Log-likelihood and maximum likelihood estimator for PDF

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Bill Foster 6 years, 3 months ago

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Name	Status	Author	Last Modified
Log-likelihood and maximum likelihood estimator for PDF	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:51
Kma's copy of Log-likelihood and maximum likelihood estimator for PDF	draft	Kma math	11/01/2017 04:26

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Question statement

The average annual wind speed, $X$, at {where} has the following probability density function with parameter $t$ which you have to estimate:

$f_X(x) = \left \{ \begin{array}{l} \phantom{{.}} \\ \phantom{{.}} \\ \phantom{{.}} \end{array} \right .$	$2txe^{-tx^2}$	$x \gt 0,$

	$0$	$\textrm{otherwise.}$

For three randomly selected years, we observe the following average wind speeds:

$x_1=\var{x1},\;\;x_2=\var{x2}$ and $x_3=\var{x3}$.

Give any introductory information the student needs.

			Name	Type	Generated Value

prod	number	75
mle	number	0.04
m	number	-1875
x1	number	2.5
sumsq	number	67.25
tol	integer	0
x3	number	6
x2	number	5
x	number	13.5
y2	number	5
where	string	Keele

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xxxxxxxxxx
 
x1*x2*x3

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Part b) Gap-fill
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Find the likelihood function for $t$ given these observations.

$L(t|\underline{x})=\;\;$Unnamed gap

Advice

\[ \begin{eqnarray*} L(t|\underline{x})&=& \var{2*x1}te^{-\var{x1}^2t}\times \var{2*x2}te^{-\var{x2}^2t} \times \var{2*x3}te^{-\var{x3}^2t}\\ &=& \var{8*prod}t^3e^{-\var{sumsq}t} \end{eqnarray*} \]
b)

The log-likelihood function is :
\[\begin{eqnarray*} l(t|\underline{x})&=&\ln\left( \var{8*prod}t^3e^{-\var{sumsq}t}\right)\\ &=&\ln(\var{8*prod})+3\ln(t)-\var{sumsq}t \end{eqnarray*} \]

We have:
\[\frac{\partial\;l}{\partial\;t}=\frac{3}{t}-\var{sumsq}\]
Now:
\[\begin{eqnarray*} \frac{\partial\;l}{\partial\;t}&=&0 \\ \Rightarrow \frac{3}{t}-\var{sumsq}&=&0\\ \Rightarrow t&=&\frac{3}{\var{sumsq}} = \var{mle} \end{eqnarray*} \] to 3 decimal places.
And putting $t=\hat{t}$ gives the MLE $\hat{t}=\var{mle}$

d)
\[\frac{\partial^2\;l}{\partial\;t^2}=-3t^{-2} \lt 0\]
when evaluated at any point including $t=\hat{t}=\var{mle}$.

Hence gives a maximum at $t=\hat{t}$.

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Log-likelihood and maximum likelihood estimator for PDF

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Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Log-likelihood and maximum likelihood estimator for PDF

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Bill Foster 6 years, 3 months ago

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Warning

Generated value: number

Parts

Gap-fill

Generated value: `number`