MATH 351 Homework 07

  1. Prove that the mean squared error of an estimator θ^\hat{\theta} can be broken down into the bias squared plus the variance MSE(θ^)=(E[θ^]−θ)2+V[θ^]\text{MSE}(\hat{\theta}) = (\mathbb{E}[\hat{\theta}] - \theta)^2 + \mathbb{V}[\hat{\theta}] Hint: there's a specific term missing on the left hand side of this equation, that exists on the right hand side. Add zero to the left hand side to account for this missing term.
  2. Try to show via a simulation that the expected value of 1N−1∑n=1N(Xn−XˉN)2\sqrt{\frac{1}{N-1}\sum_{n=1}^N (X_n - \bar{X}_N)^2}
    3. is again biased, despite the leading 1N−1\frac{1}{N-1}.
  3. Try to show via a simulation that the mean squared error of 1N∑n=1N(Xn−XˉN)2\frac{1}{N}\sum_{n=1}^N (X_n - \bar{X}_N)^2 is smaller than 1N−1∑n=1N(Xn−XˉN)2\frac{1}{N-1}\sum_{n=1}^N (X_n - \bar{X}_N)^2