魅影直播

A mathematical language that asserts that two algebraic expressions must be equal in nature is known as an equation.

\({\rm{f(x;\theta ) = }}\frac{{\rm{x}}}{{\rm{\theta }}}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{/(2\theta )}}}}{\rm{, x > 0}}\)

Allow joint pdf or pmf for random variables\({{\rm{X}}_{\rm{1}}}{\rm{,}}{{\rm{X}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{X}}_{\rm{n}}}\).

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right){\rm{, n,m}} \in {\rm{N}}\)

where\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)are unknown parameters. The likelihood function is defined as a function of parameters\({{\rm{\theta }}_{\rm{i}}}{\rm{,i = 1,2, \ldots ,m}}\)where function f is a function of parameter. The maximum likelihood estimates (mle's), or values\(\widehat {{{\rm{\theta }}_{\rm{i}}}}\)for which the likelihood function is maximised, are the maximum likelihood estimates,

\({\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{{\rm{\hat \theta }}}_{\rm{1}}}{\rm{,}}{{{\rm{\hat \theta }}}_{\rm{2}}}{\rm{, \ldots ,}}{{{\rm{\hat \theta }}}_{\rm{m}}}} \right) \ge {\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;}}{{\rm{\theta }}_{\rm{1}}}{\rm{,}}{{\rm{\theta }}_{\rm{2}}}{\rm{, \ldots ,}}{{\rm{\theta }}_{\rm{m}}}} \right)\)

As,\({\rm{i = 1,2, \ldots ,m}}\)for every\({{\rm{\theta }}_{\rm{i}}}\). Maximum likelihood estimators are derived by replacing\({{\rm{X}}_{\rm{i}}}\)with\({{\rm{x}}_{\rm{i}}}\).

Because of the independence, the likelihood function becomes,

\(\begin{array}{c}{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\theta }}} \right){\rm{ = }}\frac{{{{\rm{x}}_{\rm{1}}}}}{{\rm{\theta }}}{{\rm{e}}^{{\rm{ - x}}_{\rm{1}}^{\rm{2}}{\rm{/(2\theta )}}}}{\rm{ \times }}\frac{{{{\rm{x}}_{\rm{2}}}}}{{\rm{\theta }}}{{\rm{e}}^{{\rm{ - x}}_{\rm{2}}^{\rm{2}}{\rm{/(2\theta )}}}}{\rm{ \times \ldots \times }}\frac{{{{\rm{x}}_{\rm{n}}}}}{{\rm{\theta }}}{{\rm{e}}^{{\rm{ - x}}_{\rm{n}}^{\rm{2}}{\rm{/(2\theta )}}}}\\{\rm{ = }}\left( {{{\rm{x}}_{\rm{1}}}{\rm{ \times }}{{\rm{x}}_{\rm{2}}}{\rm{ \times \ldots \times }}{{\rm{x}}_{\rm{n}}}} \right){\rm{ \times }}\frac{{\rm{1}}}{{{{\rm{\theta }}^{\rm{n}}}}}{\rm{ \times exp}}\left\{ {{\rm{ - }}\frac{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} }}{{{\rm{2\theta }}}}} \right\}\end{array}\)

Look at the log likelihood function to determine the maximum.

\(\begin{array}{c}{\rm{lnf}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\theta }}} \right){\rm{ = ln}}\left( {\left( {{{\rm{x}}_{\rm{1}}}{\rm{ \times }}{{\rm{x}}_{\rm{2}}}{\rm{ \times \ldots \times }}{{\rm{x}}_{\rm{n}}}} \right){\rm{ \times }}\frac{{\rm{1}}}{{{{\rm{\theta }}^{\rm{n}}}}}{\rm{ \times exp}}\left\{ {{\rm{ - }}\frac{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} }}{{{\rm{2\theta }}}}} \right\}} \right)\\{\rm{ = ln}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{ \times }}{{\rm{x}}_{\rm{2}}}{\rm{ \times \ldots \times }}{{\rm{x}}_{\rm{n}}}} \right){\rm{ - nln\theta - }}\frac{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} }}{{{\rm{2\theta }}}}\end{array}\)

The maximum likelihood estimator is generated by taking the derivative of the log likelihood function in regard to\({\rm{\theta }}\)and equating it to\({\rm{0}}\).

