iid Consider data X1, X2, ... , Xn Ber(0). Consider the following two estimators of 0 + ;: 1+E, Xi Xn and ô2 2+ n 1. Compute the Bias of both 61 and 02. Which estimator has lower bias (in absolute terms) when 0#= 0? 2. Compute the variance of 01 and 02. Which estimator has lower variance? Hint: com- pute the ratio of Var[0,]/Var[02] and see if the ratio is less than, equal to, or greater than 1.
iid Consider data X1, X2, ... , Xn Ber(0). Consider the following two estimators of 0 + ;: 1+E, Xi Xn and ô2 2+ n 1. Compute the Bias of both 61 and 02. Which estimator has lower bias (in absolute terms) when 0#= 0? 2. Compute the variance of 01 and 02. Which estimator has lower variance? Hint: com- pute the ratio of Var[0,]/Var[02] and see if the ratio is less than, equal to, or greater than 1.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Text:**
Consider data \( X_1, X_2, \ldots, X_n \overset{iid}{\sim} Ber(\theta) \). Consider the following two estimators of \( \theta \neq \frac{1}{2} \):
\[
\hat{\theta}_1 = \bar{X}_n \quad \text{and} \quad \hat{\theta}_2 = \frac{1 + \sum_{i=1}^n X_i}{2 + n}
\]
1. Compute the Bias of both \( \hat{\theta}_1 \) and \( \hat{\theta}_2 \). Which estimator has lower bias (in absolute terms) when \( \theta \neq 0 \)?
2. Compute the variance of \( \hat{\theta}_1 \) and \( \hat{\theta}_2 \). Which estimator has lower variance? Hint: compute the ratio of \( Var[\hat{\theta}_1]/Var[\hat{\theta}_2] \) and see if the ratio is less than, equal to, or greater than 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F361779f5-7517-4850-b433-f5d42195c84d%2F277cd0c6-3fd5-4643-bfd2-3672991456ad%2Fpe76dr8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Text:**
Consider data \( X_1, X_2, \ldots, X_n \overset{iid}{\sim} Ber(\theta) \). Consider the following two estimators of \( \theta \neq \frac{1}{2} \):
\[
\hat{\theta}_1 = \bar{X}_n \quad \text{and} \quad \hat{\theta}_2 = \frac{1 + \sum_{i=1}^n X_i}{2 + n}
\]
1. Compute the Bias of both \( \hat{\theta}_1 \) and \( \hat{\theta}_2 \). Which estimator has lower bias (in absolute terms) when \( \theta \neq 0 \)?
2. Compute the variance of \( \hat{\theta}_1 \) and \( \hat{\theta}_2 \). Which estimator has lower variance? Hint: compute the ratio of \( Var[\hat{\theta}_1]/Var[\hat{\theta}_2] \) and see if the ratio is less than, equal to, or greater than 1.
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