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Apr
21
comment Gorgeous diophantine equation
I found this problem in the ancient unnamed academic tutorial about the number theory. And I can't let go...
Apr
21
asked Gorgeous diophantine equation
Mar
2
revised What problems are related with the following type of FDE with delay?
deleted 77 characters in body; edited title
Mar
2
revised What problems are related with the following type of FDE with delay?
added 266 characters in body
Mar
2
revised What problems are related with the following type of FDE with delay?
added 266 characters in body
Mar
1
awarded  Benefactor
Mar
1
accepted The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
Mar
1
revised What problems are related with the following type of FDE with delay?
Clarifying
Feb
26
revised What problems are related with the following type of FDE with delay?
added 11 characters in body
Feb
26
asked What problems are related with the following type of FDE with delay?
Feb
25
revised The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
clarifying
Feb
25
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
I'm a little confused by the variance formula. Actually, we don't know the true $\alpha$. Can we substitute $\alpha^*$ by the its estimated value?
Feb
24
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
Alexander, it's very good extension, thanks. The 3rd estimator is little bit surprising and at first sight it is unbiased. Could you specify the variance for it?
Feb
22
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
Alexander, what about the reliability of the suggested estimate? Is it unbiased? What is the variance?
Feb
22
revised The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
deleted 2 characters in body
Feb
22
awarded  Promoter
Feb
22
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
It's hard to believe thar PDF of $\epsilon_i(\hat \alpha)=(x_i \frac{\sum_i x_i/\delta_i}{\sum_i x_i^2/\delta_i} - 1)/\delta_i$ is most similar to $N(0,1)$. The goal of your solution is to minimize the variance of $\epsilon_i$ rather than to make it equal to 1. Furthermore, you don't take into account the mean of $\epsilon_i(\alpha)$ which should be close to zero.
Feb
22
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
I think that a good estimation of $\alpha$ shoud make PDF of $\epsilon_i$ most similar to $N(0,1)$.
Feb
22
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
The Gauss–Markov theorem for the least squares estimator (LSE) assume that residuals are i.i.d. and we don't know the true form of its distribution. In that case will get the best linear unbiased estimator for a linear regression model which gives us the lowest variance of the estimate. In our case a distribution of residuals is well known. We can't use LSE because we don't need to minimize the variance of residuals.
Feb
21
comment The $\alpha$ estimation for the model $x_i = \xi_i \cdot \alpha$
Thanks, but why are you use $\sum_i \epsilon_i^2$ as the goal function? We already know the distribution of $\epsilon_i$. We shouldn't to minimize the sample covariance of $\epsilon_i$ because it is already known. Moreover, in your case the sample mean of $\epsilon_i$ will sufficiently far from zero.