As a result, the derivative,

\(\begin{array}{c}\frac{{\rm{d}}}{{{\rm{d\theta }}}}{\rm{f}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{,}}{{\rm{x}}_{\rm{x}}}{\rm{, \ldots ,}}{{\rm{x}}_{\rm{n}}}{\rm{;\theta }}} \right){\rm{ = }}\frac{{\rm{d}}}{{{\rm{d\theta }}}}\left( {{\rm{ln}}\left( {{{\rm{x}}_{\rm{1}}}{\rm{ \times }}{{\rm{x}}_{\rm{2}}}{\rm{ \times \ldots \times }}{{\rm{x}}_{\rm{n}}}} \right){\rm{ - nln\theta - }}\frac{{\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} }}{{{\rm{2\theta }}}}} \right)\\{\rm{ = 0 - n \times }}\frac{{\rm{1}}}{{\rm{\theta }}}{\rm{ + }}\frac{{\rm{1}}}{{{\rm{2}}{{\rm{\theta }}^{\rm{2}}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} \\{\rm{ = }}\frac{{\rm{1}}}{{{\rm{2}}{{\rm{\theta }}^{\rm{2}}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} {\rm{ - n \times }}\frac{{\rm{1}}}{{\rm{\theta }}}\end{array}\)

As a result, solving equation provides the maximum likelihood estimator.

\(\begin{array}{c}\frac{{\rm{1}}}{{{\rm{2}}{{{\rm{\hat \theta }}}^{\rm{2}}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} {\rm{ - n \times }}\frac{{\rm{1}}}{{{\rm{\hat \theta }}}}{\rm{ = 0}}\\{\rm{n\hat \theta = }}\frac{{\rm{1}}}{{\rm{2}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{x}}_{\rm{i}}^{\rm{2}}} \end{array}\)

For\({\rm{\theta }}\). Hence, the maximum likelihood estimator is,

\({\rm{\hat \theta = }}\frac{{\rm{1}}}{{{\rm{2n}}}}\sum\limits_{{\rm{i = 1}}}^{\rm{n}} {{\rm{X}}_{\rm{i}}^{\rm{2}}} \)

\({\rm{F(x;\theta ) = 0}}{\rm{.5}}\)

For\({\rm{x > 0}}\), the cdf is,

\(\begin{array}{c}{\rm{ F(x;\theta ) = P(X}} \le {\rm{x)}}\\{\rm{ = }}\int_{\rm{0}}^{\rm{x}} {\rm{f}} {\rm{(t;\theta )dt}}\\{\rm{ = 1 - }}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{/(2\theta )}}}}\end{array}\)

where this cdf can be found (you can solve integral and obtain it as well). Hence,

\(\begin{array}{c}{\rm{F(x;\theta ) = 0}}{\rm{.5}}\\{\rm{1 - }}{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{/(2\theta )}}}}{\rm{ = 0}}{\rm{.5}}\\{{\rm{e}}^{{\rm{ - }}{{\rm{x}}^{\rm{2}}}{\rm{/(2\theta )}}}}{\rm{ = 0}}{\rm{.5}}\\\frac{{{\rm{ - }}{{\rm{x}}^{\rm{2}}}}}{{{\rm{2\theta }}}}{\rm{ = ln0}}{\rm{.5}}\\{{\rm{x}}^{\rm{2}}}{\rm{ = - 2\theta \times - 0}}{\rm{.6931}}\\{\rm{x = }}\sqrt {{\rm{1}}{\rm{.3863 \times \theta }}} \end{array}\)

Let\({\rm{i = 1,2,}}......{\rm{,n}}\)be the maximum likelihood estimates of the parameters\(\widehat {{{\rm{\theta }}_{\rm{i}}}}{\rm{,i = 1,2,}}......{\rm{,n}}\). The function of the mle's\(\widehat {{{\rm{\theta }}_{\rm{i}}}}\); is the mle of any function of parameters\(\widehat {{{\rm{\theta }}_{\rm{i}}}}\).

Because of the invariance principle and the fact that the median is a function of\({\rm{\theta }}\), the maximum likelihood estimator of the median is a function\(\widehat {\rm{\theta }}\)of or equally is,

\({\rm{\hat \sim x = }}\sqrt {{\rm{1}}{\rm{.3863 \times \hat \theta }}} \).