| Line 1: | Line 1: | ||
<math>\theta </math> is uniform [0,1] | <math>\theta </math> is uniform [0,1] | ||
| + | |||
| + | E[ <math>\theta^2 </math> ] = <math>\int\limits_{0}^{1} \theta^2 d\theta </math> | ||
| + | = 1/3 | ||
supp. <math>\theta^{hat}</math> = 1/3 | supp. <math>\theta^{hat}</math> = 1/3 | ||
| − | MSE = E[ (<math>\theta</math> - 1/3)^2 ] = E[ <math>\theta</math> | + | MSE = E[ (<math>\theta</math> - 1/3)^2 ] = E[ <math>\theta^2</math> - 2*(1/3)*<math>\theta</math> + (1/3)^2 ] |
= 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9 | = 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9 | ||
supp. instead <math>\theta^{hat}</math> = 1/2 | supp. instead <math>\theta^{hat}</math> = 1/2 | ||
| − | MSE = E[ (<math>\theta</math> - 1/2)^2 ] = E[ <math>\theta</math> | + | MSE = E[ (<math>\theta</math> - 1/2)^2 ] = E[ <math>\theta^2</math> - 2*(1/3)*<math>\theta</math> + (1/4) ] = 1/3 - 1/2 + 1/4 = 1/12 |
Revision as of 17:15, 7 December 2008
$ \theta $ is uniform [0,1]
E[ $ \theta^2 $ ] = $ \int\limits_{0}^{1} \theta^2 d\theta $ = 1/3
supp. $ \theta^{hat} $ = 1/3
MSE = E[ ($ \theta $ - 1/3)^2 ] = E[ $ \theta^2 $ - 2*(1/3)*$ \theta $ + (1/3)^2 ] = 1/3 - 2(1/3)(1/2) + (1/3)^2 = (1/3)^2 = 1/9
supp. instead $ \theta^{hat} $ = 1/2
MSE = E[ ($ \theta $ - 1/2)^2 ] = E[ $ \theta^2 $ - 2*(1/3)*$ \theta $ + (1/4) ] = 1/3 - 1/2 + 1/4 = 1/12
