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MLE Examples

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These are a few examples of applying the MLE method to estimate unknown parameters in different distributions.

The likelihood function is shown, and by taking logs and differentiating the maximum likelihood estimator of the parameter is calculated.

The likelihood ratio and log likelihood function is then calculated for a hypothetical value of the parameter.

Binomial

See Binomial Distrubution .

L ( ~p ) _ = _ ( ^~n_~x ) ~p^~x ( 1 - ~p )^{~n - ~x}

l ( ~p ) _ #:= _ log_~e L ( ~p ) _ = _ log_~e( ^~n_~x ) + ~x log_~e ~p + (~n - ~x) log_~e( 1 - ~p )

l #' ( ~p ) _ #:= _ ( ~x ./ ~p ) - (~n - ~x) ./ ( 1 - ~p )

est{~p} _ = _ ~x ./ ~n

LR ( ~x ) _ = _ L ( ~p_0 ) ./ L ( est{~p} ) _ = _ ( ~n~p_0 ./ ~x ) ^~x # ((~n - ~n~p_0) ./ (~n - ~x)) ^{~n - ~x}

LL ( ~x ) _ = _ -2 log_~e LR ( ~x ) _ = _ -2 ~x log_~e ( ~n~p_0 ./ ~x ) - 2 ( ~n - ~x ) log_~e ((~n - ~n~p_0) ./ (~n - ~x))

_ _ _ _ _ _ = _ 2 ~x log_~e ( ~x ./ ~n~p_0 ) + 2 ( ~n - ~x ) log_~e ((~n - ~x) ./ (~n - ~n~p_0))

 

Geometric

See Geometric Distrubution .

L ( ~p ) _ = _ ~p ( 1 - ~p ) ^~x

l ( ~p ) _ #:= _ log_~e L ( ~p ) _ = _ log_~e ~p + ~x log_~e ( 1 - ~p )

l #' ( ~p ) _ = _ 1 ./ ~p _ - _ ~x ./ ( 1 - ~p )

est{~p} _ = _ 1 ./ ( ~x + 1 )

LR ( ~x ) _ = _ L ( ~p_0 ) ./ L ( est{~p} ) _ = _ ( ~x + 1 ) ~p_0 \{ ( ~x + 1 ) ( 1 - ~p_0 ) ./ ~x \} ^~x

LL ( ~x ) _ = _ -2 log_~e LR ( ~x ) _ = _ -2 \{ ( ~x + 1 ) log_~e ( ~x + 1 ) - ~x log_~e ~x + log_~e ~p_0 + ~x log_~e ( 1 - ~p_0 ) \}

 

Poisson

See Poisson Distrubution .

L ( &mu. ) _ = _ &mu.^~x e^{-&mu.} ./ ~x#!

l ( &mu. ) _ #:= _ log_~e L ( &mu. ) _ = _ ~x log_~e &mu. - &mu. - log_~e ~x#!

l #' ( &mu. ) _ = _ ~x ./ &mu. - 1

est{&mu.} _ = _ ~x

LR ( ~x ) _ = _ L ( &mu._0 ) ./ L ( est{&mu.} ) _ = _ &mu._0^~x e^{-&mu._0} ./ ~x^~x e^{-~x}

LL ( ~x ) _ = _ -2 log_~e LR ( ~x ) _ = _ 2 &mu._0 - 2 ~x ( log_e ( &mu._0 / ~x ) + 1 )

 

Normal

See Normal Distrubution .

L ( &mu. ) _ = _ fract{1,&sqrt.${2&pi.&sigma.&powtwo.}} exp rndb{fract{- ( ~x - &mu. )^2,2&sigma.&powtwo. }}

l ( &mu. ) _ #:= _ log_~e L ( &mu. ) _ = _ log_~e fract{1,&sqrt.${2&pi.&sigma.&powtwo.}} + rndb{fract{- ( ~x - &mu. )^2,2&sigma.&powtwo. }}

l #' ( &mu. ) _ = _ ( ~x - &mu. ) / &sigma.&powtwo.

est{&mu.} _ = _ ~x _ _ (This is obvious by just looking at the likelihood function.)

LR ( ~x ) _ = _ L ( &mu._0 ) ./ L ( est{&mu.} ) _ = _ exp rndb{fract{- ( ~x - &mu._0 )^2,2&sigma.&powtwo. }}

LL ( ~x ) _ = _ -2 log_~e LR ( ~x ) _ = _ fract{( ~x - &mu._0 )^2,&sigma.&powtwo. }

 

Exponential

See Exponential Distrubution .

L ( &beta. ) _ = _ &beta. e^{- &beta.~x}

l ( &beta. ) _ #:= _ log_~e L ( &beta. ) _ = _ log_~e &beta. - &beta.~x

l #' ( &beta. ) _ = _ 1 ./ &beta. _ - _ ~x

est{&beta.} _ = _ 1 ./ ~x

LR ( ~x ) _ = _ L ( &beta._0 ) ./ L ( est{&beta.} ) _ = _ &beta._0 ~x e^{ 1 - &beta._0 ~x}

LL ( ~x ) _ = _ -2 log_~e LR ( ~x ) _ = _ 2 &beta._0 ~x - 2 log_~e &beta._0 ~x - 2

The graph on the left shows the exponential distribution for various &beta.. Note that the function with &beta. = 4 is the maximum one of these at the point ~x = 0.25.

The graph on the right shows the same function, but this time with varying &beta. but ~x fixed at 0.25. As calculated above, this has maximum at &beta. = 4.