda=\dfrac{1}{n}\sum\limits_{i=1}^n\vert x_i\vert$。 + +即$\hat{\lambda}=\dfrac{1}{n}\sum\limits_{i=1}^n\vert X_i\vert$。 + \section{置信区间} \textbf{例题:}一批零件的长度服从正态分布$N(\mu,\sigma^2)$,其中$\mu,\sigma^2$均未知。现从中随机抽取16个零件,测得样本均值$\overline{x}=20cm$,样本标准差为$s=1cm$,求$\mu$的置信水平为0.90的置信区间。 diff --git a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.pdf 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b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex index d7819af..70ed10e 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/5-mathematical-statistics/mathematical-statistics.tex @@ -284,9 +284,9 @@ $\therefore P\{F>F_\alpha(n_2,n_1)\}=\alpha$,$P\{F\leqslant F_\alpha(n_2,n_1)\ \item 如果$p(x;\theta_1,\theta_2,\cdots,\theta_k)$或$f(x;\theta_1,\theta_2,\cdots,\theta_k)$不可微,或似然方程组无解,则应由定义用其他方法求$\hat{\theta}$,如当$L(\theta)$为$\theta$的单调函数时,$\hat{\theta}$为$\theta$的取值上限或下限。 \end{enumerate} -即将概率密度或概率分布连乘,然后取对数,再求导令其为0解出$\overline{\theta}$。 +即将概率密度或概率分布连乘,然后取对数,再求导令其为0解出$\hat{\theta}$。 -\textbf{例题:}设总体$X$的概率分布为: +\textbf{例题:}设总体$X$的概率分布为:\medskip \begin{tabular}{c|cccc} \hline