برآورد درست‌نمایی بیشینه

${\displaystyle f(x_1,x_2,\dots ,x_n|\theta )=f(x_1|\theta )\cdot f(x_2|\theta )\cdots f(x_n|\theta ).}$

${\displaystyle \mathcal{L}(\theta |x_1,\dots ,x_n)=f(x_1,x_2,\dots ,x_n|\theta )=\prod _{i=1}^{n}f(x_i|\theta ).}$

${\displaystyle \ln \mathcal{L}(\theta |x_1,\dots ,x_n)=\sum _{i=1}^n\ln f(x_i|\theta ),\hat{\ell }=\frac{1}{n}\ln \mathcal{L}.}$

${\displaystyle \{{\hat{\theta }}_{\mathrm{m}\mathrm{l}\mathrm{e}}\}\subseteq \{\underset{\theta \in \mathrm{\Theta }}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}\hat{\ell }(\theta |x_1,\dots ,x_n)\}.}$

${\displaystyle P(\theta \mid x_1,x_2,...x_n)=\frac{f(x_1,x_2,...x_n\mid \theta )P(\theta )}{P(x_1,x_2,...,x_n)}}$

${\displaystyle \hat{\ell }(\theta |x)=\frac{1}{n}\sum _{i=1}^n\ln f(x_i|\theta ),}$

${\displaystyle {\hat{\theta }}_{MLE}\to {\theta }_0}$

${\displaystyle \theta \ne {\theta }_0\iff f(.\mid {\theta }_0)}$

${\displaystyle Pr[lnf(x\mid \theta )\in C^0(\mathrm{\Theta })]=1}$

${\displaystyle \theta \in \mathrm{\Theta }}$

${\displaystyle \underset{\theta \in \mathrm{\Theta }}{sup}\left|\hat{l}(\theta \mid x)-l(\theta )\right|\to 0}$

${\displaystyle \sqrt{n}({\hat{\theta }}_{mle}-{\theta }_0)⟶\mathrm{\aleph }(0,I^{-1})}$

${\displaystyle \hat{\alpha }=g(\hat{\theta })}$

${\displaystyle \overline{L}(\alpha )=\underset{\theta :\alpha =g(\theta )}{sup}L(\theta )}$

${\displaystyle f_Y(y)=\frac{f_X(x)}{\mid g^{\prime }(x)\mid }}$

${\displaystyle \sqrt{n}({\hat{\theta }}_{mle}-{\theta }_0)⟶\mathrm{\aleph }(0,I^{-1})}$

${\displaystyle I_{jk}=E_X[-\frac{{\delta }^{2}ln{f}_{{\theta }_0}(X_t)}{\delta {\theta }_j\delta {\theta }_k}]}$

${\displaystyle b_s\equiv E[({\hat{\theta }}_{mle}-{\theta }_0{)}_s]=\frac{1}{n}.I^{si}{I}^{jk}(\frac{1}{2}{K}_{ijk}+J_{j,ik})}$

${\displaystyle \frac{1}{2}{K}_{ijk}+J_{j,ik}=E_X[\frac{1}{2}\frac{{\delta }^{3}ln{f}_{{\theta }_0}(X_t)}{\delta {\theta }_i\delta {\theta }_j\delta {\theta }_k}+\frac{\delta ln{f}_{{\theta }_0}(x_t)}{\delta {\theta }_j}\frac{{\delta }^{2}ln{f}_{{\theta }_0}(X_t)}{\delta {\theta }_i\delta {\theta }_k}}$

${\displaystyle {\hat{\theta }}_{mle}^{\ast }={\hat{\theta }}_{mle}-\hat{b}}$

${\displaystyle L(p)=f_D(\mathrm{H}=49\mid p)=(\genfrac{}{}{0ex}{}{80}{49})p^{49}(1-p{)}^{31},}$

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${\displaystyle f(x\mid \mu ,{\sigma }^2)=\frac{1}{\sqrt{2\pi {\sigma }^2}}exp(-\frac{(x-\mu {)}^2}{2{\sigma }^2}),}$

${\displaystyle f(x_1,...x_n\mid \mu ,{\sigma }^2)=\prod _{i=1}^{n}f(x_i\mid \mu ,{\sigma }^2){\displaystyle =(\frac{1}{2\pi {\sigma }^2}{)}^{\frac{n}{2}}exp(-\frac{\sum _{i=1}^n(x_i-\mu {)}^2{\displaystyle }}{2{\sigma }^2})}}$

