Least square estimation with one parameter ?

[VLGsIEJRFU]

All of examples that i saw before about least square, they present two group of data (usually [xi,yi]), but these question they didn't.

Suppose X1...Xn iid of a X~U(10, 10+theta) ? How can i find a theta estimator without any other data ?

I know that looks trivial, but i really cant solve this

 

[Dason]

What estimation techniques have you learned about? Or should have learned about? Are there any techniques in your notes or your book that aren't least squares?

 

[katxt]

Can you find the mean of U(10, 10+theta)?

 

[VLGsIEJRFU]

What estimation techniques have you learned about? Or should have learned about? Are there any techniques in your notes or your book that aren't least squares?

Thats the third techinique until now, i'm economics student so probably i dont will see a lot of different techniques in college as a underg. Until now i saw Maximum likelihood and Method of moments, that i remember.

My biggest problem i cant see a demonstration or a example of how can i start develop. Usually all examples that i saw was with two gived parameters, or at least a distribuition to find these parameters and after only needed use the formula of sum(xi - x_mean)*(yi - y_mean)/sum(xi-x_mean)^2

Can you find the mean of U(10, 10+theta)?

I can see sum(10+...+ [10+theta])/n

 

[katxt]

Try drawing the distribution of U(10, 10+theta) and draw a line at the mean. If it makes things easier to see, draw U(10, 10+8) i.e. U(10, 18) and find the mean.

 

[VLGsIEJRFU]

Thank you by your time, i found a way to solve my problem.

 

[katxt]

There is a trap that you might not have thought of.
With least squares it is possible that the simple answer gives you a theta which is can't be true. For instance if the X's are 10, 10, 10, and 20, then the average is 12.5 and theta would be 10 + 2x2.5 = 15. But 20 is impossible in U(10, 15)
Interestingly, maximum likelihood gives a different answer altogether from least squares.