[R] smoothing functions

[Eugene Salinas (R)]

Dear Thomas,

The short answer is both. Some of the parameters of interest to be 
estimated are discrete while others are are continuous. Similarly for 
the support of the data variables that go into the objective.

Let me try and make this more concrete without typing all the maths 
down. Consider a set of functions that are well defined (eg exponentials 
etc) f_i (x, a) where x is some given data and a is a parameter of 
interest. Then we look at comparing different combinations of these 
functions (possibly functions of functions) but in the simplest case it 
would be 1| f_i (x,a)> f_ j (x,a)], where 1| returns 1 if the inequality 
is satisfied, 0 otherwise. Now consider building an objective function 
out of lots and lots of these indicator functions. So the issue is how 
to smooth this objective since it has no continuous elements because of 
the indicator structure?

thanks, Eugene.

Thomas W Blackwell wrote:

>Eugene  -
>
>Is the estimand in your problem (the parameter which you seek
>to estimate) discrete-valued or continuous-valued ?  If it is
>discrete-valued, then you are heading in the wrong direction,
>because no matter how smooth you make the objective function,
>you will not be able to differentiate it with respect to the
>parameter !   I think I don't have quite enough information
>to give a helpful answer to your question  . . .  but more
>important is for you to find the answer yourself.
>
>-  tom blackwell  -  u michigan medical school  -  ann arbor  -
>
>On Mon, 1 Dec 2003, Eugene Salinas (R) wrote:
>
>  
>
>>Dear all,
>>
>>I am trying to program an estimator which maximizes a likelihood type
>>objective function which is basically just lots of sums of indicator
>>functions of data and parameters. In order to make the optimization I
>>would like to smooth these functions. Since they are either 0 or 1, one
>>possibility is to use the normal cdf.
>>
>>I am wondering whether anyone is aware of a less arbitrary choice of a
>>smoothing function? (is there any theory that suggests what's best to
>>use?) Does anyone have any recommendations on what works best numerically?
>>
>>Thanks, Eugene.
>>
>>______________________________________________
>>R-help at stat.math.ethz.ch mailing list
>>https://www.stat.math.ethz.ch/mailman/listinfo/r-help
>>
>>    
>>
>
>  
>