the detail description is contained in the attachment (pdf). Any information would be appreciated!
Dear Huiping Zhuang ,
I've attached a document with some ideas to your problem. I hope it is useful to you.
Sincerely,
James D. Santos
Dear James D.Santos,
Thanks for your reply, but I could not read the pdf attachment after I downloaded. Something must be wrong. Could you please send it agian?
Zhuang
Dear James D.Santos,
I found a way to open the pdf file. The thing is that the first assuption you made may not hold in the problem I study. You see. In the new model, if we wish to use ML, the distribution of zk|xk,α maybe partially known (known mean unknown variance), as the variance must be derived under the condition of xk and α. So the key problem might be ignored by al of us. Looking forward to hearing you again.
Huiping Zhuang
Dear Zhuang,
You can find the mean and variance of zk from the given mean and variance of the wk and nk as their distribution is given(Gaussain). Express zk in terms of wk and nk and take their expected value and variance, you will get for zk.
Dear Munir,
The problem may not be easy as you suggested. The mean and variance of zk are not available simultaneously as xk|k refers to a conditional mean with respect to y1k and the variance of wk and nk are given unconditionally. The mean and variance of zk should found under the condition of y1k or under no condition. under no condition case: the mean and variance of xk are unkonwn. Under y1k condition case: the mean of zk is given by xk|k (conditional mean), but the variance, however, is also under the condition, which means the variance should a function of the condition, i.e., y1k. In the condition case, is it a valid approximation when the conditional variance of zk is replaced by the unconditional variance nuerically? Thank you!
Zhuang
Dear Huiping Zhuang,
The first assumption can be remedied. If X is the same in every moment of time, it can be estimated with the maximum likelihood method. Otherwise, I suggest using a discount factor (see the attached file). More on discount factors can be seen in
Harrison, Jeff, and Mike West. Bayesian Forecasting & Dynamic Models. Springer, 1999.
Sincerely,
James D. Santos
Dear James,
Indeed, if X is just a random variable, I can estimate the parameters via ML method. Maybe the question was too complicated. Thank you for your assistance!
Zhuang
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