Question

Suppose X1, . . . , Xn are i.i.d. random variables having a N (μ, σ2) distribution. Consider the estimator of σ2, σ2 c = c nX i=1 (Xi − ̄X)2. Recall that ˆσ2 c is the MLE if c = 1/n and is the unbiased estimator S2 if c = 1/(n − 1). a) Calculate the mean squared error of ˆσ2 c . (b) For which value of c is the mean squared error the smallest?

Name: suppose x1 xn are iid random variables having a n 2 distribution consider the estimator of 2 2 c c nx i1 xi x2 recall that 2 c is the mle if c 1n and is the unbiased estimator s2 if c 1n 1 a 30908
Uploaded: 2024-04-16T08:33:15-08:00
Duration: 3 min 20 s
Channel: Adi S
Description: suppose x1 xn are iid random variables having a n 2 distribution consider the estimator of 2 2 c c nx i1 xi x2 recall that 2 c is the mle if c 1n and is the unbiased estimator s2 if c 1n 1 a 30908

          Suppose X1, . . . , Xn are i.i.d. random variables having a N (μ, σ2) distribution. 
Consider the estimator of σ2, σ2
c = c
nX
i=1
(Xi − ̄X)2. Recall that ˆσ2
c is the MLE if c = 1/n and is the unbiased estimator S2 if c = 1/(n − 1). a) Calculate the mean squared error of ˆσ2
c .
(b) For which value of c is the mean squared error the smallest?

Added by Christopher T.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

04/16/2024

Step 1

i.d. \( N(\mu, \sigma^2) \). Estimator: \[ \hat{\sigma}^2_c = c \sum_{i=1}^n (X_i - \bar{X})^2 \] Recall: - MLE corresponds to \( c = \frac{1}{n} \). - Unbiased estimator \( S^2 \) corresponds to \( c = \frac{1}{n-1} \). --- Part (a) Calculate the mean squared Show more…

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Suppose X1, . . . , Xn are i.i.d. random variables having a N (μ, σ2) distribution. Consider the estimator of σ2, σ2 c = c nX i=1 (Xi − ̄X)2. Recall that ˆσ2 c is the MLE if c = 1/n and is the unbiased estimator S2 if c = 1/(n − 1). a) Calculate the mean squared error of ˆσ2 c . (b) For which value of c is the mean squared error the smallest?