Question

Let $X_1, X_2, X_3, X_4$ be i.i.d. observations from a distribution with mean $mu$ and variance $sigma^2$. Consider the following four estimators of $mu$: $hat{mu}_1 = X_1$, $hat{mu}_2 = (X_2 + X_3)/2$, $hat{mu}_3 = 0.1X_1 + 0.2X_2 + 0.3X_3 + 0.4X_4$, $hat{mu}_4 = ar{X}$. Show that all four estimators are unbiased. 2. Calculate the variance of each estimator. Which one has the smallest variance?

Name: let x1 x2 x3 x4 be iid observations from a distribution with mean and variance 2 consider the following four estimators of 1 x1 2 x2 x3 2 3 01x1 02x2 03x3 04x4 4 x 1 show that all four estim 00382
Uploaded: 2022-11-03T08:04:35-08:00
Duration: 6 min 12 s
Channel: Madhur L
Description: let x1 x2 x3 x4 be iid observations from a distribution with mean and variance 2 consider the following four estimators of 1 x1 2 x2 x3 2 3 01x1 02x2 03x3 04x4 4 x 1 show that all four estim 00382

          Let $X_1, X_2, X_3, X_4$ be i.i.d. observations from a distribution with mean $mu$ and variance $sigma^2$. Consider the following four estimators of $mu$: $hat{mu}_1 = X_1$, $hat{mu}_2 = (X_2 + X_3)/2$, $hat{mu}_3 = 0.1X_1 + 0.2X_2 + 0.3X_3 + 0.4X_4$, $hat{mu}_4 = ar{X}$. Show that all four estimators are unbiased. 
2. Calculate the variance of each estimator. Which one has the smallest variance?

Added by Heather S.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

11/03/2022

Step 1

The bias is calculated as the expectation of the estimator minus the true parameter value (in this case, the mean µ). Show more…

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Let $X_1, X_2, X_3, X_4$ be i.i.d. observations from a distribution with mean $mu$ and variance $sigma^2$. Consider the following four estimators of $mu$: $hat{mu}_1 = X_1$, $hat{mu}_2 = (X_2 + X_3)/2$, $hat{mu}_3 = 0.1X_1 + 0.2X_2 + 0.3X_3 + 0.4X_4$, $hat{mu}_4 = ar{X}$. Show that all four estimators are unbiased. 2. Calculate the variance of each estimator. Which one has the smallest variance?