Maximum likelyhood extimater(MLE)

[semidevil]

so by definition, the likelyhood function(w, theta) is the product of the pdf fw(w, theta) evalutated at n data points.

but I don't know how they do those calculations...

so for example:

fy(y, theta) = 1/\theta^2 ye^{-y/\theta}

L(theta) = \theta^{-2n}\prod y_{i}e^{-1/\theta}\sum y_{i}

so first of all, I'm looking at this but I don't know how they went from this to that...I look at another problme

how did they go from e^{-(y-\theta)} [\tex]to \pro e^{-(y_{i}- \theta)} [/theta]<br /> <br /> I dotn see a pattern...I compared it w/ the definitoin, but I just don&#039;t get it...<br /> I mean, when they did the L(theta) it seems that they added some &quot;n&#039; and i&#039;s somewhere...and I dotn know where they added these things.

 

[juvenal]

so by definition, the likelyhood function(w, theta) is the product of the pdf fw(w, theta) evalutated at n data points.

but I don't know how they do those calculations...

so for example:

fy(y, theta) = 1/\theta^2 ye^{-y/\theta}

L(theta) = \theta^{-2n}\prod y_{i}e^{-1/\theta}\sum y_{i}

so first of all, I'm looking at this but I don't know how they went from this to that...I look at another problme

how did they go from e^{-(y-\theta)} [\tex]to \pro e^{-(y_{i}- \theta)} [/theta]<br /> <br /> I dotn see a pattern...I compared it w/ the definitoin, but I just don&#039;t get it...<br /> I mean, when they did the L(theta) it seems that they added some &quot;n&#039; and i&#039;s somewhere...and I dotn know where they added these things.

<br /> <br /> It looks like there are some typos in your expression.<br /> <br /> You are trying to estimate \theta using n data points, labelled as y_i. The single likelihood for \theta given one data pointy_i is:<br /> <br /> 1/\theta^2 y_{i}e^{-y_{i}/\theta}<br /> <br /> In order to get the likelihood of \thetafor all n data points, then you need to multiply the single likelihoods together. And that&#039;s just a simple matter of multiplying terms and summing up what&#039;s in the exponential term.

 

[hypermorphism]

The indices are a shorthand for indeterminately long products. For example, given your definition of L, we evaluate the pdf at n values of y, {y₁, y₂, ..., y_n} and take the product. So if
f(y) = \frac{1}{\theta^2} ye^{-\frac{y}{\theta}}
we get
f(y_1)f(y_2)...f(y_n) &amp;= \prod_{i=1}^n f(y_i)

= \prod_{i=1}^n \frac{1}{\theta^2} y_i e^{-\frac{y_i}{\theta}}

= \frac{1}{\theta^{2n}} \prod_{i=1}^n y_i e^{-\frac{y_i}{\theta}}

= \frac{1}{\theta^{2n}} \left(y_1 e^{-\frac{y_1}{\theta}}...y_n e^{-\frac{y_n}{\theta}}\right)

= \frac{1}{\theta^{2n}} \left(y_1...y_n e^{-\frac{y_1+y_2+...+y_n}{\theta}}\right)

= \frac{1}{\theta^{2n}} \left(\prod_{i=1}^n y_i\right) \left(e^{-\frac{1}{\theta}\sum_{i=1}^n y_i}}\right)

 

[semidevil]

ok, thanx, now that makes a little bit more sense, but i'll think about it some more...still a bit confusing.

what about to get ln L(theta)? the book does some weird stuff an I don't know what it did.

ln L(theta) = -2n ln \theta + ln \prod yi - 1/\theta \sum yi


how did that happen?

and also, I'm not understanding where they put the n's. maybe I'm having trouble w/ the definition. like, on the first problem, why did it become \theta^{-2n}?

and for ths problem,

we have fy (y, theta) = e^{-(y-\theta)}, so L(theta) = e^{\sum y + n\theta}

 

[hypermorphism]

and also, I'm not understanding where they put the n's. maybe I'm having trouble w/ the definition. like, on the first problem, why did it become \theta^{-2n}?

Because \theta^{-2} multiplied by itself n times is \theta^{-2n}. They just took it out of the n-product by associativity.

we have fy (y, theta) = e^{-(y-\theta)}, so L(theta) = e^{\sum y + n\theta}

What happens when you multiply e^a with e^b ? That's all that's going on here. :)