Stochastic Integral

#1
Hi there,

I created an account here trying to get some help on my problem. I'm really desperate as I'm going to need this for my work on my master's degree.

So, here is the problem:

How can I express the function (process) given in pg7 of the article below in simple terms? (something familiar like a sum or a riemman integral)

https://www.stat.washington.edu/research/reports/1994/tr274.pdf

Can anyone help me "decyphering" it? I know that I can take this into a discrete sum of terms depending on my censoring... any tips or any manual for dummies on this?

Thanks,
Júlio
 
#3
You mean those integrals integrated with respect to the empirical measure \( \mathbb{Q}_n \) ?
Precisely! The function W_Lambda for example. How do I evaluate it, I mean, in terms of my data?

Any help on this would be great. Thanks for the reply.
 

BGM

TS Contributor
#4
http://en.wikipedia.org/wiki/Empirical_measure

I have not study much about empirical measure. The notation here, as the wiki page suggest, is just so called the empirical (sample) mean (see the wiki page above). If the triple \( (T_i, Y_i, Z_i) \) is observable then the computation is straight forward. But I guess censoring make this impossible (\( T_i \) is not observable with censoring) which cause the difficulties here. (I have not read the details)

Anyway, I guess the main point is that the integral can be just viewed as a Riemann-Stieltjes integral which you have the sum to approximate the integral.
 
#5
http://en.wikipedia.org/wiki/Empirical_measure

I have not study much about empirical measure. The notation here, as the wiki page suggest, is just so called the empirical (sample) mean (see the wiki page above). If the triple \( (T_i, Y_i, Z_i) \) is observable then the computation is straight forward. But I guess censoring make this impossible (\( T_i \) is not observable with censoring) which cause the difficulties here. (I have not read the details)

Anyway, I guess the main point is that the integral can be just viewed as a Riemann-Stieltjes integral which you have the sum to approximate the integral.
Thank you for the attention and help...
So, I just have to calculate the sample mean over the function? That's kinda odd, cause I'm trying to reach a sum like expression (2.5) of the article above.

Actually, the main reason I'm asking this, is because I'm trying to reach W_Lambda(t) of pg 1170 of this article (http://www3.stat.sinica.edu.tw/statistica/oldpdf/A20n310.pdf) by my own, and consequently G_Lambda(t) (that seems wrong on Ma's paper). There is this extra exp(theta*X) on the numerator of each term of the sum that doesn't make sense for me.

If you check https://www.stat.washington.edu/jaw/JAW-papers/NR/jaw-huang-97LNS-SBS.pdf , you will see the form of the sum that I should achieve with these integrals, but just the mean isn't getting me there.

Anyway, thanks for the help. Sorry for bothering!