maximum logistic function

[amintc]

how can maximum the
\( \sum\limits_{j=1}^{J}{\sum\limits_{t=1}^{T}{h_{jt}(x_t;\theta)\dfrac{e^{\alpha_j+X\beta}}{\sum\limits_{l=1}^{L}{e^{\alpha_l+X\beta}}}}}\)
where \(h_{jt}\) is known and\(\alpha\) and \(\beta=(\beta_1,\ldots,\beta_s)\) are parameter and must be estimate.

 

[Dason]

Re: maximum logestic function

http://www.talkstats.com/showthread.php/14986-How-to-use-Math-tags

 

[BGM]

Re: maximum logestic function

Do you mean \( h_{jt} \) are known functions and you need to maximize the stated objective function? and the arguments are \( \alpha, \beta \)? Also \( \theta \) is an independent parameter or just \( (\alpha, \beta) \)?

If closed-form solution exist you can use the usual calculus. Seemingly this weighted logistic function may not be so nice so you may need to rely on numerical method. For example you may use the function optim in R.

 

[amintc]

Re: maximum logestic function

\(h_{jt}\) is a known function. you can ignore it.
The \(X\) is covariate and its value is unknown.
just \((\alpha,\beta)\).
closed-form not exist!!!!
if parameter \(\beta\) has a s dimension then "optim" can do well is calculated?
What do you think about "nlminb"??

 

[BGM]

Ah sorry I forget this is in the R section, so I got to be more specific.

http://en.wikibooks.org/wiki/R_Programming/Optimization

optim(), nlm(), ucminf() (ucminf) can be used for multidimensional optimization problems.

Since you are dealing with multidimensional optimization problem, you may consider those functions.

nlminb() for constrained optimization.

Not sure if you have any constrain not listed.

 

[amintc]

ok. What does this mean?
"Not sure if you have any constrain not listed"

 

[Dason]

Sometimes there are constraints on parameters (like \(\alpha > 0\) or \(\sum \beta_i = 1\)). You didn't list any constraints but if you did then you would want to look at one of the methods that allows you to input constraints.