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Suppose we have two noisy observations y[0] = x[0] + w[0] y[1] =x[1] +w[1] where xin]; n-0, is a unknown constant, and where wO] and w[1] are statistically independ...

Question

Suppose we have two noisy observations y[0] = x[0] + w[0] y[1] =x[1] +w[1] where xin]; n-0, is a unknown constant, and where wO] and w[1] are statistically independent Gaussian random variables. Assume, w[O]~N(0,1)N(0,1) if x[1] 2 0 w[1]~ N(0,2) if x[1] < 0.Consider the estimatorsx[0] = (1/2) (y[0] y[1])x[1] = (2/3) y[0] (1/3) y[1]Calculate the Cramer-Rao bound for unbiased estimation of xin] based on [y[0J] observation of Ly[1]lShow that a minimum variance unbiased estimator does not exist:

Suppose we have two noisy observations y[0] = x[0] + w[0] y[1] =x[1] +w[1] where xin]; n-0, is a unknown constant, and where wO] and w[1] are statistically independent Gaussian random variables. Assume, w[O]~N(0,1) N(0,1) if x[1] 2 0 w[1]~ N(0,2) if x[1] < 0. Consider the estimators x[0] = (1/2) (y[0] y[1]) x[1] = (2/3) y[0] (1/3) y[1] Calculate the Cramer-Rao bound for unbiased estimation of xin] based on [y[0J] observation of Ly[1]l Show that a minimum variance unbiased estimator does not exist:



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Consider the standard simple regression model $y=\beta_{0}+\beta_{1} x+u$ under the Gauss-Markov Assumptions SLR.1, SLR.2, SLR.3, SLR.4 and SLR.5. The usual OLS estimators $\hat{\beta}_{0}$ and $\widehat{\beta}_{1}$ are unbiased for their respective population parameters. Let $\tilde{\beta}_{1}$ be the estimator of $\beta_{1}$ obtained by assuming the intercept is zero (see Section $2-6$ ). i. Find $E\left(\tilde{\beta}_{1}\right)$ in terms of the $x_{i}, \beta_{0},$ and $\beta_{1}$. Verify that $\tilde{\beta}_{1}$ is unbiased for $\beta_{1}$ when the population intercept $\left(\beta_{0}\right)$ is zero. Are there other cases where $\tilde{\beta}_{1}$ is unbiased? ii. Find the variance of $\tilde{\beta}_{1}$. (Hint: The variance does not depend on $\beta_{0} .$ ) iii. Show that $\operatorname{Var}\left(\tilde{\beta}_{1}\right) \leq \operatorname{Var}\left(\widehat{\beta}_{1}\right)$. [Hint: For any sample of data, $\sum_{i=1}^{n} x_{i}^{2} \geq \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2},$ with strict inequality unless $\bar{x}=0 .]$ iv. Comment on the tradeoff between bias and variance when choosing between $\widehat{\beta}_{1}$ and $\tilde{\beta}_{1}$.


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