${\displaystyle f(x_1,...x_n\mid \mu ,{\sigma }^2)=(\frac{1}{2\pi {\sigma }^2}{)}^{\frac{n}{2}}exp(-\frac{\sum _{i=1}^n(x_i-\overline{x}{)}^2+n(\overline{x}-\mu {)}^2{\displaystyle }}{2{\sigma }^2})}$

${\displaystyle log(L(\mu ,\sigma ))=(-n/2)log(2\pi {\sigma }^2)-\frac{1}{2{\sigma }^2}\sum _{i=1}^n(x_i-\mu {)}^2}$

${\displaystyle 0=\frac{\mathrm{\partial }}{\mathrm{\partial }\mu }log(L(\mu ,\sigma )=0-\frac{-2n(\overline{x}-\mu )}{2{\sigma }^2}}$

${\displaystyle \hat{\mu }=\overline{x}=\sum _{i=1}^n\frac{x_i}{n}}$

${\displaystyle E[\hat{\mu }]=\mu }$

${\displaystyle 0=\frac{\mathrm{\partial }}{\mathrm{\partial }\sigma }log((\frac{1}{2\pi {\sigma }^2}{)}^{n/2})exp(-\frac{\sum _{i=1}^n(x_i-\overline{x})+n(\overline{x}-\mu {)}^2}{2{\sigma }^2})}$

${\displaystyle =\frac{\mathrm{\partial }}{\mathrm{\partial }\sigma }(\frac{n}{2}log(\frac{1}{2\pi {\sigma }^2})-\frac{\sum _{i=1}^n(x_i-\overline{x})+n(\overline{x}-\mu {)}^2}{2{\sigma }^2}}$

${\displaystyle =-\frac{n}{\sigma }+\frac{\sum _{i=1}^n(x_i-\overline{x})+n(\overline{x}-\mu {)}^2}{{\sigma }^3}}$

${\displaystyle {\hat{\sigma }}^2=\frac{1}{n}\sum _{i=1}^n(x_i-\mu {)}^2}$

${\displaystyle {\hat{\sigma }}^2=\frac{1}{n}\sum _{i=1}^n(x_i-\overline{x}{)}^2=\frac{1}{n}\sum _{i=1}^n{x}_i^2-\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_j}$

${\displaystyle {\hat{\sigma }}^2=\frac{1}{n}\sum _{i=1}^n(\mu -{\delta }_i{)}^2-\frac{1}{n}\sum _{i=1}^n\sum _{j=1}^n(\mu -{\delta }_i)(\mu -{\delta }_j)}$

${\displaystyle E[{\hat{\sigma }}^2]=\frac{n-1}{n}{\sigma }^2}$

${\displaystyle \hat{\theta }=(\hat{\mu },{\hat{\sigma }}^2)}$

${\displaystyle log(L(\hat{\mu },\hat{\sigma }))=\frac{-n}{2}(log(2\pi {\sigma }^2)+1)}$

${\displaystyle f(x,y)=f(x)f(y)}$

${\displaystyle f(x_1,...,x_n)=\frac{1}{(2\pi {)}_{n/2}\sqrt{det(\mathrm{\Sigma })}}exp(-\frac{1}{2}[x_1-{\mu }_1,...,x_n-{\mu }_n]{\mathrm{\Sigma }}^{-1}[x_1-{\mu }_1,...,x_n-{\mu }_n{]}^T)}$

${\displaystyle f(x,y)-\frac{1}{2\pi {\sigma }_x{\sigma }_y\sqrt{1-{\rho }^2}}exp[-\frac{1}{2(1-{\rho }^2)}(\frac{(x-{\mu }_x{)}^2}{{\sigma }_x^2}-\frac{2\rho (x-{\mu }_x)(y-{\mu }_y)}{{\sigma }_x{\sigma }_y}+\frac{(y-{\mu }_y{)}^2}{{\sigma }_y^2})]}$

${\displaystyle {\hat{\sigma }}^2=\frac{1}{n}\sum _{i=1}^n({\hat{x}}_i-\overline{x}{)}^2}$