Gebruiker:Pjetter/klad: verschil tussen versies
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Regel 243:
:<math>\hat \beta=\frac{\sum_i^{n} x_i y_i}{\sum_i^{n} x_i^2}</math>
==Biasedness==
:<math>B(\widehat{\theta}) = \operatorname{E}(\widehat{\theta}) - \theta</math>
Regel 253 ⟶ 255:
Let us use this on the estimator in a):
:<math>
:<math> \operatorname{E}(\widehat{\beta}) = \operatorname{E}(\frac{\sum_i^{n} x_i Y_i}{\sum_i^{n} x_i^2})</math>▼
Now let us fill in the proposed population model into this equation:
Regel 265 ⟶ 263:
and you will get:
:<math>
Which is:
:<math>\widehat{\beta} = \frac{\sum_i^{n} (\beta x_i^2 + \epsilon_i x_i) }{\sum_i^{n} x_i^2}</math>
And this can be written as:
:<math>\widehat{\beta} = \frac{\sum_i^{n} \beta x_i^2} {\sum_i^{n} x_i^2} + \frac{\sum_i^{n} \epsilon_i x_i}{\sum_i^{n} x_i^2}</math>
:<math>\widehat{\beta} = \beta + \frac{\sum_i^{n} \epsilon_i x_i}{\sum_i^{n} x_i^2}</math>
According to biasedness equation above we can take the expectation from both sides according to:
▲:<math>
and as
:<math> \operatorname{E} (\sum_i^{n} \epsilon_i) = 0 </math>
you get:
:<math>\operatorname{E} (\widehat{\beta}) = \beta </math>
Therefore unbiasedness applies.
==Sigma==
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