Please see attached mathematical statistics question below.part(b)what is the mle ˆθ of θ?Additonally, follow the simple instructions below:PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM!ALLOW ANOTHER EXPERT A CHANCE TO ANSWER THIS QUESTION! Let $ S^2 $ be the sample variance of a random sample of size $ n > 1 $ from $ N(\mu, \theta) $ where $ \mu $ is known. We know $ \mathbb{E}(S^2) = \theta $.(a) What is the efficiency of $ S^2 $?(b) Under these conditions, what is the mle $ \hat{\theta} $ of $ \theta $?(c) What is the asymptotic distribution of $ \sqrt{n}(\hat{\theta} - \theta) $?

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Description: Please see attached mathematical statistics question below.part(b)what is the mle ˆθ of θ?Additonally, follow the simple instructions below:PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM!ALLOW ANOTHER EXPERT A CHANCE TO AN

b) Given that the random sample X1,X2,....,Xn follows Nμ,θ. f(x)=12πθe-12θ(x-μ)2 Consider the…

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MATLAB: An Introduction with Applications

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Author:Amos Gilat

Publisher:Amos Gilat

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Please see attached mathematical statistics question below.

part(b)

what is the mle ˆθ of θ?

Additonally, follow the simple instructions below:

PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM!

ALLOW ANOTHER EXPERT A CHANCE TO ANSWER THIS QUESTION!

Let $ S^2 $ be the sample variance of a random sample of size $ n > 1 $ from $ N(\mu, \theta) $ where $ \mu $ is known. We know $ \mathbb{E}(S^2) = \theta $.

(a) What is the efficiency of $ S^2 $?

(b) Under these conditions, what is the mle $ \hat{\theta} $ of $ \theta $?

(c) What is the asymptotic distribution of $ \sqrt{n}(\hat{\theta} - \theta) $?

Expert Solution

Step 1

Given that the random sample $X_{1}, X_{2}, . . . ., X_{n}$ follows $N (μ, θ)$ .

$f (x) = \frac{1}{\sqrt{2 π θ}} e^{\frac{- 1}{2 θ} {(x - μ)}^{2}}$

Consider the likelihood function,

$L (x, θ) = \prod_{i = 1}^{n} f (x) = \prod_{i = 1}^{n} (\frac{1}{\sqrt{2 π θ}} e^{\frac{- 1}{2 θ} {(x - μ)}^{2}}) = {(\frac{1}{\sqrt{2 π θ}})}^{n} e^{\frac{- 1}{2 θ} {\sum (x - μ)}^{2}}$

Apply log on both sides,

$\ln L (x, θ) = \ln {(\frac{1}{\sqrt{2 π θ}})}^{n} e^{\frac{- 1}{2 θ} {\sum (x - μ)}^{2}} = \frac{- n}{2} \ln (2 π θ) \frac{- 1}{2 θ} {\sum (x - μ)}^{2}$

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